
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
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%\documentclass[a4paper, 14pt, noxcolor]{article}
%\documentclass[handout, accumulate, noxcolor]{beamer} 
%\documentclass[accumulate, xcolor, 12pt]{beamer} 
%\documentclass{beamer}
\documentclass[handout]{beamer}
%\documentclass{article}
%\usepackage{beamerarticle}
\usepackage{beamerseminar}
\usepackage{beamertexpower}
\usepackage{hyperref}
%\usepackage{noxcolor}
%\usepackage{beamerfoils}
%\usepackage[francais]{babel}
%\mode<presentation>
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{
  \usetheme{}%{Boadilla} %Madrid Singapore Montpellier
  \setbeamercovered{transparent}
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%\pagestyle{plain}
%\markboth{left head}{right head}
%%%%%%%%%%%%%%%%%%%%%%%%%
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%{
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%\uncover<+(1)->{
%\hskip5pt
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%%%%%%%%%%%%%%%%%%%%%%
%\mode<article>{\usepackage{fullpage}}
%\usepackage{pdfpages}
%  \usetheme{}
%  \usetheme{Montpellier}
%  \usetheme{Antibes}
  % oder ...
  
%  \setbeamercovered{transparent}
  % oder auch nicht

\usepackage[english]{babel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage{fancybox}
\usepackage{xy}
%\usepackage{epsfig}
%%%%%%%%%%%%%%%%
\usepackage[T1]{fontenc} 
\usepackage[latin1]{inputenc}
\usepackage{amsfonts,amssymb,amsmath}
\usepackage{hyperref} 
\usepackage{graphics} 
%\usepackage{pstricks} 
\usepackage{multido, xcolor}
 
%\newcommand{\red}{\color{red}}
%\newcommand{\black}{\color{black}}
\newcommand{\magenta}{\color{magenta}}
%\newcommand{\blue}{\color{blue}}
%\newcommand{\purple}{\color{purple}}
%\newcommand{\orange}{\color{orange}}
%\newcommand{\violet}{\color{violet}}
\newcommand{\brown}{\color{brown}}
%\usepackage{showkeys} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\red}{}
\newcommand{\black}{}
%\newcommand{\magenta}{}
%\newcommand{\blue}{}
\newcommand{\purple}{}
%\newcommand{\orange}{}
\newcommand{\violet}{}
%\newcommand{\brown}{}

%\usepackage{times}
%\usepackage[T1]{fontenc}
%\usepackage{graphicx}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\BS}{\begin{frame}}
\newcommand{\ES}{\pause\end{frame}}

\newcommand{\FT}{\frametitle{}}
%\usepackage[russian]{babel}
%\usepackage[francais]{babel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{tpc-e.tex}
\newtheorem{theo}{\sf Theorem}%[section]



%\newcommand{\theo}
\renewcommand{\thesection}{\arabic{section}}
%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thesubsection}{\thesection.\arabic{subsection}}
\renewcommand{\theequation}{%\thesection.
\arabic{equation}}
\renewcommand{\thetheo}{\thesection.\arabic{theo}}
\newcommand{\Equation}{\setcounter{equation} {0}\theequation}
%\newcommand{\soussection}{\setcounter{theo} {0}\subsection}
\newcommand{\Section}{\setcounter{equation}{0} \section}
%\renewcommand{\thetheo}{\thesection\arabic{theo}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newcommand{\soussection}{\setcounter{theoc}{0} \subsection}
\renewcommand{\thesubsubsection}{\thesubsection.\arabic{subsubsection}}







%\def\rem{{\bf Remarque : }}
%\newcounter{ncours}
%\newcommand{\ncours}
% {\bigskip Cours \Numero \arabic{ncours})\addtocounter{ncours}{1} Lundi \ }
%\newtheorem{theo}{\sc Theorem}[section]
\newtheorem{coro}[theo]{\sf Corollary}
\newtheorem{defi}[theo]{\sf Definition}
\newtheorem{lemm}[theo]{\sf Lemma}
\newtheorem{prop}[theorem]{\sf Proposition}
\newtheorem{rema}[theo]{\sf Remark}
\newtheorem{rems}[theo]{\sc Remarks}
\newtheorem{main}[theo]{\sf Main Theorem}
\newtheorem{conj}[theo]{\sc Conjecture}
\newtheorem{exam}[theo]{\sc Example}


%\bibliographystyle{amsplain}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%\def\defi#1{ {\it #1 } \index{#1} }

\def\qed{ \ \hfil$\square$}
\def\pgcd{{\hskip0.3mm\rm  pgcd\hskip0.5mm}}
\def\ppcm{{\hskip0.3mm\rm  ppcm\hskip0.5mm}}
\def\Ker{{\hskip0.3mm\rm  Ker\hskip0.5mm}}
\def\Im{{\hskip0.3mm\rm  Im\hskip0.5mm}}
\def\ostar{\otimes}
\makeindex

\def \F {{\mathbb F}}
\def\Si{\Sigma}
\def\St{{\widetilde\Sigma}}
\def\Nt{{\widetilde N}}

\def\ph{\varphi}
\def\W{A}
\def\End{{\rm End}}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\is{\buildrel \sim   \over \rightarrow}

\newcommand{\Ft}{\widetilde{F}}
\newcommand{\Gat}{\widetilde{\Gamma}}
\newcommand{\Mt}{\widetilde{\mathcal M}}
\newcommand{\mt}{\tilde{m}}


\newcommand{\bi}[2]{{#1 \choose #2}}
\newcommand{\di}[1]{\displaystyle{#1}}
\newcommand{\dkr}{\delta_k^{(r)}}
\newcommand{\Dpi}{\frac{1}{2\pi i}}
\newcommand{\Kpi}{\left(\frac{-1}{4\pi}\right)}
\newcommand{\kpiy}{\frac{k}{4\pi y}}
\newcommand{\ny}{\infty}
\newcommand{\pa}{\partial}
\newcommand{\pt}{\cdot}
\newcommand{\tpa}{\tilde{\partial}}
\newcommand{\tr}{{\rm tr}}


\newcommand{\dd}{{\rm d}}
\newcommand{\cont}{{\rm cont}}
\newcommand{\Gal}{{\rm Gal}}
\newcommand{\Gl}{{\rm Gl}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\GSp}{{\rm GSp}}
\newcommand{\Hol}{{\mathcal Hol}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\M}{{{\rm M}}}
%\newcommand{\mod}{{\rm ~mod~ }}
\newcommand{\oo}{{\rm o}}
\newcommand{\Og}{{\rm O}}
\newcommand{\ord}{{\rm ord}}
\newcommand{\SL}{{\rm SL}}
\newcommand{\SO}{{\rm SO}}
\newcommand{\SP}{{\rm Sp}}
\newcommand{\Stab}{{\rm Stab}}
\newcommand{\tors}{{\rm tors}}
\newcommand{\Tr}{{\rm Tr}}
%\newcommand{\U}{{\rm U}}
\newcommand{\lcm}{{\rm LCM~\!\!}}
%\newcommand{\gcd}{{\rm GCD~\!\!}}





\newcommand{\A}{{\mathbb A}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Hb}{{\mathbb H}}
%\newcommand{\Hb}{{\frak H}}


%\newcommand{\Hm}{{\mathbb H}_m}
\newcommand{\Hm}{{\mathbb H}_m}

\newcommand{\Fb}{\hat{F}}
\newcommand{\Mb}{\bar{M}}
\newcommand{\Qb}{\overline{\mathbb Q}}
\newcommand{\Sb}{\bar{S}}
\newcommand{\chib}{\bar{\chi}}
\newcommand{\psib}{\bar{\psi}}
\newcommand{\omb}{\bar{\omega}}

\newcommand{\Ar}{{\mathcal A}}
\newcommand{\Ac}{{\mathcal A}}
\newcommand{\Br}{{\mathcal B}}
\newcommand{\Cr}{{\mathcal C}}
\newcommand{\Cc}{{\mathcal C}}
\newcommand{\Ci}{{\mathcal C}^{\infty}}
\newcommand{\Dr}{{\mathcal D}}
\newcommand{\Hr}{{\mathcal H}}
\newcommand{\Ec}{{\mathcal E}}
\newcommand{\Fc}{{\mathcal F}}
\newcommand{\Gc}{{\mathcal G}}
\newcommand{\Mc}{{\mathcal M}}
\newcommand{\Mr}{{\mathcal M}}
\newcommand{\Lr}{{\mathcal L}}
\newcommand{\Or}{{\mathcal O}}
\newcommand{\Oc}{{\mathcal O}}
\newcommand{\Sr}{{\mathcal S}}
\newcommand{\Sc}{{\mathcal S}}
\newcommand{\Tc}{{\mathcal T}}
\newcommand{\Pc}{{\mathcal P}}
\newcommand{\Qc}{{\mathcal Q}}
\newcommand{\Vr}{{\mathcal V}}
\newcommand{\Xr}{{\mathcal X}}
\newcommand{\Xc}{{\mathcal X}}
\newcommand{\Yc}{{\mathcal Y}}
\newcommand{\Zc}{{\mathcal Z}}






%\newcommand{\bi}[2]{{#1 \choose #2}}
%\newcommand{\di}[1]{\displaystyle{#1}}
%\newcommand{\dkr}{\delta_k^{(r)}}
%\newcommand{\Dpi}{\frac{1}{2\pi i}}
%\newcommand{\Kpi}{\left(\frac{-1}{4\pi}\right)}
%\newcommand{\kpiy}{\frac{k}{4\pi y}}
%\newcommand{\ny}{\infty}
%\newcommand{\pa}{\partial}
%\newcommand{\pt}{\cdot}
%\newcommand{\tpa}{\tilde{\partial}}
%\newcommand{\tr}[1]{{\rm tr}\left(#1\right)}


\newcommand{\al}{\alpha}
\newcommand{\De}{\Delta}
\newcommand{\de}{\delta}
\newcommand{\e}{\varepsilon}
\newcommand{\ep}{\varepsilon}
\newcommand{\Ga}{\Gamma}
\newcommand{\ga}{\gamma}
\newcommand{\GaN}{\Gamma_0^m(N)}
\newcommand{\ka}{\kappa}
\newcommand{\La}{\Lambda}
\newcommand{\la}{\lambda}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\vph}{\varphi}


%\newcommand{\bm}{\underline{m}}
\newcommand{\derpa}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dv}{\:|\:}
\newcommand{\fraga}[2]{\frac{\Gamma(#1)}{\Gamma(#2)}}
\newcommand{\fragab}[2]{\frac{\Gamma(\beta+#1)}{\Gamma(\beta+#2)}}
\newcommand{\fragaf}[2]{\frac{\Gamma(\alpha+#1)}{\Gamma(\alpha+#2)}}
\newcommand{\fragafb}[2]{\frac{\Gamma(\alpha+\beta+#1)}
                        {\Gamma(\alpha+\beta+#2)}}
\newcommand{\fragam}[2]{\frac{\Gamma_m(#1)}{\Gamma_m(#2)}}
\newcommand{\luv}{\lambda(u,v)}
\newcommand{\mabcd}{\left(\begin{array}{cc}a&b\\c&d\end{array}\right)}
\newcommand{\mn}{\setminus}
\newcommand{\ndv}{\not{\:|\:}}
\newcommand{\nuv}{\eta(u,v)}
\newcommand{\rabcd}{\left(\frac{az+b}{cz+d}\right)}

\font\titlefont=cmss10 scaled \magstep2
\font\mathfont=cmmi10 scaled \magstep2
\font\authorfont=cmss10 scaled \magstephalf
\font\sectionfont=cmss10 scaled \magstep1
\font\Itemfont=cmss10
\font\ninerm=cmr9

%Script (in place of caligraphic) letters:
\font\teneusm=eusm10 \font\seveneusm=eusm7 
\font\fiveeusm=eusm5 
\newfam\eusmfam 
\textfont\eusmfam=\teneusm 
\scriptfont\eusmfam=\seveneusm 
\scriptscriptfont\eusmfam=\fiveeusm 





%%%%%%%%%%%%%Notation of M.Courtieu
\def\nd{\not\hskip3.0pt\mid}
\def\nds{\not\hskip1.5pt\mid}
\def\mat #1,#2,#3,#4,{\left({#1\atop #3}{#2\atop #4}\right)}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\bra#1,{{\left\lbrace {#1}\right\rbrace}}
\def\ph{\varphi}
\def\Ph{\Phi}
\def\si{\sigma}
\def\z{\zeta}
\def \sign {{\rm sign}}
\def \sgn {{\rm sgn}}
\def \inv {{\rm inv}}
\def \diag {{\rm diag}}
%\def \rem {{\rm rem}}
\def \Inf {{\rm Inf}}
%\def \Im {{\rm Im}}
%\def \Ker {{\rm Ker}}
\def \Proj {{\rm Proj}}
\def \Mor {{\rm Mor}}
\def \Div {{\rm Div}}
\def \div {{\rm div}}
\def \Pic {{\rm Pic}}
\def \Res {{\rm Res}}
\def \vol {{\rm vol}}
\def \Ver {{\rm Ver}}
%\def \di {{\rm div}}
\def \Spin {{\rm Spin}}
\def \Spec {{\rm Spec}}
\def \Step {{\rm Step}}
\def \Spl {{\rm Spl}}
\def \M{{\rm M}}
\def \Gal{{\rm Gal}}
\def \suc{{\rm suc}}
\def \Gab{G^{\rm ab}}
\def \rk{{\rm rk}}
\def \Supp{{\rm Supp}}
\def \Id{{\rm Id}}
\def \Sp{{\rm Sp}}
\def\tm{\tilde m}
\def\tGa{\tilde{\Ga}}
\def\is{\buildrel \sim   \over \rightarrow}
\def\bs{\backslash}
\def\rr{\rangle}
\def\l1{\langle}

\newcommand{\B}{\left(\begin{array}{cc}}
\newcommand{\E}{\end{array}\right)}
%\newcommand{\e}{{\varepsilon\over N}}
\newcommand{\p}{\tilde}

%\newcommand{\f}{{\epsilon\over N}}


%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\def \fns{{${}^{*)}$}}
%%%%%%%%%%%%%%%%%%%
%\newcommand{\marginalfootnote}[1]{%
%   \fns \footnotetext{*{#1}}
%   \marginpar{\hskip0.5cm{\sf *\thefootnote}}%
%}
%\newcommand{\edit}[1]{\marginalfootnote{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EDIT
%\def\mat #1,#2,#3,#4,{\left({#1\atop #3}{#2\atop #4}\right)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\comm}[1]
%{\fns  
%\marginpar{
%$\boxed{{\scriptsize {\sf \red
%\begin{tabular} {l} 
% #1
%\end{tabular}}}
%}$
%}
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\marginalfootnote}[1]{%
   \fns \footnotetext{{#1}}
   \marginpar{\hskip0.5cm{\sf \fns}}
%\thefootnote}}%
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\edit}[1]{\marginalfootnote{#1}}
%%%%%%%%%%%%%%%%%%%
\newcommand{\comm}[1]
{\fns\marginpar{$\boxed
{\hskip-6pt
%\begin{center}
%\parbox{3
{\small {\sf 
\begin{tabular} {l}
 #1
\end{tabular}
%\end{center}
}
}
}
$
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \?  {\comm{check ?}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\def \dem  {\noindent {\it Proof}}
%%%%%%%%%%%%%%%%%%%%%
\newcommand{\abst}[1]
{\scriptsize \begin{description}
\item{\par\noindent} 
 #1
 \end{description}
\normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{euscript}
\let\scr=\EuScript
\def\scrA{{\scr A}}   \def\scrB{{\scr B}}
\def\scrC{{\scr C}}   \def\scrD{{\scr D}}
\def\scrE{{\scr E}}   \def\scrF{{\scr F}}
\def\scrG{{\scr G}}   \def\scrH{{\scr H}}
\def\scrI{{\scr I}}   \def\scrJ{{\scr J}}
\def\scrK{{\scr K}}   \def\scrL{{\scr L}}
\def\scrM{{\scr M}}   \def\scrN{{\scr N}}
\def\scrO{{\scr O}}   \def\scrP{{\scr P}}
\def\scrQ{{\scr Q}}   \def\scrR{{\scr R}}
\def\scrS{{\scr S}}   \def\scrT{{\scr T}}
\def\scrU{{\scr U}}   \def\scrV{{\scr V}}
\def\scrW{{\scr W}}   \def\scrX{{\scr X}}
\def\scrY{{\scr Y}}   \def\scrZ{{\scr Z}}
\def\Mr{{\scr M}}
\def\Hol{{\scr Hol}}
\let\mathcal=\scr           \def\Or{{\scr O}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ang#1,{{\left\langle {#1}\right\rangle}} 
\newcommand{\ds}{\displaystyle}
\def\G{{\dbG}}
\def\ca {c'est \`a dire}
\def \po {polyn\^ome }
\def \ssi {si et seulement si }
\def \dtfq { de telle fa\c con que }
\def \Mq { Montrer que }
\def \Mqe { Montrer qu'il existe }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \preuve  {\noindent {\sc Preuve}}
\def \proof  {\noindent {\it Proof}}
\def \dem  {\noindent {\it  Demonstration}}
\def \exe  {\medskip\noindent {\sc Exemple}}
\def \exo  {\medskip\noindent {\sc Exercice}}
\def \rem  {\medskip\noindent {\sc Remarque}}
\def \nota  {\medskip\noindent {\sc Notations.}\  }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\thmref}[1]{theorem{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lemref}[1]{lemma~\ref{#1}}
\newcommand{\prref}[1]{proposition~\ref{#1}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{tpc-ed.tex}

\def\qed{ \ \hfil$\square$}
\def\pgcd{{\hskip0.3mm\rm  pgcd\hskip0.5mm}}
\def\ppcm{{\hskip0.3mm\rm  ppcm\hskip0.5mm}}
\def\Ker{{\hskip0.3mm\rm  Ker\hskip0.5mm}}
\def\Im{{\hskip0.3mm\rm  Im\hskip0.5mm}}
\def\ostar{\otimes}
%\makeindex

\def \F {{\mathbb F}}
\def\Si{\Sigma}
\def\St{{\widetilde\Sigma}}
\def\Nt{{\widetilde N}}

\def\ph{\varphi}
\def\W{A}
\def\End{{\rm End}}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\is{\buildrel \sim   \over \rightarrow}
\def\Sym{{\rm Sym}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%














\font\titlefont=cmss10 scaled \magstep2
\font\mathfont=cmmi10 scaled \magstep2
\font\authorfont=cmss10 scaled \magstephalf
\font\sectionfont=cmss10 scaled \magstep1
\font\Itemfont=cmss10
\font\ninerm=cmr9

%Script (in place of caligraphic) letters:
\font\teneusm=eusm10 \font\seveneusm=eusm7 
\font\fiveeusm=eusm5 
\newfam\eusmfam 
\textfont\eusmfam=\teneusm 
\scriptfont\eusmfam=\seveneusm 
\scriptscriptfont\eusmfam=\fiveeusm 


\font\tengothic=eufm10
\font\sevengothic=eufm7
\font\fivegothic=eufm5
\newfam\Gothic
\textfont\Gothic=\tengothic
\scriptfont\Gothic=\sevengothic
\scriptscriptfont\Gothic=\fivegothic
\def\go{\fam\Gothic\tengothic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%**********
\def\a{{\mathfrak a}}%gothic
\def\b{{\go b}}%gothic
\def \c{{\go c}}%gothic
\def \d{{\go d}}%gothic
\def \n{{\go n}}%gothic
\def \m{{\go m}}%gothic
\def \mp{{\go m}^\prime}%gothic
\def \p{{\go p}}%gothic
\def \q{{\go q}}%gothic
\def \e{{\go e}}%gothic

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%Notation of M.Courtieu
\def\nd{\not\hskip3.0pt\mid}
\def\nds{\not\hskip1.5pt\mid}
\def\mat #1,#2,#3,#4,{\left({#1\atop #3}{#2\atop #4}\right)}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\bra#1,{{\left\lbrace {#1}\right\rbrace}}
\def\ph{\varphi}
\def\Ph{\Phi}
\def\si{\sigma}
\def\z{\zeta}
\def\w{\omega}
\def\eps{\varepsilon}
\def\W{\Omega}
\def \sign {{\rm sign}}
\def \sgn {{\rm sgn}}
\def \inv {{\rm inv}}
\def \diag {{\rm diag}}
%\def \rem {{\rm rem}}
\def \Inf {{\rm Inf}}
%\def \Im {{\rm Im}}
%\def \Ker {{\rm Ker}}
\def \Proj {{\rm Proj}}
\def \Mor {{\rm Mor}}
\def \Div {{\rm Div}}
\def \div {{\rm div}}
\def \Pic {{\rm Pic}}
\def \Res {{\rm Res}}
\def \Re {{\rm Re}}
\def \vol {{\rm vol}}
\def \Ver {{\rm Ver}}
%\def \di {{\rm div}}
\def \Spin {{\rm Spin}}
\def \Card {{\rm Card}\,}
\def \Step {{\rm Step}}
\def \Spl {{\rm Spl}}
\def \M{{\rm M}}
\def \Gal{{\rm Gal}}
\def \suc{{\rm suc}}
\def \Gab{G^{\rm ab}}
\def \rk{{\rm rk}}
\def \Supp{{\rm Supp}}
\def \Id{{\rm Id}}
\def \Sp{{\rm Sp}}
\def\m{{\go m}}
\def\tGa{\tilde{\Ga}}
\def\is{\buildrel \sim   \over \rightarrow}
\def\bs{\backslash}
\def\rr{\rangle}
\def\l1{\langle}
\def\Eq{\Longleftrightarrow}
%\newcommand{\B}{\left(\begin{array}{cc}}
%\newcommand{\E}{\end{array}\right)}
%\newcommand{\e}{{\epsilon\over N}}
%\newcommand{\p}{\tilde}
\def\Numero{${\rm N\sp\circ}$}




\let\scr=\EuScript
\def\scrA{{\scr A}}   \def\scrB{{\scr B}}
\def\scrC{{\scr C}}   \def\scrD{{\scr D}}
\def\scrE{{\scr E}}   \def\scrF{{\scr F}}
\def\scrG{{\scr G}}   \def\scrH{{\scr H}}
\def\scrI{{\scr I}}   \def\scrJ{{\scr J}}
\def\scrK{{\scr K}}   \def\scrL{{\scr L}}
\def\scrM{{\scr M}}   \def\scrN{{\scr N}}
\def\scrO{{\scr O}}   \def\scrP{{\scr P}}
\def\scrQ{{\scr Q}}   \def\scrR{{\scr R}}
\def\scrS{{\scr S}}   \def\scrT{{\scr T}}
\def\scrU{{\scr U}}   \def\scrV{{\scr V}}
\def\scrW{{\scr W}}   \def\scrX{{\scr X}}
\def\scrY{{\scr Y}}   \def\scrZ{{\scr Z}}
\def\Mr{{\scr M}}
\def\Hol{{\scr Hol}}
\let\mathcal=\scr           \def\Or{{\scr O}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\db=\mathbb
\def\dbA{{\db A}}   \def\dbB{{\db B}}
\def\dbC{{\db C}}   \def\dbD{{\db D}}
\def\dbE{{\db E}}   \def\dbF{{\db F}}
\def\dbG{{\db G}}   \def\dbH{{\db H}}
\def\dbI{{\db I}}   \def\dbJ{{\db J}}
\def\dbK{{\db K}}   \def\dbL{{\db L}}
\def\dbM{{\db M}}   \def\dbN{{\db N}}
\def\dbO{{\db O}}   \def\dbP{{\db P}}
\def\dbQ{{\db Q}}   \def\dbR{{\db R}}
\def\dbS{{\db S}}   \def\dbT{{\db T}}
\def\dbU{{\db U}}   \def\dbV{{\db V}}
\def\dbW{{\db W}}   \def\dbX{{\db X}}
\def\dbY{{\db Y}}   \def\dbZ{{\db Z}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\bold=\boldsymbol
\def\bal{{\bold \alpha}}  \def\bbeta{{\bold \beta}} 
\def\bga{{\bold \gamma}}  \def\bde{{\bold \delta}} 
\def\bepsilon{{\bold \epsilon}}  \def\bep{{\bold \varepsilon}} 
\def\bzeta{{\bold \zeta}}  \def\boeta{{\bold \eta}} 
\def\btheta{{\bold \theta}}  \def\bth{{\bold \vartheta}} 
\def\biota{{\bold \iota}}  \def\bkappa{{\bold \kappa}} 
\def\bvarkappa{{\bold \varkappa}}  \def\blambda{{\bold \lambda}} 
\def\bmu{{\bold \mu}}  \def\bnu{{\bold \nu}} 
\def\bomicron{{\bold \omicron}}  \def\bpi{{\bold \pi}} 
\def\brho{{\bold \rho}}  \def\bsigma{{\bold \sigma}} 
\def\btau{{\bold \tau}}  \def\bipsilon{{\bold \ipsilon}} 
\def\bphi{{\bold \phi}}  \def\bchi{{\bold \chi}} 
\def\bpsi{{\bold \psi}}  \def\bxi{{\bold \xi}} 
\def\bvarphi{{\bold \varphi}}  \def\bomega{{\bold \omega}} 
\def\Bgamma{{\bold \Gamma}}  \def\Bdelta{{\bold \delta}} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\G{{\dbG}}
\def\ca {c'est \`a dire}
\def \po {polyn\^ome }
\def \ssi {si et seulement si }
\def \dtfq { de telle fa\c con que }
\def \Mq { Montrer que }
\def \Mqe { Montrer qu'il existe }
\def\im{\Im}
\def \Pf{{\rm Pf}}
%%%%%%%%%%%%%%%%%%%%%%%%%
%\def\vin{{ \rotate[90]\in  }}
\def\vin{{ {\tiny \mid }  
%\kern-4.75pt 
\kern-7.29pt 
\bigcup }}
\def \Ann{{\rm Ann}}
\def \id{{\rm id}}
\def \Ad{{\rm Ad}}
\def \Sel{{\rm Sel}}
\def \Ext{{\rm Ext}}
\def \Coker{{\rm Coker}}
\def \bs{{\backslash}}
\def \PGL{{\rm PGL}}
\def\ang#1,{{\left\langle {#1}\right\rangle}} 
\def \sec{{'{\!}'}}
\def \cds{{\cdot{\dots}\cdot}}





%\newcommand{\f}{{\epsilon\over N}}
%\newenvironment{slide}{\rule{12cm}{.1cm}} 
%\def \preuve  {\noindent {\sc Preuve}}
%\def \exe  {\medskip\noindent {\sc Exemple.} \ }
%\def \exo  {\medskip\noindent {\sc Exercice.}\  }
%\def \rem  {\medskip\noindent {\sc Remarque.}\  }
%\def \nota  {\medskip\noindent {\sc Notations.}\  }
%\def \rep  {\medskip\noindent {\it Réponse~:}\  }
%\def \sol {\medskip\noindent {\it Solution~:}\  \small}
%\def \pgcd {{\rm pgcd}}
%\def \ppcm {{\rm ppcm}}
\def \Aut {{\rm Aut}}
\def\m{{\frak m}}


%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\kr}[2]{\left(\frac{#1},{#2},\right)}
\newcommand{\con}{\equiv}
\newcommand{\union}{\cup}
\newcommand{\Tl}{T_{\ell}}
\newcommand{\into}{\rightarrow}
\newcommand{\onto}{\rightarrow\!\!\!\!\rightarrow}
\newcommand{\intersect}{\cap}
\newcommand{\cross}{\times}
%\newcommand{\ds}{\displaystyle}
%%%%%%%%%%%%%%%%%%%%%%%%%
\newenvironment{exercises}
{\setcounter{ex}{0}\vspace{1em}\par\noindent{\sc Exercices}             
  \renewcommand{\labelenumi}{\thesection\theenumi}\small
  \begin{enumerate}
}{\end{enumerate}}\normalsize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \prodp {\mathop {\prod{}^{\displaystyle  {}^{\prime}}}}
\def \sump {\mathop {\sum{}^{\displaystyle  {}^{\prime}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\AAA}{\mathbb A}
\newcommand{\CC}{\mathbb C}
\newcommand{\FF}{\mathbb F}
\newcommand{\HH}{\mathbb H}
\newcommand{\NN}{\mathbb N}
\newcommand{\QQ}{\mathbb Q}
\newcommand{\RR}{\mathbb R}
\newcommand{\ZZ}{\mathbb Z}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\def\cb{\C}
\def\ab{\A}
\def\nb{\N}
\def\pb{\Pb}
\def\qb{\Q}
\def\rb{\R}
\def\zb{\Z}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\wh{\widehat}
\def\la{\displaystyle\mathop{\longrightarrow}}
\def\oc{\Or}
\def\cc{\Cr}
\def\dc{\Dr}
\def\hc{\Hr}
\def\lc{\Lr}
\def\mc{\Mr}
%\def\lc{\Lr}
\def\pc{\Pr}
\def\scc{\Sr}
\def\vc{\Vr}
\def\xc{\Xr}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcounter{ncours}{\setcounter{ncours} {1}}
%\newcommand{\ncours}{\bigskip Lecture \Numero %\arabic{ncours}.\addtocounter{ncours}{1} \\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\kp}{\kappa}
\newcommand{\vkp}{\varkappa}
\newcommand{\chiba}{\bar{\chi}}
\newcommand{\chibar}{\bar{\chi}}
\newcommand{\Mba}{\bar{M}}
\newcommand{\mba}{\underline{m}}
\newcommand{\omba}{\bar{\omega}}
\newcommand{\psiba}{\bar{\psi}}
\newcommand{\Sba}{\bar{S}}
\newcommand{\SE}{{\rm SE}}
\newcommand{\lb}{\lambda}
\newcommand{\vr}{}
\def\dbQbar{\overline {\dbQ}}
\def\Lb{\Lambda}
\def\exptr{{\rm exptr}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Alphabet rond``\cal"     %Alphabet double``\ Bbb"      %Alphabet gras``\bf"  
\def\ac{{\cal A}}           \def\ab{{\Bbb A}}            \def\aaf{{\bf A}}
\def\bc{{\cal B}}           \def\bbb{{\Bbb B}}           \def\bbf{{\bf B}}
\def\cc{{\cal C}}           \def\cb{{\Bbb C}}            \def\ccf{{\bf C}}
\def\dc{{\cal D}}           
                                   %\def\db{{\Bbb D}}            
                                                                          \def\ddf{{\bf D}}
\def\ec{{\cal E}}           \def\eb{{\Bbb E}}            \def\eef{{\bf E}}
\def\fc{{\cal F}}           \def\fb{{\Bbb F}}            \def\fff{{\bf F}}
\def\gc{{\cal G}}           \def\gb{{\Bbb G}}            \def\ggf{{\bf G}}
\def\hc{{\cal H}}           \def\hb{{\Bbb H}}            \def\hhf{{\bf H}}
\def\ic{{\cal I}}           \def\ib{{\Bbb I}}            \def\iif{{\bf I}}
\def\jc{{\cal J}}           \def\jb{{\Bbb J}}            \def\jjf{{\bf J}}
\def\kc{{\cal K}}           \def\kb{{\Bbb K}}            \def\kkf{{\bf K}}
\def\lc{{\cal L}}           \def\lb{{\Bbb L}}            \def\llf{{\bf L}}
\def\mc{{\cal M}}           \def\mb{{\Bbb M}}            \def\mmf{{\bf M}}
\def\nc{{\cal N}}           \def\nb{{\Bbb N}}            \def\nnf{{\bf N}}
\def\oc{{\cal O}}           \def\obb{{\Bbb O}}           \def\oof{{\bf O}}
\def\pcc{{\cal P}}          \def\pb{{\Bbb P}}            \def\ppf{{\bf P}}
\def\qc{{\cal Q}}           \def\qb{{\Bbb Q}}            \def\qqf{{\bf Q}}
\def\rcc{{\cal R}}           \def\rb{{\Bbb R}}            \def\rrf{{\bf R}}
\def\scc{{\cal S}}          \def\sbb{{\Bbb S}}           \def\ssf{{\bf S}}
\def\tcc{{\cal T}}           \def\tb{{\Bbb T}}            \def\ttf{{\bf T}}
\def\uc{{\cal U}}           \def\ub{{\Bbb U}}            \def\uuf{{\bf U}}
\def\vc{{\cal V}}           \def\vb{{\Bbb V}}            \def\vvf{{\bf V}}
\def\wc{{\cal W}}           \def\wb{{\Bbb W}}            \def\wwf{{\bf W}}
\def\xc{{\cal X}}           \def\xb{{\Bbb X}}            \def\xxf{{\bf X}}
\def\yc{{\cal Y}}           \def\yb{{\Bbb Y}}            \def\yyf{{\bf Y}}
\def\zc{{\cal Z}}           \def\zb{{\Bbb Z}}            \def\zzf{{\bf Z}}



\def\diag {\mathop{\rm diag}\nolimits}
\def\SL {\mathop{\rm SL}\nolimits}
\def\Sp {\mathop{\rm Sp}\nolimits}
\def\npn{Np^n}
\def\ol{\overline}
\def\summ{\sum}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ùù
%\input cyracc.def
\font\tenrus=wncyr10
\font\sevenrus=wncyr7
\newfam\Rus
\textfont\Rus=\tenrus
\scriptfont\Rus=\sevenrus
\def\ru{\fam\Rus\tenrus}
\def\scr{\fam\Rus\sevenrus} 

\font\tenbrus=wncyb10
\font\eightbrus=wncyb8
\newfam\Brus
\textfont\Brus=\tenbrus
\scriptfont\Brus=\eightbrus
\def\bru{\fam\Brus\tenbrus}
\def\sbr{\fam\Brus\eightbrus}

\font\tenirus=wncyi10
\font\eightirus=wncyi8
\newfam\Irus
\textfont\Irus=\tenirus
\scriptfont\Irus=\eightirus
\def\iru{\fam\Irus\tenirus}
%\def\sir{\fam\Irus\eightirus}
\font\tenrus=wncyr10
\font\sevenrus=wncyr7
\newfam\Rus
\textfont\Rus=\tenrus
\scriptfont\Rus=\sevenrus
\def\ru{\fam\Rus\tenrus}
\def\scr{\fam\Rus\sevenrus} 
\input cyracc.def
%%%%%%%%%%%%%%%%%%%%%%
\def\i1{\accent'044i}
\def\I1{\accent'044I}
\def\e1{\accent'040e}
%\newcommand{\eo}{e1}
%\newcommand{\EO}{E1}
\def\l1{l{}p1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\T}{{\mathbf{T}}}
\newcommand{\KE}{{\mathbf{e}}}
\newcommand{\KF}{{\mathbf{f}}}
%\newcommand{\Ta}{{\mathbf{T}(p)}}
%\newcommand{\Tb}{{\mathbf{T}_1(p^2)}}
%\newcommand{\Tc}{{\mathbf{T}_2(p^2)}}
%\newcommand{\Td}{{\mathbf{T}_3(p^2)}}
%\newcommand{\Te}{{[\mathbf{p}]}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\fi
%\def\qed{ \ \hfil$\square$}
\def\qed{\quad\hbox{\hskip 1pt\vrule width 4pt height 6pt
          depth 1.5pt\hskip 1pt}}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% TITRE
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[Admissible measures and Hermitian modular forms
]
{Complex and $p$-adic $L$ functions on classical groups. Admissible measures, special values.\\ \normalsize
	("Fonctions $L$ $p$-adiques et complexes sur les groupes classiques~: mesures admissibles, valeurs spéciales").
	%Constructions of $p$-adic $L$-functions and admissible measures for Hermitian modular forms 
	%Admissible measures and Hermitian modular forms 
	}
\author{Alexei PANTCHICHKINE
	\\ \small
	Institut Fourier, 
	University Grenoble-Alpes 
	%B.P.74, 38402 St.-- Martin d'H\`eres, FRANCE
}


\institute[Grenoble]{ 
	{\magenta \Large
%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Summer School on the Theory of Motives and the Theory of Numbers
%%%%%%%%%%%%%%%%%%%%%%%%%		
\normalsize (L'École d'été à LAMA)%, Chambéry (June 2018)
%Workshop "Arithmetic of automorphic forms and special $L$-values"
\\ 
\vskip0.3cm 
\large\blue	
Fonctions zetas, polyzetas, séries arithmétiques : applications aux motifs et à la théorie des
nombres.	
%Department of Mathematics Colloquium		
		%Seminar on Algebra, Geometry and Physics		
		%Organiser:  Athanasios BOUGANIS, 
		%Yuri Ivanovich Manin		
		%One day with modular forms in Clermont
		%A.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
	}
}
%\hskip0.5cm
\date[June 18-29, 2018] % (optional)
{« Le Bourget-du-Lac » Chambéry, Université Savoie Mont Blanc 
	(June 18-29, 2018)
	%Max Planck Institute for Mathematics
%Université Clermont Auvergne, 	Laboratoire de mathématiques Blaise Pascal (Unité mixte de recherche 6620 UCA et CNRS)
%Department of Mechanics and Mathematics of Lomonosov Moscow State University	
%Steklov Mathematical Institute of Russian Academy of Sciences 	
%Moscow State University Lomonosov 
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%\pgfdeclareimage[height=0.5cm]{logo-ujf}{UJF}
%\logo{\pgfuseimage{logo-ujf}}



%\AtBeginSubsection[]
\AtBeginSection[]
{
  \begin{frame}<beamer>
    \frametitle{} %\ru Soderzhanie}
    %\tableofcontents[currentsection,currentsubsection]
   \tableofcontents[currentsection]
\addtocounter{framenumber}{-1}
  \end{frame}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\FF{\mathbb F}
\def\ZZ{\mathbb Z}

\def\f{\bf f}
\def\p{\mathfrak p}
\def\q{\mathfrak q}
\def\c{\mathfrak c}
\def\h{\mathfrak h}
\def\e{\mathfrak e}
\def\r{\mathfrak r}
\def\g{\mathfrak g}
\ifarticle
\addtolength{\textwidth}{3cm}
\addtolength{\oddsidemargin}{-1.9cm}
\addtolength{\evensidemargin}{-1.9cm}
\addtolength{\textheight}{1cm}
\fi


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\def\Numero{${\rm N\sp\circ}$}
\newcounter{ncours}{\setcounter{ncours} {1}}
\newcommand{\ncours}{\bigskip Lecture \Numero \arabic{ncours}.\ 
\addtocounter{ncours}{1}}

\begin{document}
	
%\only<+(1)->{
%\setbeamertemplate{footline}{}


\maketitle 

\ 
\vskip5pt
{\blue\Large This Summer School on Theory of Motives and Number Theory}  

at the crossroad of automorphic  $L$ functions (complex and $p$-adic), zeta functions, polyzeta functions and dynamical zeta function,
was conceived as a continuation of a seris of Conferences "Zeta-functions I-VI", held in J.-V.Poncelet Laboratory UMI 2615 du CNRS, The Higher School of Economics, and Independent University in Moscow, and mainly
organized by Alexey ZYKIN, professor of the French Polynesia University in Tahiti, who tragically dissapeared in April 2017 together with his wife Tatiana MAKAROVA.  

\

The Summer Schools «Algebra and Geometry» (Yaroslavl, RUSSIA, July 2012-2016), were also largely organized by Alexey ZYKIN together with Fyodor BOGOMOLOV and~Courant Institute (New York, USA), see the video and pdf of lectures at {\tt
	http://bogomolov-lab.ru/SHKOLA2012/talks/panchishkin.html} 
 

%\end{frame}
%\end{document}

\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Alexey ZYKIN near Grenoble on June 22, 2012}
\begin{figure}[h]
	\begin{center}
		%\includegraphics[width=14cm,height=10cm]{IF-CG2-10-5}
		\includegraphics[width=16cm,height=11cm]{DSC07739}
		\caption %{\label{DSC07739}
			{Climbing the mountain Chamchaude  with Siegfried BOECHERER}
		%\includegraphics[width=14cm,height=10cm]{IMG_1153}
		%\caption{\label{IMG_1153}
		%	Freydoon Shahidi in the Conference on October 21, 2017}
	\end{center}
\end{figure}

\	

	
\end{frame}
%\end{document}
%[shrink=5]
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  INTRODUCTION
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[shrink]
	\frametitle{Contents : Mini-course of Alexei Pantchichkine (6h) :}
		
	\
	
\tableofcontents%[comma-separated option list]	
	
\

(L'intervention d'Alexei Pantchichkine:

1) Groupes classiques, le cas $\GL(n)$, le cas symplectique et unitaires. Formes modulaires et formes automorphes, exemples.

2) Formes  modulaies hermitiennes. Fonctions $L$ complexes sur les groupes classiques. Algèbres de Hecke. Methode de Rankin-Selberg

3) Distributions, mesures, congruences de Kummer.Fonction zêta $p$-aique de Kubota-Leopoldt et l'algèbre d'Iwasawa.

4) Fonctions $L$ $p$-adiques sur les groupes classiques : mesures admissibles, valeurs spéciales)


	\
	
\end{frame}	

\setbeamertemplate{footline}
{%
	\begin{beamercolorbox}{section in foot} 
		\vskip-10pt
		%%%%%%%%%%%%%%%%%%%
		\fbox{
			\insertpagenumber
			%\insertframenumber
		}
		\ {}
	\end{beamercolorbox}%
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lecture \Numero 1. Classical groups, the case  of $\GL(n)$.
The symplectic and unitary cases. Modular forms and automorphic forms.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\frametitle{\ncours 
		Classical groups, the case  of $\GL(n)$}.
	\vskip-0.8cm The sympectic and unitary cases. Modular forms and automorphic forms. ("Groupes classiques, le cas $\GL(n)$ le cas symplectique et unitaires. Formes modulaires et formes automorphes, exemples").

\begin{itemize}
\item Linear Algebraic Groups.  \S 1-6  of \cite{Bor66}

\item Radical. Parabolic subgroups. Reductive groups.

\item Structure theorems for reductive groups.
 


\S 6.5 of\cite{MaPa}, 
Automorphic Forms and The Langlands Program
                                                     
\item
\S 6.5.1 A Relation Between Classical Modular Forms and Representation Theory
\item \S I-II of \cite{Bor79}: I. Definition of the $L$-group 	

\item II. Quasi-split groups%; the unramified case (p.35)
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Radical. Parabolic subgroups. 	
	
	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\end{document}
\begin{frame}[shrink]
	\frametitle{Reductive groups}
Recall: an algebraic group is {\magenta irreducible} if and only if
it is connected. 
%To avoid confusion with the irreducibility of a linear group,
%we shall usually speak of connected algebraic groups. 
The {\magenta connected component}
of the identity of $G$ will be denoted by $G^0$. The index of $G^0$ in $G$ is finite.	
\begin{definition} Let $G$ be an algebraic group over a field $k$. The {\magenta radical} $R(G)$ of $G$ is the {\magenta greatest connected normal subgroup} of $G$; the {\magenta unipotent radical} $R_u(G)$ is the {\magenta greatest connected unipotent normal subgroup} of $G$. The group $G$ is {\magenta semisimple (resp. reductive)} if $R(G) =\{e\}$ (resp. $R_u(G) = \{e\}$).
\end{definition}
%\end{frame}
%\end{document}	
	The definitions of $R(G)$ and $R_u(G)$ make sense, because if $H, H'$ are connected normal and solvable (resp. unipotent) subgroups, then so is 
	$H \cdot H'$. 
	Both radicals are $k$-closed if $G$ is a $k$-group. 
	Clearly, $R(G) = R(G^0)$ and $R_u(G) = R_u(G^0)$. 
	The quotient G/R(G) is semisimple, and $G/R_u(G)$ is reductive. In characteristic zero, the unipotent radical has a complement; more precisely: Let $G$ be defined
	over $k$. There exists a maximal reductive $k$-subgroup $H$ of $G$ such that $G = H \cdot R_u(G)$, the product being a semidirect product of algebraic groups. If $H'$ is a reductive
	subgroup of $G$ defined over $k$, then $H'$ is conjugate over $k$ to a subgroup of $H$ of $G$ such that
	$G = H \cdot R_u(G)$,
	the product being a semidirect product of algebraic groups. If $H'$ is a reductive
	subgroup of G defined over $k$, then $H'$ is conjugate over $k$ to a subgroup of $H$.
	 	
%\end{definition}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\

	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\frametitle{}
%5.2. THEOREM. 
\begin{theorem}[5.2 of \cite{Bor66}]
	Let $G$ be an algebraic group. The following conditions are
equivalent :

(1) $G^0$ is reductive,

(2) $G^0 = S \cdot G'$, where $S$ is a central torus and $G'$ is semisimple,

%(3) $G^0$ has a locally faithful fully reducible rational representation,

%(4) If moreover the characteristic of $k is 0, all rational representations of G are fully reducible.

\end{theorem}

%5.3. THEOREM [ll 
\begin{theorem}[5.3 of \cite{Bor66}]
Let $G$ be a connected algebraic group.

(1) All maximal tori of $G$ are conjugate. Every semisimple element is contained in a torus. The centralizer of any subtorus is connected.

(2) All maximal connected solvable subgroups are conjugate. Every element of $G$ belongs to one such group.

(3) If $P$ is a closed subgroup of $G$, then $G/P$ is a projective variety if and only if $P$ contains a maximal connected solvable subgroup.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\begin{frame}[shrink]\frametitle{Characters and roots.} 
	A character of $G$ is a rational representation of degree 1;
$\chi: G\to \GL_1$. 
The set of characters of $G$ is a commutative group, denoted by
$X(G)$ or $\hat G$. The group $\hat G$ is finitely generated; it is free if $G$ is connected \cite{Bor66}, p.6. 
If one
wants to write the composition-law in $\hat G$ multiplicatively, the value at $g \in G$ of
$ \chi\in \hat G$ should be noted $\chi(g)$. 
But since one is accustomed to add roots of Lie
algebras, it is also natural to write the composition in $G$ additively. The value of $\chi$ at $g$ will then be denoted by $g^\chi$. To see the similarity between roots and characters
take $\Omega = \C$; if $X\in {\mathfrak g}$, the Lie algebra of $G$, 
$(e^x)^\chi = ed_\chi(X)$, where $d_\chi$
 is the
differential  at $e$; $d_\chi$ is a linear form over ${\mathfrak g}$. In the sequel, we not make any notational distinction between a character and its differential at $e$.

\

Let $g\in \GL(n,\Omega)$,
$g$ can be written uniquely as the product $g = g_s \cdot g_n$, 
where $g_s$ is a semisimple
matrix (i.e., g. can be made diagonal) and $g_n$ is a unipotent matrix (i.e., the only
eigenvalue of $g_n$ is 1, or equivalently $g_n - I$ is nilpotent) and 
$g_s\cdot g_n = g_n \cdot g_s$.

\

\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\begin{frame}[shrink]\frametitle{Example: the case  of $\GL(n)$}	
The rank of $G$ is the common dimension of the maximal tori, (notation rk($G$)).
A closed subgroup $P$ of $G$ is called parabolic, if $G/P$ is a projective variety.
A maximal connected closed solvable subgroup is called {\magenta Borel subgroup}.

\hskip-0.4cm{\large \blue Exemple.} $G = \GL_n$. 
{\magenta A flag} $\Fc$ in a vector space $V$
is a properly increasing sequence
of subspaces
\begin{align*}
  \Fc  :   0 \ne V_1 \subset \cdots \subset V_t \subset V_{t+ 1} = V. 
\end{align*} 
The sequence ($d_i$)
$(d_i = \dim v_i, i = 1,\cdots t))$
describes the type of the flag. If $d_i = i$ and $t = \dim V - 1$, we speak of a {\magenta full flag}.


{\magenta A parabolic subgroup} of $\GL_n$ is the stability group of a flag $\Fc$ in $\Omega^n$. $G/P$ is the manifold of flags of the same type as $F$, and is well known to be a projective variety. 
{\magenta A Borel subgroup} is the stability group of a full flag. 
In a suitable basis, it is the group of all upper triangular matrices.

\

\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\begin{frame}[shrink]\frametitle{The case of orthogonal group $G=SO(F)$} of a {\magenta nondegenerate quadratic form} $F$ on
a vector space $V_k$ (where, to be safe, one takes char $k \ne 2$). 
In a suitable basis
$$
F(x_1, \cdots, x_n) = x_1x_n + x_2x_{n-1} + \cdots + x_qx_{n-q + 1} + F_0(x_{q+ 1},  \cdots, x_{n-q})
$$
where $F_0$ does not represent zero rationally.
The {\magenta index} of $F$, the dimension of
the maximal isotropic subspaces in $V$, is equal to $q$. 
A maximal $k$-split torus $S$
is given by the set of following diagonal matrices:
$\begin{pmatrix}
s^{\lambda_1}&&&&&&&&& \\
& s^{\lambda_2} &&&&&&&& \\
&& \ddots &&&&&&&\\
&&&s^{\lambda_q}&&&&&&\\
&&&&1&&&&&\\
&&&&&\ddots&&&& \\
&&&&&&1&&& \\
&&&&&&& s^{-\lambda_q}&& \\
&&0&&&&&&s^{-\lambda_{q-1}}& \\
&&&&&&&&\ddots& \\
&&&&&&&&&s^{-\lambda_{1}}	
\end{pmatrix}
$




\

\end{frame}	
%\end{document}

\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\begin{frame}[shrink]
\frametitle{Let $SO(F_0 )$ denote the proper orthogonal group}
	 of the quadratic form $F_0$ ,
imbedded in $SO(F)$ by acting trivially on 
$x_1, \cdots, x_q, x_{n-q+ 1}, ... , x_n$· 
Then
$Z(S) = S \times SO(F_0)$. 
The minimal parabolic $k$-subgroups are the stability groups
of the full isotropic flags. 
For the above choice of $S$, and ordering of the coordinates,
the standard full isotropic flag is
$$
[e_1] \subset [e_1, e_2 ]\subset \cdots\subset[e_1, \cdots e_q ]
$$
The corresponding minimal parabolic $k$-subgroup takes then the form
$$
P=\left\{
\begin{pmatrix}
A_0&A_1&A_2\\
0&B&A_3\\
0&0&A_4
\end{pmatrix}
\right\}
$$
where $A_0$ and $A_4$ are upper triangular $q \times q$
 matrices, $B\in SO(F_0)$, with additional
relations that insure that $P \subset SO(F)$. 
The unipotent radical $U$ of $P$ is the set of
matrices in $P$, where $B = I, A_0, A_4$ are unipotent, and
\begin{align*}
&A_4={}^\sigma A_0^{-1};\  Q\cdot A_3+{}^t A_1\cdot J\cdot A_4=0,
\\
&{}^t A_4\cdot J\cdot A_2+{}^t A_3\cdot Q\cdot A_3+{}^t A_2\cdot J\cdot A_4=0
\end{align*}
where $Q$ is the matrix of the quadratic form $F_0$, $J$ is the $q \times q$ matrix with one's in the nonprincipal diagonal and zeros elsewhere, and $\sigma$ is the transposition
with respect to the same diagonal, $({}^\sigma M = J^t M J)$.	


\

\end{frame}	
\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\begin{frame}[shrink]\frametitle{Example: Unitary group}
Let us review some background and set up standard notation.
Let $E$ be a quadratic imaginary field, embedded in $\C$; $0 \le m \le n$
 and $\Lambda = \Oc_E^{n+m}$.
Let
%(1.1.1)
\begin{align} & \nonumber
 I_{n,m} =
\begin{pmatrix}
& & I_m 
\\
&I_{n-m}& \\
I_m&&
\end{pmatrix}
\end{align} 
where $I_\ell$ is the unit matrix of size $\ell$, and introduce the perfect hermitian pairing
\begin{align} & \nonumber
%(1.1.2)
 (u, v) = {}^t\bar u I_{n,m}v
\end{align}  
on $\Lambda$. 
Let
$G = GU(\Lambda, (,) )$
be the {\magenta group of unitary similitudes} of $\Lambda$, regarded as a group scheme over $\Z$; and
denote by $\nu : G \to {\bf G}_m$ the similitude character. 
For any commutative ring $R$
\begin{align*} 
G(R) = \{ g \in GL_{n+m}(\Oc_E 
\otimes R)| \forall u, v \in \Lambda\otimes R,\  
(gu, gv) = \nu(g)(u, v)
\}.
\end{align*} 
Then $G(R) = GU(n,m)$ is the {\it general unitary group of signature} $(n,m)$, and
$G(\C) \cong GL_{n+m}(\C) \otimes \C $.	


\

\end{frame}	
%\end{document}
\begin{frame}[shrink]\frametitle{Geometric algebra (see [Dieudonné], [Garrett])}
	
	- $\GL(n)$ (geometric study).
	
	- Bilinear and Hermitian forms, classical groups
	
	- Witt theorem and extensions of isometries
	
	This section is based on notions of geometric algebra.
	
	Concerning matrix notation, for a  rectangular matrix $A = (a_{ij})$ let ${}^ tA$ denote the transpose of $A$. If the entries of $A$ belong to a ring  $D$ with involution muni d'involution $\si$, let $A^\si$
	given by  $A^\si_{ ij }= a^\si_{ ij }$.
	
	%13.1 
	{\magenta Geometric study of $\GL(n)$ and its subgroups.} 
	The group $\GL(n)$ is a basic classical group showing the most interesting phenomena used in many other situations. The  general linear group $\GL(n, k)$ is the group of all invertible $ n\times n$ matrices with entries  in  a commutative field $k$.The special  linear group $\SL(n, k)$ is its subgroup
	of all  $ n\times n$  of determinant 1.
	
	For an approach less dependant of coordonnates fix a $k$-vector space
	$V$ of dimension $n$ and let  $\GL_k(V )$  be the group of all $k$-linear automorphisms of $V$ .
	Any choice of a base in $V$ gives an isomorphism $\GL_k(V ) \to \GL(n, k)$ using the  matrix of linear mapping in the chosen base. Let $e_1,\cdots , e_n$
	the  standard bases of  $k^n$ giving the isomorphhism 
	$\GL_k(k^n ) \to \GL(n, k)$.
	
	
	
	
	\
	
\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]\frametitle{Conjugation of parabolic subgroups.}
	Let $V = k^n$ and $\Fc$ the standard flag of type $(d_1, \cdots , d_m)$, the parabolic subgroup  $P_\Fc$ is represented by blocs
	$ \begin{pmatrix}
	d_1 \times d_1	& *& *&*\\
	&(d_2- d_1) \times (d_2- d_1)&*& *\\
	& &\ddots&*\\
	&\cdots&\cdots&\\
	0&0 & 0&(n - d_m) × (n - d_m)
	\end{pmatrix}	
	$
	Any $g\in P = P_\Fc $ induce
	a natural mapping  on the quotients $Vd_i/V_{d_{i-1}}$ , where $V_{d_0} = 0$ and $V_{d_{m+1}} = V$ ).
	
	Then the unipotent radical $
	R_uP$ = $\{p\in P_\Fc \ |\  p = id \text{ on  }Vd_i/V_{d_{i-1}} \text{ and } V/V_{d_m}\}
	$
	is represnted by
	$ 
	\begin{pmatrix}
	1_{d_1}	& *& *&*\\
	&1_{d_2- d_1} &*& *\\
	& &\ddots&*\\
	&\cdots&\cdots&\\
	0&0 & 0&1_{n- d_m}
	\end{pmatrix}.
	$ 
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]\frametitle{Levi components and conjugation}
	Choose  a complement $V'_{n-d_i}$ of $ V_{d_i}$ in $V$ with the property
	$V'_{n-d_m}\subset \cdots\subset V'_{n-d_1}$ (an opposit flag $\Fc'$
	of $\Fc$) with the opposit parabolic $P'=P_{\Fc'}$.
	Then {\magenta $M = P \cap P'$ is called a complementary Levi component} in $P= M\ltimes R_uP$,  a standard semi-direct product.
	
	Then the  standard Levi component is the group of matrices of the  form
	$ 
	\begin{pmatrix}
	d_1 \times d_1	& 0& 0&0\\
	&(d_2- d_1) \times (d_2- d_1) &0& 0\\
	& &\ddots&0\\
	&\cdots&\cdots&\\
	0&0 & 0&(n - d_m) × (n - d_m)
	\end{pmatrix}.
	$
	\begin{prop}
		a) All the parabolic subgroups of given type are conjugate in $\GL_k(V )$
		
		b) All the Levi components of parabolic subgroup $P$ are conjugate by elements of $P$
		
		c) All the maximal $k$-split tori are conjugate in $\GL_k(V )$.
	\end{prop}
	\qed
	
	
	
	\
	
\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]\frametitle{Extension to modules over a scew field}
	This section applies unchanged when $k$ is replaced by  a scew field (a division ring) $D$.
	Without coordinates, define a vector space $V$ of finite dimension over a scew field (a division ring) $D$ as a finitely generated  (left or right) module.
	
	If  $D$ is not commutative, there is a modification in viewing at $D$-lineair 
	endomorphisms.The the ring $\End_D(V )$ of all $D$-lineair 
	endomorphisms does not contain $D$  naturallurally. Then a choice of $D$-bases for a vector space $D$ of given  dimension gives an isomorphism
	$
	\End_D(V )$ to  $n \times n$ matrices with coeffcients in $D^{opp},
	$
	where $D^{opp}$ is the opposite ring to $D$, i.e.with the same additive group $D$ but with the multiplication $*$, given by
	$x * y = yx$
	where $yx$ is the multiplication in $D$.
	The linear  group $\GL(n,D)$ over $D$ is the  group of all the invertible $ n\times n$ matrices over $D$. A version without coordinates is $GL_D(V )$,and a choice of $D$-bases of $V$
	gives an isomorphism
	$
	\GL_D(V )\to \GL(n,D^{opp}).
	$
	Definitions concerning flags and  parabolics are identical to the commutative case. A flag $ \Fc$ in $V$ is a chain
	$\Fc = (V_{d_1}\subset V_{d_2} \subset \cdots\subset V_{d_m})$
	of subspaces.
	%, where $V_i$ is of dimension $i$.
	
	\begin{prop}
		a) All the parabolic subgroups of a given type are conjugate in $\GL_D(V )$
		
		b) All the Levi components of parabolic subgroup $P$ are conjugate by elements of $P$.
	\end{prop}
	\qed
	
	
	
	
	
	\
	
\end{frame}	
%\end{document}
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\begin{frame}[shrink]\frametitle{Bilinear, sesquilinear and Hermitian forms; classical groups}
	The classical  groups are defined as certain isomtries
	or  similitudes of "formes" on the vector spaces.
	First, orthogonal  and symplectic groups are defined.
	These can be included into more general families.
	
	\smallskip
\hskip-10pt	{\blue \large Bilinear forms, symmetric and symplectic forms}
	
	
	Let $Q(v)=\langle v, v \rangle $ be the quadratic form attached to a symmetric  bilinear form $\langle \cdot, \cdot \rangle $ on a $k$-vector space $V$.
	
	The associated {\magenta orthogonal group} $O(Q)$ is the  group  of isometries of $Q$ (or of $\langle \cdot, \cdot \rangle $), defined as
	\begin{align*}
		&
		O(Q) = O(\langle \cdot, \cdot \rangle) = \{g \in GL_k(V ) \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \langle v_1, v_2 \rangle\},  \\
		&
		\text{ and the \magenta group of orthogonal similitudes is } GO(Q) = GO(\langle \cdot, \cdot \rangle) \\ &
		= \{g \in GL_k(V ), \exists \nu(g)\in k^* \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \nu(g)\langle v_1, v_2 \rangle\}
	\end{align*}
	If $\forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = -\langle v_1, v_2 \rangle$, then the bilinear form
	$ f : V \times V \to  k, f(v_1, v_2) = \langle v_1, v_2 \rangle$ is said symplectic.The  {\magenta symplectic group} attached to  $f$ is the group of isometries of the form $f = \langle v_1, v_2 \rangle$ defined by
	$
	\Sp(f)  = \{g \in GL_k(V ) \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \langle v_1, v_2 \rangle\}, 
	$ puis\\ the {\magenta group of symplectis symilitudes}
	$
	\GSp(f)  =$\\
	$ \{g \in GL_k(V ), \exists \nu(g)\in k^*\ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \nu(g)\langle v_1, v_2 \rangle\}.
	$ 
	\
	
\end{frame}	

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]\frametitle{Structure theorems for reductive groups.} 
	{\magenta \large Root systems}. Let $V$ be a finite dimensional real vector space endowed with a positive nondegenerate scalar product. A subset $\Phi$ of $V$ is a root system when
	
	(1) $\Phi$ consists of a finite number of nonzero vectors that generate $V$, and is
	symmetric ($\Phi = -\Phi$).
	(2) for every $\alpha \in \Phi$, $s_\alpha( \Phi) =\Phi$, 
	where $s_\alpha$ denotes reflection with respect to the hyperplane perpendicular to $\alpha$.
	(3) if $\alpha, \beta \in \Phi$,  then 
	$$
	2(\alpha, \beta)/( \alpha, \alpha) \in \Z. 
	$$
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	The group generated by the symmetries
	$s_\alpha (\alpha \in \Phi)$ is called the {\magenta Weyl group} of $\Phi$ (notation $W(\Phi)$ is finite. The integers $2(\alpha, \beta)/( \alpha, \alpha)$ 
	are called the {\magenta Cartan integers} of $\Phi$. 
	The integrity condition means that for
	every $\alpha$ and $\beta$ of $\Phi$, $(s_\alpha (\beta) - \beta )$ is an integral multiple of $\alpha$, since 
	$$
	s_\alpha (\beta) = \beta - 2\alpha (\alpha, \beta)/( \alpha, \alpha).
	$$
	For the theory of reductive groups we shall have to enlarge slightly the notion of root system: if $M$ is a subspace of $V$, we say that $\Phi$ is a root system in $(N, M)$ if it generates a subspace $P$ supplementary to $M$, and is a root system in $P$.
	The Weyl group $W(\Phi))$ is then understood to act trivially on $M$.
	A root system $\Phi$ in $V$ is the direct sum of $\Phi' \subset V'$ and 
	$\Phi" \subset V"$, if $V = V' \oplus V"$, and $\Phi = \Phi' \cup \Phi "$. The root system is called {\magenta irreducible} if it is not the direct sum of two subsystems.
	
	
	
	
	\
	
\end{frame}	
%\end{document}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Weyl-chamber}A hyperplane of $V$ is called singular if it is orthogonal to a root $\alpha \in \Phi$. A Weyl-chamber $C^0$ is a connected component of the complement of the union of the singular hyperplanes.
	To a Weyl-chamber, is associated an ordering of the roots defined by:
	$\alpha > 0$, if $(\alpha, v) > 0$ for every $v$ in $C^0$.
	
	{\large\blue Roots of a reductive group}, with reference to a torus. 
	Let $G$ be a reductive group, and $S$ a torus of $G$. 
	It operates on the Lie-algebra ${\mathfrak g}$ of $G$ by the adjoint representation. 
	Since $S$ consists of semisimple elements, $\Ad_{\mathfrak g} S$
	is diagonalizable
	$$
	{\mathfrak g}= {\mathfrak g}_0^{(S)}\oplus  \coprod_\alpha {\mathfrak g}_\alpha^{(S)}
	$$
	
	The set $\Phi (G, S)$ of roots of $G$ relative to the torus $S$ is the set of nontrivial characters of S appearing in the above decomposition of the adjoint representation.
	If $T \subset S$, every root of G relative to $T$ that is not trivial on $S$ defines a root relative to $S$. 
	If $T$ is maximal $\Phi (G, T) = \Phi (G)$ is the set of roots of $G$ in the usual sense.
	
	
	\
	
\end{frame}	
%\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2

%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\begin{frame}[shrink]
	\frametitle{Positive roots for orthogonal groups} 
	To determine them one has to let $S$ operate on $U$. 
	To compute the root spaces it is easier to
	diagonalize $Q: q_{ij} = d_i \cdot \delta_{ij}$.
	{\magenta Three cases} are to be considered\\
	%\end{frame}	
	%\end{document}
	-- $ i < j \le q ; \lambda_i -\lambda_j$ is a root; the corresponding root
	space is generated by
	$ e_{ij}-e_{n-j+ 1,n-i+ 1}$; 
	the multiplicity of the root is 1.
	\\
	--$i \le q < j \le n - q; \lambda_i$; is a root with multiplicity $n - 2q$; the corresponding root space is generated by
	$
	e_{ij} - \delta_j^{-1} e_{j,n-i+ 1} \ \ \  (q + 1 \le j < n - q)
	$
	\\
	--$i < j ~ q; \lambda_i + \lambda_j$ is a root with multiplicity one; the corresponding root
	space is generated by $e_{i,n-j+1} - e_{i.n-i+1}$· 
	The simple roots are
	$
	\lambda_1 - \lambda_2, \lambda_2 - \lambda_3, \cdots, 
	\lambda_{q-1}- \lambda_q
	$
	\\
	and $\lambda_q$ if $n \ne 2q$, $\lambda_{q-1}+\lambda_q$ if $n = 2q$. 
	\\{\magenta The Weyl group} consists of all products of
	permutation matrices with symmetries with respect to a coordinate subspace
	(of any dimension if $n \ne 2q$ of even dimension if $n = 2q$).
	The group $SO(F)$ splits if and only if $q = [n/2]$.
	If it does not split, there exist roots with multiplicity > 1.
	The parabolic $k$-subgroups are the stability subgroups of rational isotropic flags. 
	The parabolic $k$-subgroups are conjugate over $k$ if and only if there exists
	an element of $G_k$ mapping one flag onto the other; 
	by Witt's theorem this is possible if and only if the two flags have the same type.
	
	\
	
\end{frame}		
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{Properties of root systems.}
	(1) Every root system is direct sum of irreducible root systems.
	
	(2) If  $\alpha$ and $\lambda\alpha \in \Phi$, then $\lambda = \pm 1, \pm (1/2)$, or  $±2$.
	
	The root system $\Phi$, is called reduced when for every $\alpha \in \Phi$, the only multiples of $\alpha$ belonging to $\Phi$ are $\pm\alpha$. 
	To every root system $\Phi$, there belongs two natural reduced systems by removing for every $\alpha \in \Phi$, the longer (or the shorter) multiple
	of $\alpha$ :
	
	(3) The only reduced irreducible root systems are the usual ones 
	(The first four are those belonging to the classical series, , with  the simple roots as follows:
	take $\R^n$ with the standard metric and basis 
	$\{{\lambda}_1, \cdots, {\lambda}_n\}$)\\
	$
	A_n (n \ge 1),\ \  \ \mathfrak {sl}_{n+1}\ \ \  \  \lambda_1 - \lambda_2, \lambda_2 - \lambda_3,\cdots,
	\lambda_{n-1} - \lambda_n,  \lambda_n-\lambda_{n+1} $ \\
	$ B_n (n \ge 2), \ \ \ \mathfrak {so}_{2n+1}, \ \  \lambda_1 - \lambda_2, \lambda_2 - \lambda_3,\cdots,
	\lambda_{n-1} - \lambda_n, \lambda_n$,  \\
	$ C_n (n \ge 3), \ \ \ \mathfrak {sp}_{2n}, \ \ \ \ \lambda_1 - \lambda_2, \lambda_2 - \lambda_3,\cdots,
	\lambda_{n-1} - \lambda_n, 2\lambda_n$ \\
	$ D_n (n \ge 4), \ \ \ \mathfrak {so}_{2n} \ \ \ \ \  \lambda_1 - \lambda_2, \lambda_2 - \lambda_3,\cdots,
	\lambda_{n-1} - \lambda_n, 	\lambda_{n-1} + \lambda_n$ \\
	\smallskip
	$G_2, F_4 , E_6 , E_7 , E_8
	$ (the exceptional root systems, see \cite{FuHa91}, \S 21).
	
%	(4) For each dimension $n$, there exists one irreducible nonreduced system, denoted by $BC_n$ (see below).
	
	EXAMPLES : 
	$B_n$: Take $\R^n$ with the standard metric and basis 
	$\{{\lambda}_1, \cdots, {\lambda}_n\}$, \\
	
	$B_n =\{\pm({\lambda}_i\pm {\lambda}_i) (i <j)  \text{ and } \pm {\lambda}_i (1 \le i \le n)\}$.
	
	$W(B_n) =  \{ s \in \GL(n, R) \text{ is a product of a permutation matrix}$\\ 
	\hskip1.5cm	with a symmetry with respect to a coordinate subspace\}
	
	
	$C_n = \{
	\pm ({\lambda}_i \pm {\lambda}_j ) \ \ (i < j) \text{ and } \pm 2{\lambda}_i (1 \le i \le n\}$,
	$W(C_n) = W(B_n)$,
	
\
	
\iffalse	
	$BC_n = \{ \pm ({\lambda}_i + {\lambda}_j)\ \   (i < j), \pm {\lambda}_i\text{ and }\pm 2{\lambda}_i (1 \le i \le n)\}$,
	
	$W(BC_n) = W(B_n)$.   
\fi	

	\
	
\end{frame}	
%\fi
\begin{frame}[shrink]
	\frametitle{Example of  $G = GL(n)$}
	$
	S =\text{ group of diagonal mattices}
	=\{\diag(s^{\lambda_1},  s^{\lambda_2},.., s^{\lambda_n}) \}
	$
	where $\lambda_i \in\hat S$ is such that $s^{\lambda_i} = s_{ii}$.
	$S$ is a split torus and is maximal.\\
	A minimal parabolic $k$-subgroup $P$ is given by the upper triangular matrices, which is in this case a Borel subgroup. 
	The unipotent radical $U$ of $P$ is given by the group of upper triangular matrices with ones in the diagonal.
	The Lie algebra ${\mathfrak gl}_{n}=\M_{n\times n}$ with a basis
	 $\{ e_{ij}\}_{i,j} $, where $e_{ij}$ is the matrix having all components zero except that with index $(i,j)$ equal to 1, 
	$$
	\Ad_G s(e_{ij}) = (s^{\lambda_i}/ s^{\lambda_j}) e_{ij}.
	$$ 
	So the positive roots are $\lambda_i - \lambda_j (i < j)$
	since the Lie algebra of $U$ is generated by $e_{ij} (i < j)$. 
	The simple roots are 
	$$
	(\lambda_1 - \lambda_2, \lambda_2 - \lambda_3,\cdots,
	\lambda_{n-1} - \lambda_n) \text{ (the root system of type } A_{n-1} )
	$$
	The Weyl group is generated by $s_\alpha$, where $\alpha$ is a positive root; since for $\alpha = \lambda_i - \lambda_j$, $s_\alpha$ permutes the $i$ and $j$ axis, 
	${}_kW ={\mathfrak S}_n$, the group of permutations of the basis elements. The parabolic subgroups are the stability groups of flags.
	
	
	\
	
\end{frame}
\begin{frame}[shrink]
	\frametitle{Matrix description of the symplectic group, see \cite{An87}}
	Let $G = \GSp_{2n}$ be the algebraic subgroup of $\GL_{2{n}}$ defined
	by
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.1)
	$\ds
	G_A = \{ \ga \in \GL_{2{n}}(A) ~~\mid~~ {}^t\! \ga J_{n} \ga =
	\nu(\ga) J_{n}, ~~ \nu(\ga) \in A^{\times} \},
	$
	for any commutative ring $A$, where
	$\ds
	J_{n} = \left( \begin{array}{cc}
	0_{n}  &  -1_{n}  \\  1_{n}  &  0_{n}
	\end{array} \right).
	$	
	The elements of $G_A$ are characterized by the conditions
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.2
	$\ds
	b {}^t\! a - a {}^t\! b = d {}^t\! c - c {}^t\! d = 0_{n}, 
	d {}^t\! a - c {}^t\! b = 1_{n},
	$
	and if
	$\ds
	\ga = \mabcd\in G_A \mbox{ then } \ga^{-1} = \nu(\ga)^{-1}
	\left( \begin{array}{cc} {}^t\! d  &  -{}^t\! b  \\  -{}^t\! c  &  {}^t\! a
	\end{array} \right).
	$\\
	\pause
	The multiplier $\nu$ defines a homomorphism $\nu : G_A \rightarrow A^{\times}$
	so that $\nu(\ga)^{2{n}} = \det(\ga)^2$ and $\ker(\nu)$ is denoted by
	$\SP_{n}(A)$.		
	
	\hskip-10pt{\large\blue Matrix description of the symplectic Lie algebra ${\mathfrak sp}_{2n}$}
 $\mathfrak {sp}_{2n}=\left\{X\in \mathfrak{gl}_{2n}| {}^tXJ_n+J_nX=0        \right\}$. Writing elements of ${\mathfrak g}=\mathfrak {sp}_{2n}$ in block form:$ X =\mat A,B,C, -{}^tA,$, 
where $A, B, C \in \M_{n\times n}$ and $B={}^tB$,  $C={}^tC$.
Note that the dimension
of $\mathfrak {sp}_{2n}$ is $n(2n + 1)$.
\iffalse
The set of diagonal matrices ${\mathfrak h}$ in $\mathfrak {sp}_{2n}$
is an abelian subalgebra of ${\mathfrak g}$ with a basis 
$h_i = e_{i,i}- e_{n+i,n+i}, 1 \le i \le n$.
\fi
A maximal torus $S$
is %given by the set of following diagonal matrices:
$\begin{pmatrix}
s^{\lambda_1}&&&&&& \\
& s^{\lambda_2}&&&&& \\
&& \ddots &&&&\\
&&&s^{\lambda_n}&&&\\
%&&&&1&&&&&\\
%&&&&&\ddots&&&& \\
%&&&&&&1&&& \\
&&&& s^{-\lambda_n}&& \\
&&0&&&s^{-\lambda_{n-1}}& \\
&&&&&\ddots& \\
&&&&&&s^{-\lambda_{1}}	
\end{pmatrix}
$	


	\
	
\end{frame}	

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Automorphic complex $L$-functions on classical groups.}
		
	
	\begin{itemize}
		
		
		\item	\S 6.5. of \cite{MaPa} 
		Automorphic Forms and The Langlands Program
		
		\item	Automorphic L-Functions
		
		\item	Analytic properties of automorphic L-functions
		
		\item  Hecke algebras.
		
		\item   	Section IV of \cite{Bor79}
		The $L$-function of an automorphic representation.
		
	\end{itemize}  	
	
	\iffalse
	
	The Langlands Functoriality Principle
	
	
	Automorphic Forms and Langlands Conjectures 	
	
	\hskip-20pt {\magenta Automorphic forms and their weights. Complex analytic weight space. Motivic weights, introduction \cite{EHLS}}.
	
	6.2.6 The Weil Group and its Representations
	(cf. [Ta79], [Wei74a]). 
	
	6.2.7 Zeta Functions, L-Functions and Motives
	(cf. [Man68], [Del79]).
	\cite{MaPa}	
	
	Algebraic characters 
	\fi	

\end{frame}	
\begin{frame}[shrink]
	\frametitle{}
	\
	
\end{frame}	
%\end{document}
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2

%\end{document}
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\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2

%\end{document}

\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Input 6.5.1, 6.5.2


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Lecture \Numero 2. 
	Classical and Hermitian modular forms. Automorphic complex $L$-functions on classical groups. Hecke algebras. The Rankin-Selberg method}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]\frametitle{\ncours Classical and Hermitian modular forms}

\begin{itemize}
	
	\item	{Classical modular forms: the case of $\GL_2$}
	
	\item{The Siegel modular case: $G=\GSp_{n}$ }
	
\item	{Geometric algebra (see [Dieudonné], [Garrett])}
	
\item	{Sesquilinear formes, Hermitian and antihermitian forms}
	
\item	Automorphic complex $L$-functions on classical groups. 
	
\item Hermitian modular forms and $L$ functions.
	
\item	Hecke algebras. 

\item The Rankin-Selberg method. 
\end{itemize}



	
\iffalse
	(Formes  modulaires hermitiennes. 
	Fonctions $L$ complexes sur les groupes classiques. Algèbres de Hecke. Methode de Rankin-Selberg).
\fi	
	
		\
		
	\end{frame}	
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\frametitle{Why study $L$-values attached to modular forms?}
\vskip-.3cm 
{{} A popular proceedure in Number Theory is the following: }

\hskip-1.5cm
\begin{tabular}{cccc}
	%%%%%%%%%%%%%%%%%%%%%%%
	\begin{tabular}{l}
		Construct a generating \\ function
		$f=\sum_{n=0}^\infty a_n q^n$ \\
		$\in {\mathbb C}[[ q]]$
		of an arithmetical \\
		function
		$n\mapsto a_n$,
		\\
		for example $a_n=p(n)$
	\end{tabular}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	 {}
	&$\rightsquigarrow$
	%\fbox{
	\begin{tabular}{l}
		Compute $f$
		via \\
		modular forms,
		\\
		for example
		\\
		$\ds\sum_{n=0}^\infty p(n) q^n$
		\\
		$=(\Delta/q)^{-1/24}$
	\end{tabular}
	%}
	&
	{}
	$\rightsquigarrow$
	\begin{tabular}{l}
		A number
		\\
		(solution)\ \ \ \
	\end{tabular}
	%{}
	\\
	{}
	\begin{tabular}{l}
		Example 1 \rm \cite{Chand70}:
		\\
		(Hardy-Ramanujan)
	\end{tabular}
	&
	{}
	$\uparrow$
	&{}
	$\uparrow$
	%\end{tabular}
	%%\end{frame}
	%\end{document}
	\\
	{}
	\tiny
	\begin{tabular}{l}
		$\ds
		p(n)=\frac{e^{\pi \sqrt{2/3({n-1/24})}}} {4\sqrt{3}\lambda_n^2}$
		\\  $+O(e^{\pi \sqrt{2/3({n-1/24})}}/
		\lambda_n^3),
		$
		%\end{tabular} & &
		%\end{tabular}
		%%\end{frame}
		%\end{document}
		\\
		$\lambda_n=\sqrt{n-1/24}$, %\quad K=\pi \sqrt{2/3}
	\end{tabular}
	%%%%%%%%%%%%%%%%%%%%%%%%%%
	&
	{}
	%%%%%%%%%%%%%%%%%%%
	\begin{tabular}{l}
		Good bases,  \\
		finite dimensions,  \\
		many relations \\ and identities
	\end{tabular}
	&
	{}
	\begin{tabular}{l}
		Values \\
		of $L$-functions, \\
		periods, \\
		congruences, \dots
	\end{tabular}
\end{tabular}
\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{}
{{} Other examples:
	Birch and Swinnerton-Dyer conjecture, \dots
	$L$-values attached to modular forms
	% (1994),
	%sm. knigu %Manin-Panchishkin,
}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Modular forms,  
		zeta functions, $L$-functions}% (complex and $p$-adic)}
	%{\large \red Examples}
	%\begin{itemize} 
	%	\item
	{\Large Eisenstein series}
	$\displaystyle E_k=1+\frac 2{\zeta(1-k)}\sum_{n= 1}^\infty\sum_{d|n}d^{k-1}q^n \in \Mc_k$, a modular forms for even weight $k\ge 4$ for $\SL_2(\Z), q=e^{2\pi i z}$), and $E_2\in \Qc\Mc $ {\magenta a quasimodular form}. 
	The ring of quasimodular forms, closed under differential operator ${\displaystyle D=q\frac{d}{dq}
		=\frac{1}{2\pi i}\frac{d}{dz}}$, used  in arithmetic, $\zeta(s)$ is the Riemann zeta function, $\displaystyle \zeta(-1)=-\frac 1{12}$, $E_2=1-24\sum_{n= 1}^\infty\sum_{d|n} d q^n$ is also  a {\magenta $p$-adic modular form} (due to J.-P.Serre,\cite{Se73}, p.211)
	%\item 
	
	\
	
	{\Large		Elliptic curves} $E: y^2=x^3+ax+b$ , $a,b\in \Z$,  {\red A.Wiles's modular forms} $\displaystyle f_E=\sum_{n=1}^\infty a_nq^n$ with $a_p=p-Card E(\F_p)$ $(p\nd 4a^3+27b^2)$,
	and the $L$-function $L(E,s)\displaystyle =\sum_{n=1}^\infty a_nn^{-s}$.
	%\end{itemize}
	%belong traditionally to the world of arithmetic, but also to the worlds of geometry, algebra and analysis.
	
\end{frame}
%\fi	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  4

\begin{frame}[shrink]
	\frametitle{Zeta-functions or $L$-functions} 
	They are attached to various mathematical objects as certain Euler products.
	
	\begin{itemize}
		\item $L$-functions link such objects to each other (a general form of functoriality); 
		
		\pause
		\item Special $L$-values answer fundamental questions about these objects in the form of a 
		{\red number (complex or $p$-adic)}.
	\end{itemize}
	\pause
	Computing these numbers use integration theory
	of Dirichlet-Hecke characters along  $p$-adic and complex valued measures.
	
	
	\pause
	This approach originates in the {\red Dirichlet class number formula} using the $L$-values in order  to compute class numbers of algebraic number fields through Dirichlet's $L$-series $L(s, \chi)$: for an imaginary quadratic field $K$ 
	of discriminant $-D<-4$, $\chi_D(n)={-D \choose n}$
	$$
	h_D
	%= \sharp \text{class group} (\Q(\sqrt{-D})
	=\frac { \sqrt{D}L(1,\chi_D)}{2\pi}= L(0,\chi)= - \frac {1} {D}
	\sum_{a=1}^{D-1}\chi_D(a)a.
	$$
	%and the generalized Bernoulli numbers 
	%$B_{k,\chi}=N^{k-1} \sum_{a=0}^{N-1} \chi(a)N^{-1}B_k(a/N)$. 
	\pause
	(Example: disc$(\Q(\sqrt{-5})))=-20$, $h_{20}=2$; in PARI/GP $\chi_{20}(n)=$  kronecker(-20,n), 
	{\tt gp > -sum(x=1,19,x*kronecker(-20,x))/20
		\% 29 = 2 
	}
	
	\pause
	Another famous example: the Millenium
	{\red BSD Conjecture} gives the rank of an elliptic curve $E$ as the order of $L(E,s)$ at s=1
	(i.e. the residue of its logarithmic derivative, see \cite{MaPa}, Ch.6).  
	
	
	\
	
\end{frame}
\begin{frame}[shrink]
	\frametitle{A short story of critical values, see \cite{YS} } 	
	Euler  discovered  $\di\zeta(2) = \frac{\pi^2}{6}$,
	and $\displaystyle \frac{2\zeta(2n)}{(2\pi i)^{2n}}$ = $\displaystyle-\frac{B_{2n}}{(2n)!}\in \Q, (n\ge 1)$.  
	%in 1735-1742, the starting result about the zeta values.
	
	\pause
	These are examples of   {\magenta critical values} (in the sense of Deligne):  for a more general zeta function 
	$\Dr (s)$ the critical values are defined using	
	its gamma factor $\Gamma_\Dr(s)$ such that the product $\Gamma_\Dr(s)\Dr(s)$ satisfies a standard functional equation under the symmetry $s \mapsto v-s$.
	Then $\Dr(n)$, $n \in \Z$ is a critical value of $\Dr(s)$ if both $\Gamma_\Dr(n)$ and	$\Gamma_\Dr(v-n)$ are finite.
	
	\smallskip\pause
	{\magenta Hurwitz \cite{Hur1899} showed a striking analogy to Euler's theorem}: 
	%\begin{align}&\nonumber
	$$
	\frac{\sum'_{\al\in\Z[i]}\al^{-4m}}{\Omega^{4m}}=\frac{H_m}{(4m)!}\in\Q,   
	\Omega=2\int_0^1 \frac{dx}{\sqrt{1-x^4}}
	%\\ & \nonumber= 
	=2.6220575542\cdots
	$$
	%\end{align}
	for $1 \le m \in \Z$, where $\al = a + ib$, $a, b \in \Z$ 
	%extends to all 
	are non-zero Gaussian integers and $H_m$ are Hurwitz numbers (recursively computed, \cite{Sl}): 
	$\di H_1, H_2, \cdots= \frac{1}{10}, \frac{3}{10}, \frac{567}{130}, \frac{43659}{170}, \frac{392931}{10},\cdots$.	
	Recall the formula: 
	
	\pause
	Let $\wp$ be the Weierstrass $\wp$-function satisfying $\wp'^2 = 4\wp^3 - 4\wp$. Then
	$\di\wp(z)
	= \frac 1{z^2}+ \sum_{n=1}^\infty\frac {2^{4n}H_nz^{4n-2}}{4n(4n-2)!}$.
	%$$ \sum_{ (r, s) \ne (0, 0) } 1/(r+si)^{4n} = (2w)^{4n}H_n/(4n)!
	%$$ where 
	%$w = 2 \int_0^1 dx/\sqrt{1-x^4}$.
	\pause
	{\red A rapid computation of these values}: take
	the Fourier expansion of the Eisenstein series at $z=i$, $q=e^{-2\pi}$:   %=0.0018674427: $ 
	$\di G_{4m}(z)=\sum^{}_{a,b}{{}'}(az+b)^{-4m}= 2\zeta(4m)+
	\frac{2(2\pi)^{4m}}{(4m-1)!}
	\sum_{d\ge 1}\frac{d^{4m-1}q^d}{(1-q^d)}$, 
	\pause
	{\red $\displaystyle\frac{G_{4m}(i)}{\Omega^{4m}}=\displaystyle \frac{H_m }{(4m)!}$}, $\pi,\Omega$ --  {\magenta periods} of $\zeta(s)$ and of
	$E:y^2=4x^3-4x$.
	%	$\zeta(s)$ and  %$\di\Dr(s)=
	%	$\sum_{\al}\bar\al^{m}N(\al)^{-s}$).
	
	\
	
	
	%\smallskip\pause	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\fi  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  6





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink=3]
	\frametitle{Classical modular forms: definition;}
		Let $\Gamma$ be a
		subgroup of finite index in the modular group 
		$\SL_2(\Z)$.
	A holomorphic function $f : {{\mathbb H}} \to \C$
	is called a
	modular form\index{Modular form}
	of (integral) weight $k$ with respect to $\Ga$
	iff the conditions a) and b) are satisfied:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.84)
	%\medskip
	\begin{itemize}
		\item{a)} {\it Automorphy condition}
		\begin{align}\label{ii43.2}
		f((a_\ga z + b_\ga)/(c_\ga z + d_\ga))
		=
		(c_\ga z + d_\ga)^k f(z)
		% \eqno(3.2)
		\end{align}
		for all elements $\ga \in \Ga$;
		
		\item{b)} {\it Regularity at cusps}:
		$f$ is regular at
		cusps\index{Cusp}
		$z \in \Q \cup i\infty$
		(the cusps can be viewed as fixed points of parabolic elements of 
		$\Ga$);
		this means that for each element $\si = \mat a, b, c, d, \in 
		\SL_2(\Z)$
		the function $\ds (cz + d)^{-k}f\left(\frac {az + b} {cz + d} \right )$
		admits a Fourier expansion
		over non--negative powers of $q^{1/N} = e(z/N)$
		for a natural number $N$.
		One writes traditionally 
		$$
		q = e(z) = \exp (2\pi i z).
		$$
		A modular form
		$ \ds
		f(z) = \sum _{n=0}^\infty a(n) e(nz/N)
		$
		is called a
		cusp form\index{Cusp!-- form}
		if $f$ vanishes at all cusps
		(i.e. if the above Fourier expansion
		contains only positive powers of $q^{1/N}$),
		see 
		\cite{LangMF}, 
		\cite{MaPa}
	\end{itemize}
	
	
\	
	
	
\end{frame}

\begin{frame}[shrink=3]
\frametitle{The complex vector space of all modular (resp. cusp) forms}
	of weight $k$ with respect to $\Ga$
	is denoted by $\Mc_k(\Ga)$ (resp.  $\Sc_k(\Ga)$).
	
	A basic fact from the theory of modular forms
	is that the spaces of modular forms are finite dimensional.
	Also, one has $\Mc_k(\Ga)\Mc_l(\Ga) \subset \Mc_{k + l}(\Ga)$.
	The direct sum
	$$
	\Mc (\Ga)
	=
	\bigoplus_{k=0}^\infty \Mc _k(\Ga)
	$$
	turns out to be a graded algebra over $\C$
	with a finite number of generators.
	
	An example of a modular form with respect to $\SL_2(\Z)$ of weight $k 
	\ge 4$
	is given by the
	{\it Eisenstein series}\index{Eisenstein series}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.85)
	\begin{align}\label{ii43.3}
	G_k(z)
	=
	\sum_{m_1, m_2 \in \Z}\!\!\!\!{}^{\displaystyle {}'} 
	(m_1+ m_2 z)^{-k}
	% \eqno(3.3)
	\end{align}
	(prime denoting
	$(m_1, m_2) \not= (0, 0)$).
	For these series the automorphy condition (\ref{ii43.2})
	can be deduced straight from the definition.
	One has $G_k(z) \equiv 0$ for odd $k$ and 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    (6.86)
	\begin{align}\label{ii43.4}
	G_k(z)
	=
	\frac{ 2(2\pi i)^k}{ (k-1)!}
	\left[ - \frac {B_k} {2k}  + \sum_{n=1}^\infty \si_{k-1}(n) e(nz) 
	\right ],
	% \eqno(3.4)
	\end{align}
	where $\si_{k-1} (n) = \sum_{d|n}d^{k-1}$
	and $B_k$ is the $k^{\rm th}$
	{\it Bernoulli number}.\index{Bernoulli number}
	
	The graded algebra $\Mc (\SL_2(\Z))$
	is isomorphic to the polynomial ring
	of the (independent) variables $G_4$ and $G_6$.
	
	\
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile=singleslide,shrink=3]
	\frametitle{Examples} 
	%Eisenstein series and Congruences for the Ramanujan  function
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	Recall that $B_k$ denote the Bernoullli numbers defined by the development
	$$
	\frac x {e^x-1} =\sum_{k=0}^\infty B_k\frac {x^k} {k!}, 
	$$
	One has for even $k\ge 2$,  
	$\ds
	2\z(k)=-\frac {(2\pi i)^{k} B_k} {k!}, \ 
	G_k(z)= \frac {2 (2\pi i)^k} {(k-1)!} 
	\left[ -\frac {B_k} {2 k} + \sum_{n=1}^\infty
	\si_{k-1}(n)q^n\right], \
		E_k(z) =  1 -\frac  {2 k}{B_k} \sum_{n=1}^\infty
		\si_{k-1}(n)q^n
	$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{align*}
	&
	E_4(z)=1+240\sum_{n=1}^\infty\si_{3}(n)q^n \in \Mc_4(\SL(2, \Z)),\cr &
	E_6(z)=1-504\sum_{n=1}^\infty\si_{5}(n)q^n \in \Mc_6(\SL(2, \Z)),\cr &
	E_8(z)=1+480\sum_{n=1}^\infty\si_{7}(n)q^n\in \Mc_8(\SL(2, \Z)),\cr &
	E_{10}(z)=1-264\sum_{n=1}^\infty\si_{9}(n)q^n\in \Mc_{10}(\SL(2, \Z)),\cr &
	E_{12}(z)=1+\frac {65520} {691} \sum_{n=1}^\infty\si_{11}(n)q^n
	\in \Mc_{12}(\SL(2, \Z)),\cr &
	E_{14}(z)=1 - 24\sum_{n=1}^\infty\si_{13}(n)q^n\in \Mc_{14}(\SL(2, \Z)).
	(\text{Proof  see in  }\cite{Se70}).
	\end{align*}
	\begin{verbatim}with PARI/GP:
	gp > k=14;Ek=1-(2*k)/bernfrac(k)*sum(d=1,20,d^(k-1)*q^d/(1-q^d)+O(q^4>
	% = 1 - 24*q - 196632*q^2 - 38263776*q^3 + O(q^4)
	\end{verbatim}
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile=singleslide,shrink=20]
	\frametitle{Fast computation of the Ramanujan function $\tau(n)$}
	Put
	$ \ds
	h_k:=\sum_{n=1}^\infty \sum_{d|n}d^{k-1}q^n
	=\sum_{d=1}^\infty \frac {d^{k-1}q^d}{1-q^d}.
	$ 
	The classical fact is that 
	$\ds\De=\frac{(E_4^3-E_6^2)}{1728}= q\prod_{n\ge 1}(1-q^n)^{24}= \sum_{n\ge 1}\tau(n)
	q^n$,  
	($E_4= 1+240h_4$,
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	$ E_6= 1 - 504h_6$). 
	%\end{frame}
	%\end{document} 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\begin{frame}[fragile=singleslide,shrink=20]
	%\frametitle
	
	\
	
	{\large\blue Computing with PARI-GP} 
	see
	\rm \cite{BBBCO}, 
	The PARI/GP number theory system),
	{\tt http://pari.math.u-bordeaux.fr}
	$ \ds
	h_k:=\sum_{n=1}^\infty \sum_{d|n}d^{k-1}q^n
	=\sum_{d=1}^\infty \frac {d^{k-1}q^d}{1-q^d}
	\Longrightarrow
	$ 
\begin{verbatim}
	gp > h6=sum(d=1,20,d^5*q^d/(1-q^d)+O(q^20))
	gp > h4=sum(d=1,20,d^3*q^d/(1-q^d)+O(q^20)
	gp > Delta=((1+240*h4)^3-(1-504*h6)^2)/1728
\end{verbatim}


{\large \magenta Congruence of Ramanujan}
	$\ds
	\tau (n) \equiv \sum_{d|n}d^{11} \ \bmod \ 691:
	$
	
	\small
	\begin{verbatim}
	gp > (Delta-h12)/691
	% = -3*q^2 - 256*q^3 - 6075*q^4 - 70656*q^5 - 525300*q^6 + O(q^7)
	\end{verbatim}
	
	{\large \blue More programs of computing  $\tau(n)$ (see \cite{Sl})}
	\begin{verbatim}
	(MAGMA) M12:=ModularForms(Gamma0(1), 12); t1:=Basis(M12)[2]; 
	PowerSeries(t1[1], 100); Coefficients($1);	
	(PARI) a(n)=if(n<1, 0, polcoeff(x*eta(x+x*O(x^n))^24, n))	
	(PARI) {tau(n)=if(n<1, 0, polcoeff(x*(sum(i=1, (sqrtint(8*n-7)+1)\2,
	(-1)^i*(2*i-1)*x^((i^2-i)/2), O(x^n)))^8, n));} 
	gp > tau(6911)
	%3 = -615012709514736031488
	\end{verbatim}
	%\end{tabular}
	
	
	
	\end{frame}	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
	
	\begin{frame}[shrink]\frametitle{Multiplicativity and Hecke theory, \cite{An87}}
	
	The example of the Eisenstein series, of the Ramanujan function $\tau(n)$ 
	\iffalse
	as well as the numerous
	examples of theta-series of positive definite integral quadratic forms in an
	even number of variables, which were analysed by the number theorists of
	the classical period, 
	\fi
	demonstrate an interesting phenomenon: often the Fourier coefficients $a(n)$ of modular forms for $\SL_2(\Z)$ are multiplicative number-theoretical functions or linear combinations of such functions. The reason for this is to be found in Hecke's theory as follows. Let
	$ \ds
	f(z) = \sum _{n=0}^\infty a(n) e(nz)\in \Mc_k 
	$
The "multiplicativity" of the function $a(n)$ should mean
that there is a regular connection between $a(n$) and $a(nm)$ for any fixed $m$.
The numbers $a(nm)= (n = 1, 2, \dots )$ are the Fourier coefficients of the
function\\	
$ \ds
f_m(z) = \sum _{n=0}^\infty a(nm) e(nz)= \frac 1 m \sum_{b=0}^{m-1}
 f\left(\frac{z+b} m \right)=m^{k-1}\sum_{b=0}^{m-1}f|_k\mat 1,b,0,m, 
$ where  $\ds f|_k M = (cz + d)^{-k} f \left(\frac{az+b} {cz+d} \right).$  	
	
	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%		
\begin{frame}[shrink]\frametitle{Defining Hecke operators through $U(m)$, $V(m)$}	
		If the operator $f\to f_m : = f|U(m)$ were to carry the space $\Mc_k$ into itself, then one
could hope to find its eigenfunctions in $\Mc_k$ but for such functions
 $f_m = \lambda_mf$, that is, $ a(nm) = \lambda_ma(n) (n = 1, 2, \dots )$, and the desired multiplicativity property would follow.
 
 To develop this idea let us use for any positive integer  $m$ the operators $f\mapsto f|U(m)$, $f\mapsto f|V(m)$: 	
$ \magenta\ds
f|U(m) (z)= \sum _{n=0}^\infty a(mn) e(nz)=m^{k-1}\sum_{b=0}^{m-1}f|_k
\mat 1,b,0,m, $
\\
$ \magenta\ds
f|V(m) (z)= \sum _{n=0}^\infty a(n) e(mnz)=m^{k-1}f|_k\mat m,0,0,1,$ \\	
In the general case note that the matrices
$\mat 1,b,0,m, $
in the definition 
of $U(m)$ form part of a complete system of right coset representatives for
$\ds SL_2(\Z)\bs \Delta_m\left\{ \ga = \mat a,b,0,d,|  ad=m, ,b=0\dots, d-1  \right\}$, 
where $\Delta_m$ denotes the set
$\ds \Delta_m= \left\{ \ga = \mat a,b,c,d,|  \det \ga=m  \right\}$
 which is invariant under right multiplication by elements of $\SL_2(\Z)$. The action of this coset representatives produce the Hecke operator 			
$\magenta \ds f|T(m)=\sum_{a|m}a^{k-1}f|U(m/a)V(a)$.			
			
\			
			
\end{frame}	
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%		
\begin{frame}[shrink]\frametitle{The Siegel modular case: $G=\GSp_{n}$ }
Let $\HH_{n}$ denote the Siegel upper half plane of genus ${n}$,
$\ds
\HH_{n} = \{ z \in \M_{n}(\CC) ~~\mid~~ {}^t\! z = z = x+iy, ~~ y>0 \},
$
so that $\HH_{n}$ is a complex analytic variety whose dimension is denoted by
$\langle {n} \rangle = {n}({n}+1)/2$.	\\	
Put
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.3)
$\ds
G_{\infty} = G_{\RR}$, $G_{\infty}^+$ =$ \{ \ga \in G_{\infty} \mid \nu(\ga) > 0 \},$ $G_{\QQ}^+ = G_{\infty}^+ \cap G_{\QQ}.
$
\\ \smallskip
The group $G_{\infty}^+$ acts transitively on  the Siegel upper half plane $\HH_{n}$ by
$ \ds
z \longmapsto \ga(z) = (az+b)(cz+d)^{-1}, z \in \HH_{n}$, $\ga = \mabcd \in G_{\infty}^+.
$
so that the scalar matrices act trivially, and $\HH_{n}$ can be identified with
a homogeneous space of the group $\SP_{n}(\RR)$.
Let $K_{n}$ denote the stabilizer of the point $i1_{n} \in \HH_{n}$ in
the group $\SP_{n}(\RR)$,
$\ds K_{n} = \{ \ga \in \SP_{n}(\RR)\mid\ga(i1_{n}) = i1_{n} \},  $
then there is a bijection $\SP_{n}(\RR)/K_{n} \simeq \HH_{n}$ and
$K_{n} = \SP_{n}(\RR) \cap \SO_{2{n}}$.
The group $K_{n}$ is a maximal compact subgroup
of the Lie group $\SP_{n}(\RR)$ and it can be identified with the group $U({n})$
of all unitary ${n} \times {n}$ -matrices via the map $\ga = \mabcd
\longmapsto a+ib$.		
		
\end{frame}	
%\end{document}		
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]
\frametitle{The Siegel modular group ${\Gamma}^{n} = \SP_{n}(\ZZ)$} 
Let ${\Gamma} \subset G_{\QQ}^+$ be its arbitrary congruence subgroup.
%\end{frame}	
%\end{document}
The general definition of modular forms is given for a rational representation
$\rho : \GL_{n}(\CC) \rightarrow \GL_r(\CC)$ a rational representation also denoted by $\rho$. For $\ga = \mabcd \in G_{\infty}^+$ and for
any complex valued function $f : \HH_{n} \rightarrow \CC^r$ we use the notation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.5)
$  \ds
f|_{\rho} \ga(z) = \rho(cz+d)^{-1} f(\ga(z)).
$
 
\
%\end{frame}	
%\end{document}
\hskip-10pt{\blue\large Definition}
	A function $f : \HH_{n} \rightarrow \CC^r$ is called a holomorphic modular
	form of weight $\rho$ on ${\Gamma}$ if the following conditions
 are satisfied:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.6)
	\begin{align*}
	&    f|_{\rho} = f,   \\
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.7)
	&    f \mbox{ is holomorphic on } \HH_{n},  \\
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.8)
	&    \mbox{if } {n} =1 \mbox{ then } f \mbox{ is holomorphic at cusps
		of  } {\Gamma}.  
	\end{align*}

Let $\Mc_{\rho}({\Gamma})$ be the complex vector space of functions satisfying
the above conditions. For each $f \in \Mc_{\rho}({\Gamma})$ there is the
following Fourier expansion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.9)
$ \ds
f(z) = \sum_{\xi} c(\xi) e_{n}(\xi z),  
$
where $c(\xi) \in \CC^r$, $\xi$ run over all $\xi = {}^t\! \xi \in \M_{n}(\QQ)$,
$\xi \ge 0$ (for ${n} > 1$ the last condition automatically follows by the
Koecher principle). 

\

\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]\frametitle{The Fourier expansions of Siegel modular forms}
Let $M$ be the smallest integer such
that
$ {\Gamma} \supset \left\{ \left( \begin{array}{cc}  1_m  &  Mu  \\
0_m  &  1_m
\end{array}  \right)
\mid u \in \M_m(\ZZ), {}^t\! u = u \right\}  
$
and we put
$$ \begin{array}{rcccl}
A  &  =  &  A_m  &  =  &  \left\{ \xi = (\xi_{ij}) \in \M_m(\RR)
~~\mid~~ \xi = {}^t\!\xi, \xi_{ij}, 2\xi_{ii} \in \ZZ \right\},  \\
B  &  =  &  B_m  &  =  &  \left\{ \xi \in A ~~\mid~~ \xi \geq 0
\right\},  \\
C  &  =  &  C_m  &  =  &  \left\{ \xi \in A ~~\mid~~ \xi > 0 \right\}.
\end{array}  $$

Then $A_m$ is a lattice of half-integral matrices in the $\RR$-vector
space of symmetric matrices
$V = \{ x \in \M_m(\RR) ~~\mid~~ {}^t\! x = x \}$ dual to the lattice
$L = \M_m(\ZZ) \cap V$ with respect to the action
$(\xi ,x) \longmapsto e_m(\xi x)$ and for each $f \in \Mc_{\rho} ({\Gamma})$
there is the following Fourier expansion
\begin{align}\label{EQdevFoufrhosig}
f(z) = \sum_{\xi \in M^{-1}B} c(\xi ) e_m(\xi z),
\end{align}
Moreover, for each $\ga \in G_{\QQ}^+$ we have that
$f|_{\rho} \ga \in \Mc_{\rho} ({\Gamma}(\ga))$, where ${\Gamma}(\ga)$
is a congruence subgroup,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.10)
\begin{align*}  \label{EQdevFoufrhosig}
(f|_{\rho} \ga)(z) = \sum_{\xi \in M_{\ga}^{-1}B}
c_{\ga}(\xi) e_m(\xi z),
\end{align*}
with $c_{\ga}(\xi ) \in \CC^r, M_{\ga} \in \NN$. A form $f$ is called
a cusp form if for all $\xi$ with $\det(\xi) = 0$ in expansion
(\ref{EQdevFoufrhosig}) one has $c_{\ga}(\xi ) = 0$ for all
$\ga \in G_{\QQ}^+$ that is
$$ (f|_{\rho} \ga)(z) = \sum_{\xi \in M_{\ga}^{-1} C}
c_{\ga}(\xi) e_m(\xi z).  $$
We denote by $\Sc_{\rho}({\Gamma}) \subset \Mc_{\rho}({\Gamma})$ the subspace
of cusp forms. 
 
 
 \
 
 
 
 \end{frame}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
 
 \begin{frame}[shrink]\frametitle{The Petersson scalar product}
 	For $f \in \Sr_{n}^k(N,\psi)$ and $h \in \Mc_{n}^k(N,\psi)$ the Petersson
 	scalar product is defined by
 	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.14)
 	\begin{equation}  \label{EQpetersprod}
 	\langle f,h \rangle_N = \int_{\Phi_0(N)} \overline{f(z)} h(z)
 	\det(y)^k \dd^{\times}z,
 	\end{equation}
 	where $\Phi_0(N) = {\Gamma}_0^{n}(N) \backslash \HH_{n}$ is a fundamental domain
 	for the group ${\Gamma}_0^{n}(N)$ with
 	 the notations
 	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.4)
 $  \di{ \dd x = \prod_{i \leq j} \dd x_{ij},   ~~~~ \dd y =
 		\prod_{i \leq j}y_{ij}, ~~~~ \dd z = \dd x \dd y }$,  \\
 	$  \di{ \dd^{\times}y = \det(y)^{-\kp}\dd y, ~~~~ \dd^{\times}z =
 		\det(y)^{-\kp} \dd z }, \nonumber
$ 
 	where $z = x+iy$, $x = (x_{ij}) = {}^t\! x$, $y = (y_{ij}) = {}^t\! y > 0$, $\ds \kp=\frac{n(n+1)}{2}$. Then
 	$\dd^{\times}z$ is a differential on $\HH_{n}$ invariant under the action of the
 	group $G_{\infty}^+$, and the measure $\dd^{\times}y$ is invariant under the
 	action of elements $a \in \GL_{n}(\RR)$ on
 	$
 	Y = \{ y \in \M_{n}(\RR) ~~\mid~~ {}^t\! y = y > 0 \}
 	$
 	defined by the rule $y \longmapsto {}^t\! aya$.
 	
 	
 	
 	
 \
 
 
 
 \end{frame}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
  
\begin{frame}[shrink]\frametitle{Sesquilinear formes, Hermitian and antihermitian forms}
 
Let $K$ be a quadratic extension of $k$, its subfield fixed be the involution $\si$.
 
%Définition 13.2.1
\begin{defi}
a) A $k$-bilinear form 
$f : V \times V \to K, f(v_1, v_2) = \langle v_1, v_2 \rangle $ on a $K$-vectorspace $V$ of finite dimension  is said sesquilinear (with an implicite  reference to $\si$) if
$\langle \al v_1, \beta v_2 \rangle= \al^\si \beta \langle v_1, v_2 \rangle$
($\forall \al, \beta \in K$ and  $v_1, v_2 \in V$ ).

b) A sesquilinear form $f(v_1, v_2) = \langle v_1, v_2\rangle$ on a $K$-vector space $V$ of finite dimension
is said hermitian if  $\forall v_1, v_2 \in V, \langle v_2, v_1\rangle=\langle v_1, v_2\rangle^\si$. 
 \end{defi}
  
 The  {\magenta unitary group} $U(f)$ is the  group  of isometries  $f$ (or of $\langle \cdot, \cdot \rangle $), defined as
 \begin{align*}
 &
 U(f) = U(\langle \cdot, \cdot \rangle) = \{g \in GL_K(V ) \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \langle v_1, v_2 \rangle\},  \\
 &
 \text{ and the {\magenta group of unitary similitudes} is } GU(f) = GU(\langle \cdot, \cdot \rangle) \\ &
 = \{g \in GL_K(V ), \exists \nu(g)\in K^* \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \nu(g)\langle v_1, v_2 \rangle\}
 \end{align*} 
  
  
  
 \
 
 \end{frame}	
 %\end{document}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
 \begin{frame}[shrink]
\frametitle{Simultaneuos treatment of general isometries groups} 
    Over a division algebra $D$ with an anti-involution $\si$.  
 	Note that $\si: D\to D$ 
 	satisfies the properties
$$
\forall \al, \beta\in D, \al^{\si \si} = \al,\  (\al + \beta)^\si=\al^\si +\beta^\si \text{ and } (\al\beta)^\si = \beta^\si\al^\si
$$  
Let $Z$ be the center of $D$. Suppose that $D$ is of finite dimension over
$Z$, and that
$k = \{x \in Z | x^\si = x\}.$
Let $V$ be a $D$-vector space of finite dimension, and fix $\ep = \pm 1$ . 
Let
$f = \langle\cdot,\cdot\rangle, f : V\times V \to D$
a $k$-bilinear form with values in $D$ on $V$  such that
$
\forall \al,\beta\in D$, $\forall v_1,v_2\in V, \langle v_2,v_1\rangle = \ep \langle v_1,v_2\rangle^\si, \  
\langle \al v_1,\beta v_2\rangle = \al^\si\langle v_1,v_2\rangle\beta.
$
Such a form is said $\ep$-hermitian on $V$, and such space $V$ (endowed
with $\langle \cdot, \cdot\rangle$) is called a $(D, \si, \ep)$-space.

The  {\magenta group  of isometries} $U(f)$  of  $f$ (or of $\langle \cdot, \cdot \rangle $), is defined as
\begin{align*}
&
U(f) = U(\langle \cdot, \cdot \rangle) = \{g \in GL_D(V ) \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \langle v_1, v_2 \rangle\},  \\
&
\text{ and the {\magenta group of isometry similitudes} is } GU(f) = GU(\langle \cdot, \cdot \rangle) \\ &
= \{g \in GL_D (V ), \exists \nu(g)\in k^* \ |\  \forall v_1, v_2 \in  V, \langle gv_1, gv_2 \rangle = \nu(g)\langle v_1, v_2 \rangle\}
\end{align*}


 
 \
 
\end{frame}	
%\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{Orthogonalisation and  isotropy vectors}
	A $D$ - vector subspace $U$  in a $(D, \ep, \si)$ - vector space admits an  orthogonal complement
	$U^\perp = \{u'\in V \langle u', u\rangle = 0, \forall u \in U\}.$
	Note that $U \cap U^\perp = 0$ is not valid in general. 
	The kernel $V$ is denoted $V^\perp$. The form is called non degenerate if $V^\perp = 0$. Suppose for simplicity that the space $V$ is non-degenerate.
	
If $V_1, V_2$ are two  $(D, \ep, \si)$ - vector spaces endowed with forms, respectively, $\langle \cdot, \cdot\rangle_1$, $\langle \cdot, \cdot\rangle_2$, then
	the direct sum $V_1\oplus V_2$ of $D$- vector spaces  is a 
	$(D, \ep, \si)$ - vector space with the form
$$
	\langle  v_1 + v_2, v'_1 + v_2'\rangle=
	\langle  v_1, v'_1 \rangle_1
	\langle   v_2, v_2'\rangle_2
$$	
called the orthogonal sum. In general, two subspaces $V_1, V_2$ of a $(D, \ep, \si)$ - vector space $V$ are orthogonal if
$V_1 \subset V_2^\perp$
or equivalently, if $V_2 \subset V_1^\perp$.

If $\langle  v, v \rangle = 0$ for $v \in V$, then  $v$ is called an isotropic vecteur. If $\langle  v, v' \rangle = 0$  for all 
$v, v' \in U$ for a subspace  $U$ of $V$ then $U$ is a (totally) isotropic.
If there is no  isotropic  non zero vector in $U$, then $U$ is said {\magenta anisotropic}.

%Proposition 13.2.3
\begin{prop}
Let $V$ be a  $(D, \ep, \si)$- non degenerate vector space with a subspace $U$.
Then $U$ is  non degenerate iff $V = U \oplus U^\perp$, with $U^\perp$
non degenerate.
\end{prop} 





\

\end{frame}	
%\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{Orthogonalisation in $(D,\sigma, \epsilon)$-spaces}
This is used for classification of  orthogonal and hermitians spaces
%Proposition 13.2.4 
\begin{prop}
Let $V$ be a  $(D, \ep, \si)$- non degenerate vector space. Suppose that the case where $\ep = -1$, $D = k$,
and $\si$ trivial is excluded. If the product $\langle  \cdot,\cdot \rangle$ does not vanish identically then there exists $v \in V$
with  $\langle  v, v \rangle\ne 0$. If $V$ is non degenerate then it has an orthogonal basis.
\end{prop}

{\blue Proof.} Suppose that $\langle  v, v \rangle= 0$ for all $v \in V$. 
Then 
$$
0 = \langle  x + y, x + y \rangle  = \langle  x , x  \rangle +
\langle  y, y \rangle +\langle  x , y \rangle +\ep \langle  x , y \rangle^\si.
$$
%\end{frame}
%\end{document}
If $\ep = 1$ 
and the product $\langle  x + y, x + y \rangle$ does not vanish identically, then there exist $x, y$ such that  $\langle  x , y \rangle = 1$.
%Then $0 =\langle  x , y \rangle + \ep\langle  x , y \rangle\si = 1+1$.
Contradiction. 
%\end{frame}
%\end{document}
Suppose that  $\ep =-1$ and $\si$ non trivial on $D$. 
Then there exists $\al\in D$ such that $\al\ne \al^\si$, with
$\w  = \al -\al^\si$, $\w  =-\w^\si$. 
%\end{frame}
%\end{document}
If $\langle  x , y \rangle$  does not vanish identically then there existe $x, y$ such that $\langle  x , y \rangle = 1$. Then one has
$0 =\langle  \w x , y \rangle + \ep \langle  \w x , y \rangle^\si =\w^\si \langle  x , y \rangle + \langle  x , y \rangle^\si\w =-\w +\ep\w = -2\w,$
Contradiction.~\qed

In order to construct an orthogonal basis, one uses induction on dimension. If the dimension of a non degenerate vector space $V$ is 1, then any non-zero forme admits an orthogonal basis. In general, one finds $v \in V$ such that  $\langle  v, v \rangle\ne 0$. Then $Dv^\perp$  is non degenerate and $V$ is the orthogonal direct sum of $ Dv$ and$Dv^\perp$, by the previous proposition.


\iffalse	
		Over a division ring		
		This section applies unchanged when the division ring K is commutative. More specific terminology then also applies: the division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions.
		Definition
		
		A ?-sesquilinear form over a right K-module M is a bi-additive map ? : M × M ? K with an associated anti-automorphism ? of a division ring K such that, for all x, y ? M and all ?, ? ? K,
		
		? ( x ? , y ? ) = ? ( ? ) ? ( x , y ) ? . {\displaystyle \varphi (x\alpha ,y\beta )=\sigma (\alpha )\,\varphi (x,y)\,\beta .} {\displaystyle \varphi (x\alpha ,y\beta )=\sigma (\alpha )\,\varphi (x,y)\,\beta .}
		
		The associated anti-automorphism ? for any nonzero sequilinear form ? is uniquely determined by ?.			
\fi

	\
	
\end{frame}	
	\begin{frame}[shrink]
		\frametitle{A Hermitian modular form of weight $\ell$ with
			character $\sigma$} is  a  holomorphic function $F$ on $\Hr_{n}$ $({n}\ge 2)$ such that 
		$F(g\langle Z\rangle) = \sigma(g)F(Z)j(g,Z)^\ell$ for any 
		$g \in \Gamma_{n,K}$. 
		Here $\sigma$ be a character of $\Gamma_K^{({n})}$, trivial on
		$
		\left\{\mat 1_{n},B,0,1_{n},
		\right\},
		$
		and for $Z \in \Hr_{n}$, put
		$g\langle Z\rangle = (AZ + B)(CZ + D)^{-1}$, 
		$j(g,Z) = \det(CZ +D)$.
		
		\smallskip\pause
		{\magenta Fourier expansions}:  a semi-integral Hermitian matrix is a Hermitian matrix $H \in (\sqrt{-D_K})^{-1}M_{n}(\Oc)$
		whose diagonal entries are integral.
		
		Denote the set of semi-integral Hermitian matrices by $\Lambda_{n}(\Oc)$,\\
		the subset of its positive definite elements  is  $\Lambda_{n}(\Oc)^+$.
		%For $ H \in \Lambda_{n}(\Oc)$, put
		%$\gamma(H) = (-D_K)^{[{n}/2]} \det H$, $\gamma(H)\in\Z$. 
		
		\smallskip
		{\magenta A Hermitian modular form $F$ is called a cusp form}  
		if it has a Fourier expansion of the form
		$\di
		F(Z)=\sum_{H\in\Lambda_{n}(\Oc)^+}A(H) q^H.
		$
		Denote the space of cusp forms of weight $\ell$ with character $\sigma$ by $\Sr_\ell(\Gamma_{n,K},\sigma)$.
		
		\
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
	

%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]\frametitle{Hermitian modular forms and standard zeta functions.}
	%\vskip -0.5cm  
		
	Automorphic complex $L$-functions on classical groups. 
	Hecke algebras. The Rankin-Selberg method. 
	
	

	
		{\sl Hermitian modular group $\Gamma_{n,K}$ and the standard zeta
		function $\Zc(s; \f)$ (definitions)}
	
	Let $K=\Q(\sqrt{-D_K})$ be an imaginary quadratic field, $\theta=\theta_K$ its quadratic character, $n\in \N,  n'=\left[\frac n 2 \right]$. {\magenta The Hermitian group} 
	\begin{align*}
	& \nonumber
	\Gamma_{{n},K}=
	\left \{  M=\mat A,B,C,D,\in  \GL_{2{{n}}}(\Oc_K) | M\eta_{n}M^*=\eta_{n} \right\}, \eta_{n}=\mat 0_{{n}},-I_{{n}},I_{{n}},0_{{n}},
	\pause
	\\ &
	\nonumber
	\Zc(s,  {\bf f} ) =\left(\prod_{i=1}^{2n}
	L(2s -i + 1, \theta^{i-1})\right)
	\sum_{\a}
	\lambda(\a)N(\a)^{-s}, 
	\\ & \nonumber
	(\text{via Hecke's eigenvalues: }{\f}|T(\a) = \lambda(\a)\f, \a\subset {\Oc_{\it K}})	
	\\ &
	=\prod_\q \Zc_\q(N(\q)^{-s})^{-1}
	(\text{an Euler product over primes } \q \subset\Oc_K,
	\\ & \text{ with }
	\deg \Zc_\q(X)= 2n, 
	\text{ the Satake parameters }  
	t_{i,\q} , i=1, \cdots, n), 
	\pause
	\\ &
	\Dr(s,{\f})=\Zc(s-\frac \ell 2+\frac{1}{2},{\f})\ \ 
	(\text{Motivically normalized standard zeta function} \\
	& 	
	\text{ with a functional equation }s \mapsto \ell -s; \rk = 4n)
	\end{align*}
	
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8		
	\begin{frame}[shrink]
		\frametitle{The standard zeta function of  a Hermitian modular form}
		
		Fix  an integral ideal ${\mathfrak c}$ of $\Oc_K$. 
		Denote by $C\subset \Gamma_{n,K}$ the congruence subgroup of level $\c$; the group is essentially a principal congruence subgroup; it is an analogue of the group $\Gamma_0(N)$ in the elliptic modular case.  
		Write $T(\a)$ for the Hecke operator associated to it as it is defined in \cite{Shi00}, page 162, using the action of double cosets $C\xi C$ with
		$\xi=\diag(\hat D,  D)$, $(\det (D))=(\al)$, $\hat D=(D^*)^{-1}$.
		
		\smallskip\pause
		Consider a non-zero Hermitian modular form ${\f} \in \Mc_k(C,\psi)$ and assume ${\f}|T(\a) = \lambda(\a)\f$ with $\lambda(\a) \in \CC$ for all integral ideals $\a \in\Oc$. Then
		%For  a Hecke ideal character $\chi$ of $K$ of conductor ${\mathfrak f}$, Consider the Dirichlet series: for  $Re(s) \gg 0$,
		\begin{align}
			\nonumber
			\Zc(s,  {\f} ) =\left(\prod_{i=1}^{2n}
			L_\c(2s -i + 1, \theta^{i-1})\right)
			\sum_{\a}
			\lambda(\a)N(\a)^{-s}, 
		\end{align}
		the sum is over all integral ideals of $\Oc_K$. 
		%It is shown in [\cite{Shi00}, page 171] that 
		
		\smallskip\pause
		This series has an Euler product representation
		%, which we write as 
		$\Zc(s, {\f}) =
		\prod_{\q} (\Zc_\q (N(\q)^{-s})^{-1}$, where the product is over all prime ideals of $\Oc_K$,
		{\red  $\Zc_\q (X)$ is the numerator
			of the series 
			$\sum_{r\ge 0} \lambda(\q^r)X^r\in \CC(X)$, computed by Shimura (see \cite{Shi00}, p. 171).}
		
		\
		
	\end{frame}
\begin{frame}[shrink]
	\frametitle{Recalls about  Hecke algebras  (\cite{An87}, \cite{Shi71})}
For any subgroup $\Gamma$ of a semigroup $S$ 
	consider the $\Q$-vector space $L_{\Q}(\Gamma , S)$ generated by all left classes $\Gamma M$.
If each double class  $(M)=\Gamma M\Gamma\subset S$ is a finite union of left  classes, that is 
	$\ds(M)=\mathop\cup_j\Gamma M_j$, 
	one defines the Hecke algebra ${\Dr}_{ \Q}(\Gamma , S)= L_{\Q}(\Gamma ,S)^\Gamma$
as the  $\Q$-espace vectiriel  of elements in $L_{\Q}(\Gamma ,S)^\Gamma$ fixed by $\Gamma$; 
	the multiplication in ${\Dr}_{ \Q}(\Gamma , S)$ being defined by:
	$$
	(\sum_j a_j(\Gamma g_j))(\sum_{j'} a'_{j'}(\Gamma g'_{j'}))=
	\sum_{j,j'} a_j a'_{j'}(\Gamma g_jg'_{j'}).
	$$
	\pause
For two subgroups $\Ga$ and $\Ga'$ 	write $\Ga\sim\Ga'$ if $\Ga$ and $\Ga'$ are {\magenta c}, i.e. $\Ga\cap\Ga'$ is of finite index in both $\Ga$ and $\Ga'$. 
The subgroup $\tilde\Ga$=$\{ M \in S\ | \ M \Ga M^{-1} \sim \Ga\}$ is the commensurator of $\Ga$ in $S$. 
By Prop.3.1 \cite{Shi71}, for a family $\{ \Ga_\lambda \}_\lambda $ of subgroups 
commensurables to a fixed $\Ga\subset S$	 the following hold
$\ds \Ga_\lambda M \Ga_\mu=\mathop\cup_{i=1}^d \Ga_\lambda M_i$ with 
$d= [\Ga_\mu:\Ga_\mu\cap M^{-1}\Ga_\mu M]$. There is the decomposition
$\ds \Ga_\mu=\mathop\cup_{i=1}^d(\Ga_\mu\cap  M^{-1} \Ga_\lambda M ) P_i$, 
$M_i=M P_i$.	



\

\end{frame}
\begin{frame}[shrink]\frametitle{The Hecke algebra in the symplectic Siegel modular case}	%$\Lr = \Lr_q^{n}(N) = \Dr_{\QQ}({\Gamma}, \De)$ }  %\label{SSECalgHec}
	For a prime $q$, $q \nmid N$, let $\De = \De_q^{n}(N)$ = 
	$$ \Big\{ \ga = 
	\mat a,b,c,d,
	\in G_{\QQ}^+\cap \GL_{2{n}}(\ZZ[q^{-1}]) \Big| 	\nu(\ga)^{\pm}
	\in \ZZ[q^{-1}], c \equiv 0_{n} \bmod{N} \Big  \}
	$$
	is a subgroup in $G_{\QQ}^+$ containing ${\Gamma} = {\Gamma}_0^{n}(N)$.	
	The local Hecke
	algebra 
	$$
	\Lr = \Lr_q^{n}(N) = \Dr_{\QQ}({\Gamma}, \De) \text{ over } \QQ
	$$ is 
	defined  as a $\QQ$-linear space generated by the double cosets
	$(g) = ({\Gamma} g {\Gamma})$, $g \in \De$ of the group $\De$, with respect
	to the subgroup ${\Gamma}$ for which multiplication is defined by the standard
	rule (see \cite{An87}, \cite{Shi71}). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The operators of diagonal  matrices $\mathbf{T}({q})$, $\mathbf{T}_i({q}^2)$ 
%\begin{align}&
$
\mathbf{T}({q})= \mathbf{T}(\underbrace{1, \cdots, 1}_{n},\underbrace{{q}, \cdots, {q}}_{n})$\ \ generate it (see p.149 of \cite{An87}),\\
$\mathbf{T}_i({q}^2)= \mathbf{T}(\underbrace{1, \cdots, 1}_{n-i},\underbrace{{q}, \cdots, {q}}_{i},\underbrace{{q}^2, \cdots, {q}^2}_{n-i},\underbrace{{q}, \cdots, {q}}_{i}) , i=1,  \cdots, n$.

%\end{align}
	
		
		\
		
	\end{frame}
	\begin{frame}[shrink]
		\frametitle{The structure of the local Hecke algebras via Satake isomorphism}
	For 	$\Lr = \Lr_q^{n}(N)$, ($q \nmid N$)
	for each $j$, $1 \leq j \leq {n}$ us above let us denote by $\omega_j$ an automorphism
	of the algebra $\QQ[x_0^{\pm 1}, x_1^{\pm 1}, \cdots, x_{n}^{\pm 1} ]$
	defined on its generators by the rule:
	$$
	x_0 \longmapsto x_0x_j, ~~x_j \longmapsto x_j^{-1}, ~~ x_i \longmapsto
	x_i ~~~~(\mbox{where } 1 \leq i \leq {n}, ~~ i \neq j ).
	$$
	Then the automorphisms $\omega_j$ and the permutation group $S_{n}$ of the
	variables $x_i$ ($1 \leq i \leq {n}$) generate together the Weyl group $W = W_{n}$,
	and there is the {\it Satake isomorphism}
	\index{Satake isomorphism}:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.20)
	\begin{equation}  \label{EQisomdeSat}
	Sat : \Lr \stackrel{\sim}{\longrightarrow}
	\QQ[x_0^{\pm 1}, x_1^{\pm 1}, \cdots, x_{n}^{\pm 1} ]^{W_{n}}
	\end{equation}
	where $W_{n}$ indicates the subalgebra of elements fixed by $W_{n}$.
	For any commutative $\QQ$-algebra $A$ the group $W_{n}$  acts on the set
	$(A^{\times})^{{n}+1}$, therefore any  homomorphism of $\QQ$-algebras
	$\lambda : \Lr \mapsto A$ can be identified  with some element
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.21)
	\begin{equation}  \label{EQparamSatake}
	(\al_0, \al_1, \cdots, \al_{n}) \in [(A^{\times})^{{n}+1}]^{W_{n}}.
	\end{equation}
	
	\
		
	\end{frame}	
	%\end{document}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{frame}[shrink]
		\frametitle{Explicit action of the Hecke operators}
	
	Any double coset  $(g) = ({\Gamma} g {\Gamma})$ ($g \in \De = \De_q^{n}(N)$)
	can be represented as a disjoint union of  left cosets:
	$ \ds
	(g) = \bigcup_{i = 1}^{t(g)} {\Gamma} g_i,
	$
	therefore any element $X \in \Lr$ of the Hecke algebra $\Lr$ takes the form of
	a finite linear combination
	$\ds
	X = \sum_{i=1}^{t(X)} \mu_i({\Gamma} g_i),
	$
	$\mu_i \in \QQ$, $g_i \in \De$.
	
	In order to define explicit action of the Hecke operators
	on the complex vector space $\Mc_{n}^k(N,\psi)$ ($q\nmid N$)
for any  $g = \mat a,b, c, d, \in \De$
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.22)
%	\begin{equation}  \label{EQfkpsi}
$\ds 
\magenta	(f|_{k,\psi} g)(z) = \det(g)^{k-\kp} \psi(\det(a)) \det(cz+d)^{-k}
	f(g(z))
$
%	\end{equation}
	(the convenient notation by Petersson and Andrianov)
	%, is
	%especially useful when dealing with Hecke operators in their
	%normalized form, compare with (\ref{EQautomorphy})). 
	With this notation
	the automorphy condition can be rewritten as follows
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.23)
	\begin{align*}  \label{EQactionofHecke}
	(f|_{k,\psi} \ga)(z) = f \mbox{ for all } \ga \in {\Gamma} = {\Gamma}_0^{n}(N).
	\end{align*}
	In this case for any
	$\ds
	X = \sum_{i=1}^{t(X)} \mu_i ({\Gamma} g_i) \in \Lr
	$
this explicit expression
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.24)
	%\begin{equation}  \label{EQactiondeopdeHec}
	$\ds f|X = \sum_{i=1}^{t(X)} \mu_i f|_{k,\psi} g_i$,
	%\end{equation}
is	well defined
	% and $f|X \in \Mc_{n}^k(N,\psi)$ so that the formula
	%(\ref{EQactiondeopdeHec}) gives a representation of the Hecke algebra
	%$\Lr = \Lr_q^{n}(N)$.
	
\


\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Hecke polynomials in the Siegel modular case}  %\label{SSECpolyHec}
%Following A.N.Andrianov (see \cite{And79}), let us 
Consider the polynomials
$\tilde{Q}(z) \in \QQ[x_0, \cdots, x_{n}][z]$ and  \\ \smallskip
$\tilde{R}(z) \in \QQ[x_0^{\pm 1}, \cdots, x_{n}^{\pm 1}][z]$ :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.25)
%\begin{eqnarray}

\smallskip
$ \ds
\tilde{Q}(z)    =    %\tilde{Q}(x_0, x_1, \cdots, x_{n} ; z)
%\label{EQpolyQtilde}    
(1 - x_0 z) \prod_{r = 1}^{n} \prod_{1 \leq i_1 < \cdots < i_r \leq {n}}
(1 - x_0 x_{i_1} \cdots x_{i_r} z )$,   \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.26)
$\tilde{R}(z)    =   \prod_{i = 1}^{n} (1 - x_i^{-1} z) (1 - x_i z).
$
%\label{EQpolyRtilde}
%\end{eqnarray}

%From the definition it follows that 
\smallskip
The coefficients of the powers of the
variable $z$ all belong to the subring
$\QQ[x_0^{\pm 1}, \cdots, x_{n}^{\pm 1}]^{W_{n}}$.
Therefore under the
Satake isomorphism
\index{Satake isomorphism}
(\ref{EQisomdeSat}) there exist polynomials over %the associative commutative ring 
$\Lr = \Lr_q^{n}(N)$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.27)
%\begin{equation}  \label{EQpolyQzRz}
$
Q(z) = \sum_{i=0}^{2^{n}} (-1)^i T_i z^i$, 
$R(z) = \sum_{i=0}^{2{n}}  (-1)^i R_i z^i \in \Lr[z]$,
%\end{equation}
such that

$\ds \tilde{Q}(z) = \sum_{i=0}^{2^{n}} (-1)^i \tilde{T}_i z^i, \
\tilde{R}(z) = \sum_{i=0}^{2{n}}  (-1)^i \tilde{R}_i z^i,  
\text{ and }
\tilde{X} = Sat X  \in \Lr.
$

Moreover, the polynomials
$\De_M '{}^{\pm 1}$, $R_i$ ($1 \leq i \leq {n}-1$) and $T_1$ with
$ 
\tilde{\De}_M'{}^{\pm 1}    =    x_0^2 x_1 \cdots x_{n}$,  \\
$\tilde R_i    =    S_i(x_1, \cdots, x_{n} ; x_1^{-1}, \cdots, x_{n}^ {-1})$,\\
$\tilde{T_1}    =    \di{ x_0 \sum_{i=1}^{n} S_i(x_1, \cdots, x_{n}) }
= \di{ x_0 \prod_{i=1}^{n} (1 + x_i) },  
$
generate of the {\it Hecke algebra}
\index{Hecke algebra}, where $S_i$ denotes the
elementary symmetric polynomial of degree $i$ of the corresponding set of
variables.

	\
		
\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{The spinor zeta function and the standard zeta function}

Let $f \in \Mc_{n}^k(N,\psi)$ be an eigenfunction of all {\it Hecke operators}
\index{Hecke operators}
$f \longmapsto f|X$, $X \in \Lr_q^{n}(N)$ with $q$ being prime numbers,
$q \nmid N$, so that $f|X = \lambda_f(X) f$. Then the numbers
$\lambda_f(X) \in \CC$ define a homomorphism
$\lambda_f : \Lr \longrightarrow \CC$, %(see \ref{SSECalgHec}), 
 determined by a $({n} + 1)$-tuple of numbers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.28)
%\begin{equation}
$\ds
(\al_0, \al_1, \cdots, \al_{n})
% =(\al_{0,f}(q), \al_{1,f}(q), \cdots, \al_{{n},f}(q)) \in
\left[ (\CC^{\times})^{{n}+1} \right]^{W_{n}}
$
%\end{equation}
(the {\magenta Satake parameters}).

Now let the variables $x_0$, $x_1$, $\cdots$, $x_{n}$ 
%in (\ref{EQpolyQtilde}), (\ref{EQpolyRtilde}) be equal to the corresponding
be  { Satake $q$-parameters}
\index{Satake $q$-parameters}
$\al_{0,f}(q)$, $\al_{1,f}(q)$, $\cdots$, $\al_{{n},f}(q)$, then
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.29)
%\begin{eqnarray}
\medskip

$Q_{f,q}(z)    =    (1 - \al_0 z) \prod_{r=1}^{n}
\prod_{1 \leq i_1 < \cdots < i_r \leq {n}} (1 - \al_0 \al_{i_1}
\cdots \al_{i_r} z)$,  \label{EQpolyQfq}  \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.30)
$R_{f,q}(z)    =    \prod_{i=1}^{n} (1 - \al_i^{-1} z)
(1 - \al_i z) \in \QQ[ \al_0^{\pm 1}, \cdots, \al_{n}^{\pm 1}][z].
$
%\end{eqnarray}

%From the description of the {\it Hecke polynomials}
%\index{Hecke polynomial}
%in \ref{SSECpolyHec} 
\pause
It follows
then that the coefficients of  (\ref{EQpolyQfq}) can be expressed in terms
of the eigenvalues $\lambda_f(X)$ of the Hecke operators $X = T_i$, $R_i$.
%from (\ref{EQpolyQzRz}).
Next let us  put
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.31)
%\begin{eqnarray}
$\ds Z^{(q)}(s,f)    =    Q_{f,q}(q^{-s})^{-1}  = $
\\   
$\ds \left[ (1 - \al_0(q) q^{-s}) \prod_{r=1}^{n}
\prod_{1 \leq i_1 < \cdots < i_r \leq {n}}
(1 - \al_0(q) \al_{i_1}(q) \cdots \al_{i_r}(q) q^{-s}
\right]^{-1}
$
%end{eqnarray}
\\
 {\magenta The spinor zeta function $Z(s,f)$} \index{spinor zeta function}
of 
$f \in \Mc_{n}^k(N,\psi)$ is the following product
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (2.32)
$\ds 
Z(s,f) = \prod_{q \nmid N} Z^{(q)}(s,f).
$
\\
{\magenta The standard zeta function}\index{standard zeta function}
$\Dr(s,f,\chi)$ of $f \in \Mc_{n}^k(N,\psi)$ is the product
$ \ds
\Dr(s,f,\chi) = \prod_{d \nd N} \Dr^{(q)}(s,f,\chi)
$
with
$
\Dr^{(q)}(s,f,\chi)$ = $(1 - \chi(q) \psi(q) q^{-s})^{-1}
R_{f,q}(\chi(q)\psi(q)q^{-s})^{-1}$  For a Dirichlet character  $\chi$ modulo $M$, \ $\Dr(s,f,\chi)$ =

$\ds \prod_q \left[ \left( 1 - \frac{\chi\psi(q)}{q^s} \right)
\prod_{l=1}^{n} \left( 1 - \frac{\chi\psi(q) \al_l(q)}{q^s}
\right) \left( 1 - \frac{\chi\psi(q)\al_l(q)^{-1}}{q^s}
\right) \right]^{-1} .
$
	
	\
	
		
\end{frame}	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Examples of subgroup families $\{ \Ga_\lambda \}_\lambda $ in $p$-adic constructions}
	
Let us fix $\Ga$ and for $\lambda = (v), v\ge 1$ put 
$$
\Ga_\lambda = \Ga_{(v)}= \Ga\cap  M^{-v} \Ga M^v.
$$
For example if  $\Ga=\SL_2(\Z)$, $M = \mat p,0,0,1,$, the congruence subgroups 
$\Gamma_0(N)=\left\{ \mat a,b,Nc,d , \in %\SL_2(\Z)   \right\}$
\SL_2(\Z) \cap \mat N,0,0,1,^{-1}  \SL_2 (\Z){\mat N,0,0,1,}  \right \}
$ 
produce the family $\{ \Ga_{(v)}\}_{(v)}$ attached to $M$  and a fixed  $\Ga_{(0)}=\Ga_0(C)\subset \Ga$: 
$$
\Ga_{(v)}= \Ga_{(0)}\cap  M^{-v} \Ga M^{v}= \Gamma_0(p^v).
$$
Notice the acion of the lowering operator (generalized Atkin's operator):  
for $v'> v$, 
$$
U_M^v=(\Ga_{(v')} M^{v}(\Ga_{(v'-v)}): \Mc_k(\Ga_{(v')})\to \Mc_k\Ga_{(v'-v)}), 
$$
compare with \S 4 of \cite{Hi85}. Indeed, 
$\forall \ga\in \Ga_{(v')}, h|\ga=h \Rightarrow \forall \ga\in \Ga_{(v'-v)} f|U_M^v|\ga= f|U_M^v$






\end{frame}

%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Examples of Hermitian cusp forms} 
	{\blue\Large The Hermitian Ikeda lift, \cite{Ike08}}.
	Assume 
	${n} = 2n'$ even. Let
	$\di
	f(\tau) =
	\sum_{N=1}^\infty
	a(N)q^N \in  \Sr_{2k+1}(\Gamma_0(D_K), \chi)
	$
	be a primitive form, whose $L$-function is given by
	$$
	L(f, s) =
	\prod_{p \nd D_K}
	(1 - a(p)p^{-s} + {\theta}(p)p^{2k-2s})^{-1}
	\prod_{p|D_K} (1 - a(p)p^{-s})^{-1}.
	$$
	For each prime $p \nd D_K$, define the Satake parameter 
	$\{\al_p, \beta_p\} = \{\al_p, {\theta}(p)\al_p^{-1}\}$  by
	$$
	(1 - a(p)X + {\theta}(p)p^{2k}X^2)=
	(1-p^k\al_pX)(1-p^k\beta_pX)
	$$
	For $p | D_K$, we put $\al_p = p^{-k} a(p)$. 
	Put
	\begin{align}
	& \nonumber
	A(H)=|\gamma(H)|^{k}
	\prod_{p|\gamma(H)}\tilde F_p(H;\al_p), H \in \Lambda_{n}(\Oc)^+\\ & \nonumber
	F(H)=\sum_{H\in\Lambda_{n}(\Or)^+}A(H) q^H, Z\in\Hr_{2n}.
	\end{align}	
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{The first Hermitian lift  (even case)} 
	{\Large \blue Theorem 5.1 (Case E) of \cite{Ike08}} Assume that ${n} = 2n'$ is even. Let $f(\tau )$, $A(H)$ and $F(Z)$ be as above. Then we have
	$F\in \Sr_{2k+2n'}(\Gamma_K^{(n)},\det^{-k-n'}).$
	
	
	\smallskip
	In the case when ${n}$ is odd, consider a similar lifting for a normalized Hecke eigenform ${n} = 2n'+1$ is odd. Let
	$\di
	f(\tau) =
	\sum_{N=1}^\infty
	a(N)q^N$ $\in  \Sr_{2k}(\SL_2(\Z))
	$
	be a primitive form, whose $L$-function is given by
	$$
	L(f, s) =
	\prod_{p}
	(1 - a(p)p^{-s} + p^{2k-1-2s})^{-1}.
	$$
	For each prime $p $, define the Satake parameter 
	$\{\al_p, \al_p^{-1}\}$  by
	$$
	(1 - a(p)X + p^{2k-1}X^2)=
	(1-p^{k-(1/2)}\al_pX)(1-p^{k-(1/2)}\al^{-1}X).
	$$
	Put
	\begin{align}
	& \nonumber
	A(H)=|\gamma(H)|^{k-(1/2)}
	\prod_{p|\gamma(H)}\tilde F_p(H;\al_p), H \in \Lambda_{n}(\Oc)^+\\ &
	\nonumber
	F(H)=\sum_{H\in\Lambda_{n}(\Or)^+}A(H) q^H, Z\in\Hr_{n}.
	\end{align}
	
	
	
	
	\
	
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{The second Hermitian lift (odd case)}
	{\blue \Large Theorem 5.2 (Case O) of \cite{Ike08}}.
	Assume that ${n} = 2n'+1$ is odd. Let $f(\tau )$, $A(H)$ and $F(Z)$ be as above. Then we have
	$F\in \Sr_{2k+2n'}(\Gamma_K^{(n)},\det^{-k-n'}).$
	
	\smallskip
	The lift $Lift^{({n})}(f)$ of $f$ is a common Hecke eigenform of all
	Hecke operators of the unitary group, if it is not identically zero (Theorem 13.6). 
	
	\smallskip
	{\blue \Large Theorem 18.1 of \cite{Ike08}}. Let ${n}$, $n'$, and $f$ be as in Theorem 5.1 or as in Theorem 5.2. 
	Assume that $Lift^{({n})}(f)\ne 0$. 
	Let $L(s, Lift^{({n})}(f), st)$ be the $L$-function of $Lift^{({n})}(\f)$ associated to $st : {}^L{\cal G} \to \GL_{4{n}}(\CC)$. 
	Then up to bad Euler factors, $L(s, Lift^{({n})}(f), st)$ is equal to
	$$
	\prod_{i=1}^{n}
	L(s + k + n' - i + \frac 1 2, f)L(s + k + n' - i + \frac 1 2, f, {\theta}).
	$$
	Moreover, the %Satake parameters and
	$4{n}$ charcteristic roots of
	$L(s, Lift^{({n})}(f), st)$ given
	as follows:  for $i=1, \cdots ,{n}$ 
	\begin{align} & \nonumber
	\al_p p^{-k-n'+i-\frac 1 2}, \al_p^{-1}p^{-k-n'+i-\frac 1 2},
	{\theta}(p) \al_pp^{-k-n'+i-\frac 1 2}, {\theta}(p)\al_p^{-1}p^{-k-n'+i-\frac 1 2}
	%\  \ (i=1, \cdots ,{n}) %\\ &
	%\{\al_p, \beta_p\} = \{\al_p, {\theta}(p)\al_p^{-1}\}
	\end{align}
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Sho}
\begin{frame}[shrink]
	\frametitle{Functional equation of the lift (Sho Takemori)}	
	There are two cases \cite{Ike08}: the even case (E) and the odd case (O): 
	$
	\begin{cases}
	f \in S_{2k+1}(\Gamma_0(D), \theta),  F=Lift^{(n)}(f) &(E) \\
	(\text{the lift is of even degree }n = 2n'\text{ and of weight } 2k + 2n')&\\
	f \in S_{2k}(\SL(\Z)), F=Lift^{(n)}(f) & (O)  \\
	(\text{the lift is of odd degree }n = 2n'+1 \text{ and of weight } 2k + 2n').&
	\end{cases}
	$ 
	Then,
	up to bad Euler factors, the standard $L$-function of $ F=Lift^{(n)}(f)$ is given by	
	$
	\prod_{i=1}^{n}
	L(s + k + n' - i + \frac 1 2, f)L(s + k + n' - i + \frac 1 2, f, {\theta})
	$
	$=\begin{cases}
	\prod_{i=1}^{2n'}
	L(s + k + n' - i + \frac 1 2, f)L(s + k + n' - i + \frac 1 2, f, {\theta})&  (E) \\
	\prod_{i=1}^{n'}
	L(t(s,i), f)L(t(s,2n'+1-i), f) & \\ 
	L(t(s,i), f, {\theta})L(t(s,2n'+1-i), f, {\theta}) & \\
	\prod_{i=1}^{2n'+1}
	L(s + k + n' - i + \frac 1 2, f) &
	\\
	\times L(s + k + n' - i + \frac 1 2, f, {\theta})  & (O) \\
	= L(s + k- \frac 1 2, f)L(s + k - \frac 1 2, f, {\theta})&\\ 
	\prod_{i=1}^{n'}
	L(t(s,i), f)L(t(s,2n'+2-i), f) & \\
	L(t(s,i), f, {\theta})L(t(s,2n'+2-i), f, {\theta})&
	\end{cases}
	$
	where $t(s,i)=s + k + n' - i + \frac 1 2$.	
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Sho2}
\begin{frame}[shrink]
	\frametitle{The Gamma factor $\Gamma_\Zc(s)$ of Ikeda's lift}
	In the even case since $(2k +1)-t(s, i)$ = $t(1-s, 2n'+1-i)$, using the Hecke functional equation in the symmetric terms of the product, gives the functional equation of the standard $L$ function of the form $s \mapsto 1 - s$, and the gamma factor is given by 
	$$
	\prod_{i=1}^{n}\Gamma_\CC(s + k + n' - i + 1/2)^2 =\Gamma_\Dr(s+n'+  \frac 1 2).
	$$	
	In the odd case when $f \in S_{2k}(SL_2(\Z))$, the lift is of degree $n = 2n' + 1$ and of weight $2k + 2n'$. By $2k - t(s, i)$ = $t(1 - s, 2n + 2 - i)$, the standard $L$ functions has functional equation of the
	form $s \mapsto 1 - s$ and the gamma factor is the same.	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	Hence the Gamma factor of Ikeda's lifting, denoted by ${\f}$, of an elliptic modular form $f$ and used as a pattern,  extends to a general (not necessarily lifted) Hermitian modular form ${\f}$ of even weight $\ell$, which equals in the lifted case to 
	$\ell=2k+2n'$, where 
	$k=(\ell-2n')/2$ $=\ell /2-n'$=$\ell/2-n'$, when 
	the Gamma factor of the standard zeta function with the symmetry $s\mapsto 1-s$ becomes (see p.\pageref{Sho}) 
	\smallskip
	$\prod_{i=1}^n 
	\Gamma_\CC (s+\ell/2 -n'+n'-i+(1/2))^2
	=\prod_{i=1}^n 
	\Gamma_\CC (s+\ell/2-i+(1/2))^2
	=\prod_{i=0}^{n-1} 
	\Gamma_\CC (s+\ell/2-i-(1/2))^2$.
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[shrink]
	\frametitle{Eisenstein series and the Rankin-Selberg method} 
	The (Siegel-Hermite) Eisenstein series $E_{2\ell}^{({n})}(Z)$ of weight $2\ell$, character $\det^{-\ell}$,  is defined by
	$\di
	E_{2\ell}^{({n})}(Z)=
	\sum_{g\in\Gamma_{K,\infty}^{({n})}\bs \Gamma_K^{({n})}}
	(\det g)^{\ell}j(g,Z)^{-2\ell} .
	$
	The series  converges absolutely for $\ell >{n}$. 
	Define the normalized Eisenstein series ${\cal E}_{2\ell}^{({n})}(Z)$ by
	$
	{\cal E}_{2\ell}^{({n})}(Z)=$
	$2^{-{n}}\prod_{i=1}^{{n}}L(i-2\ell, \theta^{i-1})\cdot  E_{2\ell}^{({n})}(Z)
	$
	If $H\in\Lambda_{n}(\Oc)^+$, then the $H$-th Fourier coefficient of ${\cal E}_{2\ell}^{({n})}(Z)$ is polynomial over $\Z$ in $\{p^{\ell-({n}/2)}\}_p$, and equals 
	$$
	|\gamma(H)|^{\ell-({n}/2)}
	\prod_{p|\gamma(H)}\tilde F_p(H,p^{-\ell+({n}/2)}),
	\gamma(H) = (-D_K)^{[{n}/2]} \det H.
	$$	
	Here, $\tilde F_p(H,X)$ is a certain Laurent polynomial in the variables $\{X_p=p^{-s}, X_p^{-1}\}_p$ over $\Z$. % arising from the Siegel series for $H$.
	This polynomial is a {\red key point in proving congruences} for
	the modular forms in a {\red Rankin-Selberg integral}. 
	
	\smallskip
	Also, we set , for $s\in\CC$ and a Hecke ideal character $\psi \bmod \c$,
	$$ \magenta
	E(Z,s,\ell,\psi)=
	\sum_{g\in C_\infty\bs C}
	\psi(g) (\det g)^{\ell}j(g,Z)^{-2\ell}|(\det g)j(g,Z)|^{-s} .
	$$
	
	\	
	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{A Rankin-Selberg method: the simplest  case}
	Let us recall a Rankin-Selberg integral representation in the simplest elliptic modular case of $\GL_2$.	
	
	Let $H=\{ a=x+iy\ | y >0  \}$ be complex upper half plane on which  
	the group $\GL_2^+(\R)$  acts (of real $2\times 2$ matrices  $\ga =\mat 
   {a_\ga},{b_\ga},{c_\ga},{d_\ga},$ of positive determinant).   
  For a natural $k$ the action of $\GL_2^+(\R)$ on a function $f: H\to \C$ is defined by 
  $\ds (f|_k\ga) (z)$= $\det\ga^{k/2} f(\ga(z))({c_\ga}z+{d_\ga})^{-k}$
   
	   For a primitive cusp form $\ds f=\sum_{n=1}^\infty a(n)e(nz)$ of conductor $C_f|N$, weight $k$ and Dirichlet character $\psi \bmod N$ (that is, a normalized new cusp  Hecke 
	    eigenform $f\in \Sr_k^{new}(N,\psi)\subset \Sr_k(\Gamma_1(N)))$, $a(1)=1$, 
	    and another modular form $\ds g=\sum_{n=1}^\infty b(n)e(nz)\in \Mc_\ell(N,\w)$ of level $N$, weight $\ell$ and Dirichlet character $\w \bmod N$.
	    
	  	    
	  	    \	
	  	    
	  	  \end{frame}
	  	  %\end{document}
	  	  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	  	  \begin{frame}[shrink]
	  	  	\frametitle{Classical convolution of Rankin-Selberg}
	For two  Hecke eigenforms \\
	$\ds
	f=\sum_{n=1}^\infty a(n)e(nz)\in \Sr_k(N,\psi)\ g=\sum_{n=1}^\infty b(n)e(nz)\in \Mc_\ell(N,\w)
	$\\
their convolution is defined by $L(s,f,g)=	\sum_{n=1}^\infty a(n)b(n)n^{-s}$.
Moreover, one obtains an Euler product of degree 4 if	
\begin{align*}&
	L(s,f)=\sum_{n=1}^\infty a(n)n^{-s}=\prod_{primes\, q}
	[(1-\al_qq^{-s})(1-\al'_qq^{-s})]^{-1} \\
	&
	L(s,g)=\sum_{n=1}^\infty b(n)n^{-s}=\prod_{primes\, q}[(1-\beta_qq^{-s})(1-\beta'_qq^{-s})]^{-1},
	\end{align*} 
one has $\ds \Dr(s,f,g)=L_N(2s+2-k-\ell,\w\psi)L(s,f,g)$, where 
$\ds \Dr(s,f,g)$= \\ $\ds \prod_{primes\, q}
[(1-\al_q\beta_q q^{-s})(1-\al'_q\beta_qq^{-s})(1-\al_q\beta'_qq^{-s})(1-\al'_q\beta'_qq^{-s})]^{-1}$ (Rankin's lemma).	 	  	
	  	  	
	    
	    \	
	    
	  \end{frame}
	  %\end{document}
	  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	  \begin{frame}[shrink]
	  	\frametitle{Classical  Rankin-Selberg integral representation}	uses the Petersson product $\langle f,h\rangle_N$
	  	 on %the congruence subgroup 
	  	 $\Gamma_0(N)=\{ \mat a,b,Nc,d , \in \SL_2(\Z) \}$
	  	 %\SL_2(\Z) \cap \mat 1,0,0,N,  \SL_2 (\Z){\mat %1,0,0,N,}^{-1}$  	  	  
	\begin{align*}	&    
	  2 (4\pi)^{-s}  \Gamma (s) \Dr(s,f,g)=\langle f^\rho, gE(z,s-k+1)\rangle_N,
	 \text{where}\\& E(z,s)=y^{s}\sum^{}{}'_{(c,d)}(\psi\w)(d) (Ncz+d)^{-(k-\ell)}|Ncz+d|^{-2s}.
	\end{align*}	    
	This integral produces $\C$-analytic continuation 
	 from the right half plane $\{s\in \C| \Re(2s-k-\ell)>2\}$ where both the Eisestein series and $\Dr(s,f,g)$ converge, to the whole $\C$.
	 
	 \
	 
Also, it produces a $\C_p$-analytic continuation (or $p$-adic interpolatoon) of the critical values 
$ \ds \frac{\Dr(\ell+r,f,g(\chi))}{\pi^{1+\ell+2r}\langle f,f\rangle_N} $ (twisted by Dirichlet's characters $\chi\bmod p^v$) which are algebraic numbers for all integers $r=0,\cdots, k-\ell-1$ assuming $k>\ell+2$, see     
\cite{Hi85}, 



	\	
	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{A Hermitian  integral of Rankin-Selberg type} 
	%\iffalse	
	%Let us specialize to $F=\Q$
%	{\magenta Include reading in the simplest elliptic modular case of $\GL_2$}
	
%	The integral representation of Rankin-Selberg type in the Hermitian modular case: % is used
% (\cite{Bou16}, p.13):
	%"The following theorem (see also [25, Theorem 7.8]) is obtained by combining results of
	%Shimura [31] and Klosin [25]. For details we refer to [6, Section 4], 
	
	{\blue \Large Theorem 4.1 (Shimura, Klosin)}, see \cite{Bou16}, p.13.\\
	Let $0 \ne {\f} \in\Mc_\ell(C,\psi ))$ of scalar weight $\ell$, $\psi
	\bmod \c$, such that $\forall \a, {\f} |T(\a) = \lambda(\a){\f}$, and assume that $2\ell\ge n$, then there exists ${\cal T} \in S_+\cap \GL_n(K)$ and ${\cal R} \in \GL_n(K)$ such that	
	\begin{align}\nonumber
	& \Gamma ((s)) \psi(\det({\cal T}))%c_{\f} (\tau, r)
	\Zc(s + 3n/2, {\f} , \chi) =	\\ & \nonumber
	\Lambda_{\c}(s + 3n/2, \theta\psi\chi)\cdot
	%\\ \nonumber
	%&  \left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2} C_0
	%\Lambda_\c(s + 3n/2, \theta(\psi\chi)_1)\cdot
	%\\ & \nonumber
	%\left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}
	C_0 \langle {\f} , \theta_{\cal T}(\chi)
	%\tilde E_\A(x, 
	{\Ec}(\bar s + n, \ell - \ell_\theta, \chi^\rho\psi) \rangle_{C\sec},
	\end{align}
	where ${\Ec}(Z, s, \ell-\ell_\theta, \psi)_{C\sec}$ is a
	normalized
	group theoretic %adelic
	Eisenstein series with components as above of level $\c\sec$ divisible by $\c$, and weight $\ell-\ell_\theta$. 	
	Here $\langle \cdot, \cdot \rangle_{C\sec}$ is the  
	%group theoretic %adelic
	normalized
	Petersson inner product associated to the congruence subgroup 
	$C\sec$ of level $\c\sec$.
	%$D[{\e}, {\e}{\h}]$,
	% as defined for example in [30, Equation 10.9.6],	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\
	$\ds \Gamma((s))=
	%\prod_{v\in {\bf a}}
	(4\pi)^{-n(s+h)}\Gamma^\iota_n(s + h), 
	{\Gamma}^\iota_n(s) = \pi^{\frac{n(n-1)}2} \prod_{j=0}^{n-1} {\Gamma}(s-j)$,\\ 
	where $h=0	\text{ or } 1, \ 
	C_0$
	%:=\frac{[\Gamma_0(\c\sec):\Gamma]}{\sharp X}.	
	\ a subgroup index.
	%\fi{}
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Euler factors of the standard zeta function, \cite{Shi00}, p. 171 }
	%	{\magenta Euler factors of the stanard zeta function.}
	%	Following %\cite{Bou16}, p.12, and 
	%	\cite{Shi00}, page 171, 
	The Euler factors $\Zc_{\q}(X)$ in the Hermitian modular case
	at the prime ideal $\q$ of $\Oc_K$ are
	\smallskip\pause
	\begin{align} \nonumber
	\text{ (i) } &\Zc_\q(X) =\prod_{i=1}^{n}
	{\left((1- N(\q)^{n-1}t_{\q,i}X)(1- N(\q)^{n}t_{\q,i}^{-1} X)\right)}^{-1},\\ & 
	\nonumber
	\text{ if } \q^{\rho} = \q \text{ and }\q \nd \c, \text{(\magenta the inert case outside level }\c),\\
	\smallskip\pause
	\text{(ii) } \nonumber & \Zc_{\q_1}(X_1) \Zc_{\q_2}(X_2) =
	\prod_{i=1}^{2n}
	\left((1- N(\q_1)^{2n}t_{\q_1\q_2,i}^{-1}X_1)
	(1- N(\q_2)^{-1}t_{\q_1\q_2,i}X_2)\right)^{-1},  \\ &  \nonumber
	\text{ if }
	\q_1 \ne \q_2, \q_1^{\rho}= \q_2 \text{ and } \q_i \nd \c \text{
		for } i = 1,2\text{ \magenta (the split case outside  level) }, \\
	\nonumber
	\smallskip\pause
	\text{(iii) } &\Zc_\q(X) = \prod_{i=1}^{n} 
	\left(1- N(\q)^{n-1}t_{q,i}X\right)^{-1}, \text{ if } \q^{\rho} = \q \text{ and } \q|\c 	\text{   \magenta  (inert level divisors )},\\  \nonumber
	\text{(iv) } & \Zc_{\q_1}(X_1) \Zc_{\q_2}(X_2) =
	\prod_{i=1}^{n}
	\left((1- N(\q_1)^{n-1}t_{\q_1\q_2,i}^{-1}X_1)
	(1- N(\q_2)^{n-1}t_{\q_1\q_2,i}X_2)\right)^{-1},  \\ &  \nonumber
	\text{ if } \q_{1} \ne \q_{2},  \q_i | \c \text{ for } i = 1,2
	\text{   \magenta (split level divisors).  } 
	\end{align}
	where the $t_{?,i}$ above for $? = \q, \q_{1}\q_{2}$, are the {\magenta Satake parameters } of the eigenform $\f$.
	
	\
	
	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
\begin{frame}[shrink]
	\frametitle{Notice the important dychotomy for the $L$-factors} 
	in the Siegel modular case (that is, of symplectic type)  vs. the Hermite modular case (of unitary type). 
	In these cases the correspopnding complex 
	component of the Langlands $L$-group is either $GSpinO(2n+1)(\CC)$,
	with the Euler factors of degree $2n+1$ (the standard representation of $GO(2n+1)$,  resp. of  degree $2^n$ (the spinor representation of the  $L$-group) (the symplectic case),
	or, in the Hermite case,  the complex component of the $L$-group is
	$GL_{2n}(\CC)\times GL_{2n}(\CC)$,  with the Euler factors of degree  $4n$  (the standard representation of the $L$-group),
	see also 16.16,  p.133, in particular, formula (16.16.2) at p.134 of \cite{Shi97a} 
	or \cite{Shi97b} for a concise exposition.
	
	\
	
	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fi
\begin{frame}[shrink]
	\frametitle{The standard motivic-normalized zeta
		$\Dr (s,\f,\chi)$}
	The standard zeta	function of $\f$ is defined by means of the
	$p$-parameters as the
	following Euler product:
	$$ 
	\Dr(s, {\bf f}, \chi ) = \prod_p \prod_{i=1}^{2n}
	\left\{  \left( 1 - \frac{\chi(p)\al_i(p)}
	{p^s} \right) 
	\left( 1 - \frac{\chi(p)\al_{4n-i}(p)}
	{p^s} \right) \right\}^{-1} ,  
	$$
	where $\chi$ is an arbitrary Dirichlet character.
	The  $p$--parameters  
	$\alpha _{1}(p),  \dots,  \alpha _{4{n}}(p)$ of $\Dr (s,\f,\chi)$ for $p$ not dividing the level $C$ of the form $\f$ are related to the
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%******
	the $4{n}$ characteristic numbers
	$$
	\alpha _{1}(p),\   \cdots,\   \alpha _{2{n}}(p),\  \alpha_{2{n}+1}(p),\  \cdots,\   \alpha _{4n}(p)
	$$
	of the product of all $\q$-factors $\Zc_\q( N\q^{(n'+\frac{1}{2}) }X)^{-1}$ for all $\q|p$, which is a polynomial of degree 4${n}$ of the variable  $X=p^{-s}$ (for almost all $p$) with coefficients in a number field $T=T(\f)$ .	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\smallskip
	There is a relation between the two normalizations 
	$\magenta \Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})=\Dr(s,{\f})$ 
	explained below, 
	see \cite{Ha97} for general zeta functions
	$\Zc(s,\f)$ of type introduced in \cite{Shi00}, 
	using reprsentation theory of unitary groups and 
	Deligne's motivic $L$-functions.
	
	\
	
	\smallskip\pause
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]
	\frametitle{Description of the Main theorem} 		
	Let $\Omega_{\f}$ be a period attached to  an Hermitian cusp eigenform ${\f}$, $\magenta \Dr(s,{\f})=\Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})$ the standard zeta function, and
	$$
	\alpha _{{\f}}=\alpha _{{\f},p}=\left(\prod_{\q|p}\prod_{i=1}^n t_{\q,i}\right) p^{-n(n+1)}, \ \
	h 
	= \ord_{p}(\alpha _{{\f},p}),
	$$
	The number $\alpha_{\f}$	turns out to be an eigenvalue of Atkin's type
	operator $U_p : \sum_{H} A_H q^H\mapsto \sum_{H} A_{pH} q^H$ on some $\f_0$,
	and $h=P_{N}(\frac {d}2)-P_{H}(\frac {d}2)$.
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\pause
	\hskip-10pt {\Large \blue Definition.} 
	Let $M$ be a $\Oc$-module of finite rank where $\Oc\subset \C_p$.
	For $h\ge 1$, consider the following $\C_p$-vector spaces of functions on $\Z_p^*$ : $\Cc^h\subset \Cc^{loc-an}\subset \Cc$.
	Then
	
	- a continuous homomorphism $\mu: \Cc\to M$  is called a {\magenta (bounded) measure} $M$-valued measure on $\Z_p^*$.
	
	- $\mu: \Cc^h\to M $ is called an {\magenta $h$ admissible measure} $M$-valued measure on $\Z_p^*$ measure
	if the following growth condition  is satisfied	
	$$
	\left |\int_{a+(p^v)}(x-a)^j d\mu\right|_p \le p^{-v(h-j)}
	$$	 
	for $j=0,1,...,h-1$, and et  $\Yc_p=Hom_{cont}(\Z_p^*, \cb_p^*)$ be the space
	of definition of  {\magenta $p$-adic Mellin transform}
	
	\hskip-10pt{\Large\blue  Theorem} (\cite{Am-V}, \cite{MTT}) For an $h$-admissible measure
	$\mu$, the Mellin transform $\Lr_\mu: \Yc_p\to\C_p$ exists and has growth $o(\log^h)$ (with infinitely many zeros).	
	
	
	\
	
	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]
	\frametitle{Main Theorem.}
	Let  ${\f} $ be a Hermitian cusp eigenform of 
	degree $n\ge 2$ and of weight $\ell > 4n+2$.
	There exist  distributions  
	$\mu_{\Dr, s}$ for $s=n,\cdots, \ell-n$ 
	%for $s=1-k+n,\cdots,2n'+1-n+k$
	% a ${\CC}_{p}$--analytic function
	%$L^{(\ep  _{0})}_{(p)}$ on ${\cal Y}_{p}$ 
	with the properties:
	\par
	\medskip
	i) 
	for all pairs $(s, \chi)$ such that $s \in  {\Z}$ with
	$n\le s\le \ell-n$,  
	%and  for $s=1$ the character $\chi ^{2}$ is non--trivial,
	$$
	\int_{\Z_p^*}\chi  d\mu_{\Dr,s} = 
	%G(\chi )^{-{n}-1}
	A_{p}(s,\chi)% = \prod_{\q | p}A_{\q}(\c,s, \chi) 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\frac{{\cal D}^{*}(s,  {\f},\overline \chi )}{\Omega_{\f}}
	$$
	(under the inclusion $i_p$), with elementary factors $A_{p}(s,\chi) = \prod_{\q | p}A_{\q}(s, \chi)$
	including a finite Euler product, gaussian sums, the conductor 
	of $\chi$; the integral is a finite sum. 
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%			
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
\begin{frame}[shrink]
	\frametitle{}
	%{Main theorem  (continued) for 
	%$ \Dr(s,{\f})=\Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})$ }	
	%\begin{theo}
	(ii) if				
	$\ord_{p}
	\left((\prod_{\q|p}\prod_{i=1}^n t_{\q,i}) 
	p^{-n(n+1)}\right)=0$
	then the above distributions $\mu_{\Dr,s}$ are bounded measures, we set  $\mu_\Dr=\mu_{\Dr,s^*}$ and the  integral is defined for all continuous characters 
	$y\in\Hom(\Z_p^*, \C_p^*) =:\Yc_p$.
	%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\smallskip
	Their {\magenta  Mellin transforms} 
	$\Lr_{\mu_\Dr}(y)=
	\int_{\Z_p^*}yd \mu_\Dr$, 
	$\Lr_{\mu_\Dr}: \Yc_p\to \CC_p$, \\
	% $\Lr_{\mu_\Dr^-}(y)=$
	% $\int_{\Z_p^*}yd \mu_\Dr^-$, 
	% $\Lr_{\mu_\Dr^-}: \Yc_p\to \CC_p$,\\
	give bounded $p$-adic analytic interpolation   of the above $L$-values
	to %bounded ${\CC}_{p} $--analytic function 
	on the $\CC_p$-analytic group $\Yc_p$;
	and these distributions are related by: 
	$\ds \int_X \chi d\mu_{\Dr,s}=\int_X \chi x^{s^*-s}d\mu_\Dr^*$, $X=\Z_p^*$, 
	where $s^*=\ell-n$, $s_*=n$. 
	
	\smallskip\pause
	(iii) in the  {\magenta admissible}  case assume that $0<h \le 
	\ds \frac { s^*-s_*+1} 2= \frac{\ell+1-2n}2$, where
	$h=\ord_{p}\left((\prod_{\q|p}\prod_{i=1}^n t_{\q,i}) 
	p^{-n(n+1)}\right)>0$,
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	Then there exist $h$--admissible measures 
	$\mu_\Dr$ whose integrals 
	$\int_{\Z_p^*} \chi x_p^s d\mu_\Dr$ are
	given by $ \di i_p\left(	A_{p}(s,\chi)% = \prod_{\q | p}A_{\q}(\c,s, \chi) 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\frac{{\cal D}^{*}(s,  {\f},\overline \chi )}{\Omega_{\f}}\right)\in \CC_p$ with $A_{p}(s,\chi)$ as in (i);
	their Mellin transforms
	$\Lr_{\Dr}(y)=\int_{\Z_p^*} y d\mu_\Dr$,
	%$\Lr_{\Dr^\pm}:\Yc_p\to\CC_p$
	belong to the type
	$o(\log  x^{h}_{p})$.
	\iffalse 
	with 
	and it is given by the Mellin transform of an $h$--admissible measure;
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%Their  {\magenta Mellin transforms} 
	$\Lr_{\Dr^\pm}(y)=\int_{\Z_p^*} y d\mu^\pm_\Dr$,
	$\Lr_{\Dr^\pm}:\Yc_p\to\CC_p$
	give $p$-adic analytic interpolation  of growth  $\log_p^h(\cdot)$ of the $L$-values:   
	the values $\Lr_{\Dr^\pm}(\chi x_p^m)$ are integrals given by $ \di i_p\left(\frac
	{{\cal D}^{*}(m,  {\f},\chi )}{\Omega_{\f}}\right)\in \CC_p$.
	\fi
	\\
	\smallskip\pause
	(iv) %if $h \le  (s^{*} - s_{*} + 1)/2 = k - {n}$ then 
	the functions $\Lr_{\Dr}$ are determined by
	% the above conditions
	(i)-(iii).
	%\end{theo}
	
	\hskip-10pt{\blue\Large Remarks.} \\
	(a) Interpretation of  $s^{*}$: 
	the smallest of the "big slopes" of
	$P_{H}$ 
	
	(b) Interpretation of  $s_{*}-1$:
	the biggest of the "small slopes" of
	$P_{H}$.
	%	(c) We have  $\Dr^*(s_0,{\bf f}, \chi)=
	%	\Lambda_\Dr(s_0,{\bf f},\chi)$, 
	%	because in this case  $h^{\ell,\ell}=0$, and all the critical points $s_0$ %are	$s_{*} + 1 \le s_0\le  s^{*}$. %(no parity condition).
	
	\
	
\end{frame}
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{Eisenstein series and congruences \red  (KEY POINT!)} The (Siegel-Hermite)Eisenstein series $E_{2\ell}^{({n})}(Z)$ of weight $2\ell$, character $\det^{-\ell}$,  is defined by
	$\di
	E_{2\ell}^{({n})}(Z)=
	\sum_{g\in\Gamma_{K,\infty}^{({n})}\bs \Gamma_K^{({n})}}
	(\det g)^{\ell}j(g,Z)^{-2\ell} .
	$
	The series  converges absolutely for $\ell >{n}$. 
	Define the normalized Eisenstein series ${\cal E}_{2\ell}^{({n})}(Z)$ by
	$
	{\cal E}_{2\ell}^{({n})}(Z)=$
	$2^{-{n}}\prod_{i=1}^{{n}}L(i-2\ell, \theta^{i-1})\cdot  E_{2\ell}^{({n})}(Z)
	$
	If $H\in\Lambda_{n}(\Oc)^+$, then the $H$-th Fourier coefficient of ${\cal E}_{2\ell}^{({n})}(Z)$ is polynomial over $\Z$ in $\{p^{\ell-({n}/2)}\}_p$, and equals 
	$$
	|\gamma(H)|^{\ell-({n}/2)}
	\prod_{p|\gamma(H)}\tilde F_p(H,p^{-\ell+({n}/2)}),
	\gamma(H) = (-D_K)^{[{n}/2]} \det H.
	$$	
	Here, $\tilde F_p(H,X)$ is a certain Laurent polynomial in the variables $\{X_p=p^{-s}, X_p^{-1}\}_p$ over $\Z$. % arising from the Siegel series for $H$.
	This polynomial is a {\red key point in proving congruences} for
	the modular forms in a {\red Rankin-Selberg integral}. 
	
	\smallskip
	Also, we set , for $s\in\CC$ and a Hecke ideal character $\psi \bmod \c$,
	$$ \magenta
	E(Z,s,\ell,\psi)=
	\sum_{g\in C_\infty\bs C}
	\psi(g) (\det g)^{\ell}j(g,Z)^{-2\ell}|(\det g)j(g,Z)|^{-s} .
	$$
	
	\	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{An integral representation of Rankin-Selberg type} 
	%\iffalse	
	%Let us specialize to $F=\Q$
	The integral representation of Rankin-Selberg type in the Hermitian modular case: % is used (\cite{Bou16}, p.13):
	%"The following theorem (see also [25, Theorem 7.8]) is obtained by combining results of
	%Shimura [31] and Klosin [25]. For details we refer to [6, Section 4], 
	
	{\blue \Large Theorem 4.1 (Shimura, Klosin)}, see \cite{Bou16}, p.13.\\
	Let $0 \ne {\f} \in\Mc_\ell(C,\psi ))$ of scalar weight $\ell$, $\psi
	\bmod \c$, such that $\forall \a, {\f} |T(\a) = \lambda(\a){\f}$, and assume that $2\ell\ge n$, then there exists ${\cal T} \in S_+\cap \GL_n(K)$ and ${\cal R} \in \GL_n(K)$ such that	
	\begin{align}\nonumber
	& \Gamma ((s)) \psi(\det({\cal T}))%c_{\f} (\tau, r)
	\Zc(s + 3n/2, {\f} , \chi) =	\\ & \nonumber
	\Lambda_{\c}(s + 3n/2, \theta\psi\chi)\cdot
	%\\ \nonumber
	%&  \left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2} C_0
	%\Lambda_\c(s + 3n/2, \theta(\psi\chi)_1)\cdot
	%\\ & \nonumber
	%\left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}
	C_0 \langle {\f} , \theta_{\cal T}(\chi)
	%\tilde E_\A(x, 
	{\bf E}(\bar s + n, \ell - \ell_\theta, \chi^\rho\psi) \rangle_{C\sec},
	\end{align}
	where ${\bf E}(Z, s, \ell-\ell_\theta, \psi)_{C\sec}$ is a
	normalized
	group theoretic %adelic
	Eisenstein series with components as above of level $\c\sec$ divisible by $\c$, and weight $\ell-\ell_\theta$. 	
	Here $\langle \cdot, \cdot \rangle_{C\sec}$ is the  
	%group theoretic %adelic
	normalized
	Petersson inner product associated to the congruence subgroup 
	$C\sec$ of level $\c\sec$.
	%$D[{\e}, {\e}{\h}]$,
	% as defined for example in [30, Equation 10.9.6],	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\
	$\ds \Gamma((s))=
	%\prod_{v\in {\bf a}}
	(4\pi)^{-n(s+h)}\Gamma^\iota_n(s + h), 
	{\Gamma}^\iota_n(s) = \pi^{\frac{n(n-1)}2} \prod_{j=0}^{n-1} {\Gamma}(s-j)$,\\ 
	where $h=0	\text{ or } 1, \ 
	C_0$
	%:=\frac{[\Gamma_0(\c\sec):\Gamma]}{\sharp X}.	
	\ a subgroup index.
	%\fi
	
	\
	
\end{frame}

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\fi



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%\end{document}
\begin{frame}[shrink]
	\frametitle{}
	
\
	
\end{frame}	
\begin{frame}[shrink]
	\frametitle{}

\
	
\end{frame}	
\begin{frame}[shrink]
	\frametitle{}
	
	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lecture \Numero 3. Distributions, measures, Kummer congruences.
 Kubota-Leopoldt $p$-adic zeta function and Iwasawa algebra.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   3
\begin{frame}[shrink]
	\frametitle{\ncours Distributions, measures, Kummer congruences.}
		\vskip -0.4cm  
		{\magenta Kubota-Leopoldt $p$-adic zeta function} and {\magenta Iwasawa algebra}.
	%("Distributions, mesures, congruences de Kummer. Fonction zêta $p$-aique de Kubota-Leopoldt et l'algèbre d'Iwasawa".)	
	
	\hskip-0.5cm {\Large \blue Zeta values  and Bernoulli Numbers}	
	A key result in number theory  is the Euler product expansion of the
	Riemann zeta $\zeta(s)$ :
	$$
	\zeta(s) = \prod_p  (1 - p^{-s})^{-1}
	= \sum_{n = 1}^{\infty}
	n^{-s} \hspace{1cm} (\text{defined for }\Re(s) > 1 ).  
	$$
	\pause
	The set of arguments $s$ for which $\zeta(s)$ is defined was extended by Riemann
	to all $s \in \CC$, $s \neq 1$.
	%, and we may view $\CC$ as the group of all
	%continuous homomorphisms
	%$$
	%\CC = \Hom_{\rm cont}(\RR_+^{\times}, \CC^{\times}), ~~~~ y \longmapsto y^s  
	%$$
	%of $\RR_+^{\times}$. 
	The special values $\zeta(1 - k)$ at negative integers are
	rational numbers:
	$ 
	\di\zeta(1 - k) = -\frac{B_k}{k},  
	$
	satifying certain Kummer congruences $\bmod p^m$, 
	where $B_k$ are Bernoulli  numbers, defined by the 
	\begin{align*}&
	%"e^{Bt}" = 
	\sum_{n = 0}^{\infty } \frac{B_n t^n}{n!}
	= \frac{t e^t}{e^t - 1} ;  
	B_0=1, \ B_1=-\frac 1 2 ,  \ B_2=\frac 1 6 ,  \ B_3=B_5=\cdots=0, B_4=-\frac 1 {30},\\ & 
	\ B_6= \frac 1 {42} , 
	\ B_8=-\frac 1 {30} ,B_{10}=\frac {5} {66}, 
	B_{12}=\frac {691} {2730},\ B_{14} =-\frac 7 6 ,
	%\ B_{16}=\frac {3617} {510}, \cdots, 
	%B_{18} = -\frac {43867} {798} , \  \dots.
	{\displaystyle\z(2k)=- \frac {(2\pi i)^{2k}B_{2k}} {2(2k)!}}, 
	\end{align*}
	The denominators of $B_{k}$ are small (Sylvester-Lipschitz):   
	$ 
\ds	\forall c \in \ZZ \Longrightarrow c^k (c^k - 1) \frac{B_k}{k} \in \ZZ$\ \  (\text{see in }\cite{Mil-Sta74}),  
	%using the known formula for the sum of $k$-th powers via 
	Bernoulli polynomials 
	$\ds B_{k}(x)=\sum_{i=0}^k{k \choose
		i}B_ix^{k-i} ="(x+B)^k"$ \\
	$ 
	\ds S_k(N) = \sum_{n=1}^{N-1} n^k = \frac{1}{k+1} \left[ B_{k+1}(N) -
	B_{k+1} \right]$,   
	$B_{1}(x)=x-\frac 1 2,  B_{2}(x)=x^2-x+\frac 1 6$,
	
	\
	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{Bernoulli numbers and Kummer congruences}
	Kubota and Leopoldt constructed \cite{KuLe64} a $p$-adic interpolation of these special values, explained by Mazur via a $p$-adic measure $\mu_c$ on $\Z_p$  and Kummer congruences for the Bernoulli numbers, see \cite{Ka78} 
	($p$ is a prime number,
	$c > 1$ an integer prime to $p$). Writing the normalized values
	$$ 
	\zeta_{(p)}^{(c)}(-k) = (1 - p^k)(1 - c^{k+1}) \zeta(-k) = \int_{\Z_p^*}x^{k}d\mu_c(x)
	$$
	produces the {\magenta Kummer congruences} in the form: for any polynomial 
	$h(x) = \sum_{i = 0}^{n}\al_i x^i$ over $\Z$,
	$$ 
	\forall x\in \ZZ_p, \sum_{i = 0}^{n} \al_i x^i  \in p^m \ZZ_p\Longrightarrow \sum_{i=0}^{n} \al_i \zeta_{(p)}^{(c)}(-i) \in p^m \ZZ_p,  
	$$
	\pause
	Indeed, integrating the above polynomial $h(x)$ over 
	$\mu_c$ produces the congruences.
	%\pause
	%This property expresses the fact that the numbers $\zeta_{(p)}^{(c)}(-k)$
	%depend continuously (and even analytically) on $k$ in the $p$-adic sense; 
	The existence of $\mu_c$  is deduced from the above formula for the sum of $k$-th powers $S_k(p^r)$ for $r\to \infty$, 
	%$$ S_k(N) = \sum_{n=1}^{N-1} n^k = \frac{1}{k+1} \left[ B_{k+1}(N) -
	%B_{k+1} \right]  $$
	%in which  $B_{k}(x) = (x+B)^k = \sum_{i=0}^k \binom{k}{i} B_ix^{k-i}$
	%denotes the Bernoulli polynomial. Indeed, all summands in $S_k(N)$
	%depend $p$-adic analytically on $k$, 
	restricted to numbers $n$,
	prime to $p$.
	%The desired congruence follows if we express the numbers
	%$\zeta_{(p)}^{(c)}(-k)$ in terms of Bernoulli numbers. % (see \cite{Ka78}).
	
	%\end{frame}
	%\begin{frame}[shrink]
	%	\frametitle
	%	{\magenta Mazur's $p$-adic integral} : 
	%	there exists a $p$-adic measure 
	%	$\mu_c$ on $\Z_p^*$ , such that the special values
	%	$$
	%	\zeta(1-k)(1-p^{k-1})=%(1-\chi(c)c^{k})^{-1}
	%	{\int_{\Z_p^*}x^{k-1}d\mu_c(x) \over 1-c^{k}}
	%	\in \Q, k \ge 2\mbox{ even},  c=1+p.	$$
	\pause
	In order to define such a measure $\mu_c$  it suffices for any continuous function $\phi : \Z_p \to \Z_p$ to define its integral
	$\int_{\Z_p} \phi(x)d\mu_c$.
	Approximating $\phi (x)$ by a polynomial (when the integral is already defined),  pass to the limit (which is well defined due to Kummer congruences).
	
	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{\blue Kubota-Leopoldt $p$-adic zeta-function} 
	The domain of definition of $p$-adic zeta functions is the $p$-adic
	analytic group  $\Yc_p=Hom_{cont}(\Z_p^*, \cb_p^*)$
	of all continuous $p$-adic characters of the profinite group $\ZZ_p^{\times}$,
	where $\CC_p = \widehat{\overline{\QQ}}_p$ denotes the Tate field
	(completion of an algebraic closure of the $p$-adic field $\QQ_p)$
	(over complex numbers $\C=Hom_{cont}(\R_+^*,\C^*)$, 
	$y$ run the characters $t\mapsto t^s$ . 
	\pause
	
	Define
	$\zeta_p: \Yc_p \to \cb_p
	$ 
	on the space
	as the  {\magenta $p$-adic Mellin transform} 
	%of  $\mu_c$: for any integer $c>1$ not divisible by $p$, 
	$$
	\zeta_p(y)=%(1-cy(c))^{-1}
	{\int_{\Z_p^*}y(x)d\mu_c (x)\over 1-cy(c)}= {\Lr_{\mu_c}(y)\over 1-cy(c)},
	$$ 
	%(notice that
	%$\int(1+t)^z d\mu(z)=\sum_{n=0}^\infty\int{z\choose n}d\mu(z)$).
	\\	with a single simple pole at 
	$y=y_p^{-1}\in \Yc_p$, 
	where  
	$y_p(x)=x$ the inclusion character $\Z_p^* \hookrightarrow \C_p^*$ and $y(x)=\chi(x)x^{k-1}$  is a typical  arithmetical character ($y=y_p^{-1}$ becomes $k=0$, $s=1-k=1$). 
	
	
	\smallskip\pause
	{\large \magenta Explicitly}: Mazur's measure is given by $\mu_c(a+p^v\Z_p) = $\\
	$
	\frac {1}c\left[ \frac {ca}{p^v}\right] + 
	\frac {1-c}{2c}=\frac{1}c B_1(\{\frac {ca}{p^v}\})-B_1(\frac {a}{p^v})$,~$B_1(x)=x-\frac 1 2$,
	(\cite{LangMF}, Ch.XIII), we see the zeta distribution $\mu_s|_{s=0} (a+(N))=-B_1(\frac{a}{N})$.
	{\red \large Then the binomial formula 
		$\int_Z(1+t)^z d\mu_c= \sum_{n=0}^\infty t^n \int_Z{z\choose n}d\mu_c$, 
		gives the analyticity of $\zeta_p(y)$ on $t=y(1+p)-1$ in the unit disc
		$\{t\in \C_p\| \ |t|_p<1\}$.}
	
	%\fi
	%and given by the {\red constant term of Eisenstein series}.	
	
	\
	
\end{frame}	
\begin{frame}[shrink]
	\frametitle{The abstract Kummer congruences, \\
		$p$-adic Mellin transform and the Iwasawa algebra}
A useful criterion for the existence of a measure with given properties is:
%PROPOSITION 3.2.2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{prop}[The abstract Kummer congruences, 
	see \cite{Ka78}]\label{PROPabKum} 
	Let $\{ f_i \}$ be a system of continuous
	functions $f_i \in \Cr({\Xc_p},\Or_p)$
	in the ring $\Cr({\Xc_p},\Oc_p)$ of all continuous functions on the compact
	totally disconnected group ${\Xc_p}$ with values in the ring of integers $\Or_p$
	of $\C$ such that ${\C_p}$-linear span of $\{ f_i \}$ is dense in $\Cr({\Xc_p},{\C_p})$.
	Let also $\{ a_i \}$ be any system of elements $a_i \in {\Oc}_p$. Then the
	existence of an ${\Oc}_p$-valued measure $\mu$ on ${\Xc_p}$ with the property
	$$ \int_{\Xc_p} f_i d\mu = a_i  $$
	is equivalent to the following congruences, for an arbitrary choice of
	elements $b_i \in {\C_p}$ almost all of which vanish
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.31)
	\begin{equation}  \label{EQabKum}
	\sum_i b_i f_i(x) \in p^n {\Oc}_p \mbox{ for all } x \in {\Xc_p}
	\mbox{ implies } \sum_i b_i a_i \in p^n {\Oc}_p.
	\end{equation}
\end{prop}
%REMARK 1.8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%\begin{rema}
\hskip-10pt{\blue \large Remark.}	Since ${\C_p}$-measures are characterised as bounded ${\C_p}$-valued distributions,
	every ${\C_p}$-measures on $Y$ becomes a ${\Oc}_p$-valued measure after
	multiplication by some non-zero constant.
	
	
	\
	
	
%\end{rema}
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=2]
	\frametitle{Proof of proposition \ref{PROPabKum}.}
	The necessity is obvious since
	$$ \begin{array}{rcccl}
	\di{\sum_i b_i a_i}  &  =  &  \di{\int_{\Xc_p} (p^n {\Oc}_p-
		\mbox{valued function})d\mu}  &  =  &  \\  &  =   &
	\di{p^n \int_{\Xc_p} ({\Oc}_p- \mbox{valued function})d\mu}  &  \in  &
	p^n {\Oc}_p.
	\end{array}  $$
	In order to prove the sufficiency we need to construct a measure $\mu$ from
	the numbers $a_i$. For a function $f \in \Cr({\Xc_p}, {\Oc}_p)$ and a positive
	integer $n$ there exist elements $b_i \in {\C}$ such that only a finite
	number of $b_i$ does not vanish, and
	$$ f - \sum_i b_i f_i \in p^n \Cr({\Xc_p}, {\Oc}_p),  $$
	according to the density of the ${\C}$-span of $\{ f_i \}$ in $\Cr({\Xc_p},{\C})$.
	By the assumption (\ref{EQabKum}) the value $\sum_i a_i b_i$ belongs to
	${\Oc}_p$ and is well defined modulo $p^n$ (i.e. does not depend on the choice
	of $b_i$). Following N.M. Katz (\cite{Ka78}), we denote this value by
	``$\int_{\Xc_p} f d\mu \bmod p^n$ ''. Then we have that the limit procedure
	$$ \int_{\Xc_p} f d\mu = \lim_{n \rightarrow \infty}
	\mbox{``$\di{\int_{\Xc_p} f d\mu \bmod p^n}$ ''}
	\in \lim_{\stackrel{\longleftarrow}{n}} {\Oc}_p/p^n{\Oc}_p = {\Oc}_p,  $$
	gives the measure $\mu$.
	
	\
	
	
	%\end{proof}
	%'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
	%-----------------------------PARAGRAPHE 1.3.3.-------------------------------
	%\subsection{The $S$-adic Mazur measure}
\end{frame}%[shrink=2]
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Mazur's measure over $\Xc_S=\Z_S$}
	
	Let  $c > 1$ be a positive integer coprime to
	$ \ds M_0 = \prod_{q \in S} q  $
	with $S$ being a fixed set of primes containing $p$. Using the criterion of the	proposition \ref{PROPabKum} we show that the $\QQ$ -valued distribution defined
	by the formula
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.33)
	\begin{equation}  \label{EQmesEk}
	E_k^c(f) = E_k(f) - c^k E_k(f_c), ~~~~ f_c(x) = f(cx),
	\end{equation}
	turns out to be a measure where $E_k(f)$ are defined in \cite{LangMF},
	$f \in \Step(\Xc,\Q_p)$ and the field $\QQ$ is viewed as a subfield of $\C_p$.
	
	Define the generalized Bernoulli polynomials $B_{k,f}^{(M)}(X)$ as
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.33)
	\begin{equation}  \label{EQpolyBer}
	\sum_{k=0}^{\infty} B_{k,f}^{(M)}(X) \frac{t^k}{k!} =
	\sum_{a=0}^{M-1} f(a) \frac{t e^{(a+X)t}}{e^{Mt}-1} ,
	\end{equation}
	and the generalized sums of powers
	$$ S_{k,f}(M) = \sum_{a=0}^{M-1} f(a) a^k.  $$
	
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=5]
	Then the definition (\ref{EQpolyBer}) formally implies that
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.34)
	\begin{equation}  \label{EQsumpolyBer}
	\frac{1}{k} [ B_{k,f}^{(M)}(M) -  B_{k,f}^{(M)}(0)] = S_{k-1,f}(M),
	\end{equation}
	and also we see that
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.35)
	\begin{equation}  \label{EQBerbino}
	B_{k,f}^{(M)}(X) = \sum_{i=0}^k \binom{k}{i} B_{i,f} X^{k-i} =
	B_{k,f} + kB_{k-1,f}X + \cdots + B_{0,f}X^k.
	\end{equation}
	The last identity can be rewritten symbolically as
	$$ B_{k,f}(X) = (B_f + X)^k.  $$
	
	The equality (\ref{EQsumpolyBer}) enables us to calculate the (generalized)
	sums of powers in terms of the (generalized) Bernoulli numbers. In particular
	this equality implies that the Bernoulli numbers $B_{k,f}$ can be obtained by
	the following $p$-adic limit procedure (see \cite{LangMF}):
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.36)
	\begin{equation}  \label{EQBerlim}
	B_{k,f} = \lim_{n \rightarrow \infty} \frac{1}{Mp^n} S_{k,f}(Mp^n)
	~~~~(\mbox{a }p\mbox{-adic limit}),
	\end{equation}
	where $f$ is a $\C_p$-valued function on ${\Xc_p} = \ZZ_S$. Indeed, if we replace $M$
	in (\ref{EQsumpolyBer}) by $Mp^n$ with growing $n$ and let $D$ be the common
	denominator of all coefficients of the polynomial $B_{k,f}^{(M)}(X)$. Then we
	have from (\ref{EQBerbino}) that
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.37)
	\begin{equation}  \label{EQBermod}
	\frac{1}{k} \left[ B_{k,f}^{(Mp^n)}(M) - B_{k,f}^{(M)}(0) \right]
	\equiv B_{k-1,f}(Mp^n) \pmod{\frac{1}{kD} p^2n}.
	\end{equation}
	The proof of (\ref{EQBerlim}) is accomplished by division of (\ref{EQBermod})
	by $Mp^n$ and by application of the formula (\ref{EQsumpolyBer}).
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=2]
	
	Now we can directly show that the distribution $E_k^c$ defined by
	(\ref{EQmesEk}) are in fact bounded measures. If we use (\ref{EQabKum}) and
	take the functions $\{f_i\}$ to be all of the functions in $\Step({\Xc_p},\Or_p)$.
	Let $\{b_i\}$ be a system of elements $b_i \in \C_p$ such that for all $x \in {\Xc_p}$
	the congruence
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.38)
	\begin{equation}
	\sum_i b_i f_i(x) \equiv 0 \pmod{p^n}
	\end{equation}
	holds. Set $f = \sum_i b_i f_i$ and assume (without loss of generality) that
	the number $n$ is large enough so that for all $i$ with $b_i \neq 0$ the
	congruence
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (1.39)
	\begin{equation}
	B_{k,f_i} \equiv \frac{1}{M p^n} S_{k,f_i} (M p^n) \pmod{p^n}
	\end{equation}
	is valid in accordance with (\ref{EQBerlim}). Then we see that
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.40)
	\begin{equation}
	B_{k,f} \equiv (Mp^n)^{-1} \sum_i \sum_{a=0}^{Mp^n-1} b_i f_i(a) a^k
	\pmod{p^n},
	\end{equation}
	hence we get by definition (\ref{EQmesEk}):
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.41)
	\begin{eqnarray}     E_k^c(f)  & = &\label{EQmesEkBer}  B_{k,f} - c^k B_{k,f_c}  \\   & \equiv  &
	(Mp^n)^{-1} \sum_i \sum_{a=0}^{Mp^n-1} b_i
	\left[f_i(a) a^k - f_i(ac)(ac)^k \right] \pmod{p^n}.  \nonumber
	\end{eqnarray}
	
	
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=5] 
	Let $a_c \in \{ 0, 1, \cdots, Mp ^n-1 \}$, such that $a_c \equiv ac
	\pmod{Mp^n}$, then the map $a \longmapsto a_c$ is well defined and acts as a
	permutation of the set $\{0, 1, \cdots, Mp^n-1\}$, hence (\ref{EQmesEkBer}) is
	equivalent to the congruence
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%    (1.42)
	\begin{equation}  \label{EQmesEkBermod}
	E_k^c(f) = B_{k,f} - c^k B_{k,f_c} \equiv \sum_i \frac{a_c^k - (ac)^k}
	{Mp^n} \sum_{a=0}^{Mp^n-1} b_i f_i(a) a^k \pmod{p^n}.
	\end{equation}
	Now the assumption (\ref{EQBermod}) formally inplies that
	$E_k^c(f) \equiv 0 \pmod{p^n}$, completing the proof of the abstact
	congruences and the construction of measures $E_k^c$.
	
	%REMARK 1.9  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	\begin{rema}
		The formula $(\ref{EQmesEkBer})$ also implies that for all $f \in \Cr({\Xc_p},\C_p)$
		the following holds
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.43)  
		\begin{equation}  \label{EQEkf}
		E_k^c(f) = k E_1^c(x_p^{k-1}f)
		\end{equation}
		where $x_p : {\Xc_p} \longrightarrow \C_p \in \Cr({\Xc_p},\C_p)$ is the composition of
		the projection ${\Xc_p} \longrightarrow \Z_p$ and the embedding $\Z_p
		\hookrightarrow \C_p$.
	\end{rema}
	
	Indeed if we put $a_c = ac + Mp^nt$ for some $t \in \ZZ$ then we see that
	$$ a_c^k - (ac)^k = (ac + Mp^nt)^k - (ac)^k \equiv kMp^nt(ac)^{k-1}
	\pmod{(Mp^n)^2},  $$
	and we get that in (\ref{EQmesEkBermod}):
	$$ \frac{a_c^k - (ac)^k}{Mp^n} \equiv k(ac)^{k-1} \frac{a_c - ac}{Mp^n}
	\pmod{Mp^n}.  $$
	The last congruence is equivalent to saying that the abstract Kummer
	congruences (\ref{EQabKum}) are satisfied by all functions of the type
	$x_p^{k-1} f_i$ for the measure $E_1^c$ with $f_i \in \Step({\Xc_p},\C_p)$
	establishing the identity (\ref{EQEkf}).
\end{frame}%[shrink=2]
%\iffalse
%*************************************************************************
%---------------------------SECTION 1.4.------------------------------------
%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%-------------------------PARAGRAPHE 1.4.1.----------------------------------
%\subsection{Iwasawa algebra and the non-Archimedean Mellin transform}
%\label{SECIwasawa}
%\begin{frame}%[shrink=3]
%\frametitle{
%Distributions, measures, and the abstract Kummer congruences.
%$p$-adic $L$-functions,  Iwasawa algebra,  Mellin transform. 
% and relations with modular forms}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=5]
	\frametitle{The domain of definition  of the  non-Archimedean zeta functions}
	
	In the classical case the set on which zeta functions are defined is the set of
	complex numbers $\CC$ which may be viewed equally as the set of all
	continuous characters (more precisely, quasicharacters) via the following
	isomorphism:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.44)  
	\begin{eqnarray}
	\CC  &  \stackrel{\sim}{\longrightarrow}  &  \Hom_{\cont} (\RR_+^{\times},
	\CC^{\times})  \\  \nonumber
	s    &  \longmapsto                       &  (x \longmapsto x^s)
	\end{eqnarray}
	
	\pause
	The construction which associates to a function $h(x)$ on $\RR_+^{\times}$
	(with certain growth conditions as $x \rightarrow \infty$ and
	$x \rightarrow 0$) the following integral
	$$
	L_h(s) = \int_{\RR_+^{\times}} h(x) x^s \frac{dx}{x}  
	$$
	(which converges probably not for all values of $s$) is called
	the {\it Mellin transform}\index{Mellin transform}.
	
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=5]
	\frametitle{The case of the Riemann
		zeta function}
	For example, if $\zeta(s) = \sum_{n \geq 1} n^{-s}$ 
%	is the Riemann zeta function, 
	then the function $\zeta(s) \Ga(s)$ is the Mellin transform
	of the function $h(x) = 1/(1-e^{-x})$:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.45)  
	\begin{equation}  \label{EQtransMell}
	\zeta(s) \Ga(s) = \int_0^{\infty} \frac{1}{1-e^{-x}} x^s \frac{dx}{x},
	\end{equation}
	so that the integral and the series are absolutely convergent for $\Re(s) > 1$.
	For an arbitrary function of type
	$\ds  f(z) = \sum_{n=1}^{\infty} a(n) e^{2i\pi n z}  $
	with $z = x + iy \in \HH$ in the upper half plane $\HH$ and with the growth
	condition $a(n) = \Or(n^c)$ ($c>0$) on its Fourier coefficients, we see that
	the zeta function
	$ \ds
	L(s,f) = \sum_{n=1}^{\infty} a(n) n^{-s} ,  
	$
	essentially coincides with the Mellin transform of $f(z)$, that is, for
	$\Re(s) > 1+c$
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.46)  
	\begin{equation}  \label{EQgaLMel} 
	\frac{\Ga(s)}{(2\pi)^s} L(s,f) = \int_0^{\infty} f(iy) y^s \frac{dy}{y}
	\text{ using }\Ga(s) = \int_0^{\infty} e^{-y} y^s \frac{dy}{y}
	\end{equation}
\end{frame} 
%[shrink=2]

\iffalse
\begin{frame}%[shrink=5]  
	
	Both sides of the equality (\ref{EQgaLMel}) converge absolutely for
	$\Re(s) > 1+c$. The identities (\ref{EQtransMell}) and (\ref{EQgaLMel}) are
	immediately deduced from the well known integral representation for
	the gamma-function
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.47)  
% \label{EQgam}
$	\Ga(s) = \int_0^{\infty} e^{-y} y^s \frac{dy}{y},
$
	where $\frac{dy}{y}$ is a measure on the group $\RR_+^{\times}$ which is
	invariant under the group translations (Haar measure). The integral
	(\ref{EQgam}) is absolutely convergent for $\Re(s) > 0$ and it can be
	interpreted as the integral of the product of an additive character
	$y  \mapsto e^{-y}$ of the group $\RR^{(+)}$ restricted to
	$\RR_+^{\times}$, and of the multiplicative character $y \mapsto y^s$,
	integration is taken with 
	respect to the Haar measure $\dd y/y$ on the group $\RR_+^{\times}$.
	
\
	
\end{frame} 
\fi
\begin{frame}[shrink]
	\frametitle{$p$-adic Mellin transform}
	In the theory of the non-Archimedean integration one considers the group
	$\ZZ_S^{\times}$ (the group of units of the $S$-adic completion of the ring of
	integers $\ZZ$) instead of the group $\RR_+^{\times}$, and the Tate field
	$\C_p = \hat{\Qb}_p$ (the completion of an algebraic closure of $\Q_p$)
	instead of the complex field $\CC$. The domain of definition of the $p$-adic
	zeta functions is the $p$-adic analytic group
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.48)  
	\begin{equation}
	{\Yc}_S = \Hom_{\cont}(\ZZ_S^{\times},\C_p^{\times}) = {\Yc}(\ZZ_S^{\times}),
	\end{equation}
	where:
	$$
	\ZZ_S^{\times} \cong \oplus_{q \in S} \ZZ_q^{\times},  
	$$
	and the symbol
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.49)  
	\begin{equation}
	{\Yc}(G) = \Hom_{\cont}(G,\C_p^{\times})
	\end{equation}
	denotes the functor of all $p$-adic characters of a topological group $G$.
%	(see \cite{Vis76}).
\end{frame}
%[shrink=2]
%\iffalse
%\subsection{The analytic structure of $\Yc_S$ and
%	$p$-adic Mellin transform}
\begin{frame}[shrink=8]
	\frametitle{The analytic structure of $\Yc_S$}
	Let us consider in detail the structure of the topological group ${\Yc}_S$.
	Define
	$$ U_p = \{ x \in \Z_p^{\times} ~~\mid~~ x \equiv 1 \pmod{p^{\nu}} \},  $$
	where $\nu = 1$ or $\nu = 2$ according as $p > 2$ or $p = 2$. Then we have the
	natural decomposition
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.50)  
	\begin{equation}  \label{EQdecdeXS}
	{\Yc}_S = {\Yc} \left( (\ZZ/p^{\nu}\ZZ)^{\times} \times \prod_{q \neq p}
	\ZZ_q^{\times} \right) \times {\Yc}(U_p).
	\end{equation}
	The analytic structure on ${\Yc}(U_p)$ is defined by the following isomorphism
	(which is equivalent to a special choice of a local parameter):
	$$ \varphi : {\Yc}(U_p) \stackrel{\sim}{\longrightarrow} T =
	\{ z \in \C_p^{\times} ~~\mid~~ |z-1|_p < 1 \},  $$
	where $\varphi(x) = x(1+p^{\nu})$, $1+p^{\nu}$ being a topoplogical generator
	of the multiplicative group $U_p \cong \Z_p$. An arbitrary character
	$\chi \in {\Yc}_S$ can be uniquely represented in the form $\chi = \chi_0 \chi_1$
	where $\chi_0$ is trivial on the component $U_p$, and $\chi_1$ is trivial
	on the other component
	$$
	(\ZZ/p^{\nu}\ZZ)^{\times} \times \prod_{q\neq p} \ZZ_q^{\times}.  
	$$ 
\end{frame}
%\iffalse
\begin{frame}[shrink=5]  
	The character $\chi_0$ is called the
	{\it tame component}\index{tame component}, and $\chi_1$ the
	{\it wild component}\index{wild component} of the character $\chi$. We
	denote by the symbol $\chi_{(t)}$ the
	(wild) character which is uniquely determined by the condition
	$$ \chi_{(t)}(1+p^{\nu}) = t  $$
	with $t \in \C_p$, $|t|_p < 1$.
	
	In some cases it is convenient to use another local coordinate $s$ which is
	analogous to the classical argument $s$ of the Dirichlet series:
	$$ \begin{array}{rcl}
	\Oc_p  &  \longrightarrow  &  \Yc_S  \\
	s      &  \longmapsto      &  \chi^{(s)} ,
	\end{array}  $$
	where $\chi^{(s)}$ is given by
	$\chi^{(s)}((1+p^{\nu})^{\al}) = (1+p^{\nu})^{\al s} =
	\exp(\al s \log(1+p^{\nu}))$.
	The character $\chi^{(s)}$ is defined only for such $s$ for which the series
	$\exp$ is $p$-adically convergent (i.e. for $|s|_p < p^{\nu-1/(p-1)}$).
	In this domain of values of the argument we have that $t = (1+p^{\nu})^s-1$.
	But, for example, for $(1+t)^{p^n} = 1$ there is certainly no such value of
	$s$ (because $t \neq 1$), so that the $s$-coordonate parametrizes a
	smaller neighborhood of the trivial character than the $t$-coordinate (which
	parametrizes all wild characters) (see \cite{Ma73}, \cite{Ma76}).
\end{frame}
%[shrink=2]
%\iffalse
%\subsection{$p$-adic Mellin transform}
\begin{frame}[shrink=5]
%	\frametitle{$p$-adic analytic structure on ${\Yc}_S$}
	Recall that an analytic function $F : T \longrightarrow \C_p$ 
	($T = \{ z \in \C_p^{\times} ~~\mid~~ |z-1|_p < 1 \}$), is defined as the
	sum of a series of the type $\sum_{i \geq 0} a_i (t-1)^i$ ($a_i \in \_Cp$),
	which is assumed to be absolutely convergent for all $t \in T$. The notion of
	an analytic function is then obviously extended to the whole group ${\Yc}_S$ by
	shifts. The function
	$$ F(t) = \sum_{i=0}^{\infty} a_i (t-1)^i  $$
	is bounded on $T$ iff all its coefficients $a_i$ are universally bounded. This
	last fact can be easily deduced for example from the basic properties of the
	Newton polygon of the series $F(t)$ (see \cite{Ko80}, \cite{Am-V}). 
	If we apply to these series the Weierstrass preparation theorem
	(see  \cite{Ko80}, \cite{Ma73}), % and theorem \ref{THEOWeier})
	we see that in this case the function $F$ has only a finite number of zeroes
	on $T$ (if it is not identically zero). 
\end{frame} 
%[shrink=2]

%\iffalse
\begin{frame}
	\frametitle{$p$-adic analytic functions on ${\Yc}_S$}
Consider the torsion
	subgroup ${\Yc}_S^{\tors} \subset {\Yc}_S$. This subgroup is discrete in ${\Yc}_S$ and its
	elements $\chi \in {\Yc}_S^{\tors}$ can be obviously identified with primitive
	Dirichlet characters $\chi \mod M$ such that the support $S(\chi) = S(M)$ of
	the conductor of $\chi$ is containded in $S$. This identification is provided
	by a fixed embedding denoted
	$$
	i_p : \Qb^{\times} \hookrightarrow \C_p^{\times}  
	$$
	if we note that each character $\chi \in {\Yc}_S^{\tors}$ can be factored through
	some finite factor group $(\ZZ/M\ZZ)^{\times}$:
	$$
	\chi : \ZZ_S^{\times} \rightarrow (\ZZ/M\ZZ)^{\times}
	\rightarrow  \Qb^{\times} \stackrel{i_p}{\hookrightarrow}
	\C_p^{\times},  
	$$
	and the smallest number $M$ with the above condition is the conductor of
	$\chi \in {\Yc}_S^{\tors}$.
	
	The symbol $x_p$ will denote the composition of the natural projection
	$\ZZ_S^{\times} \rightarrow \Z_p^{\times}$ and of the natural embedding
	$\Z_p^{\times} \rightarrow \C_p^{\times}$, so that $x_p \in {\Yc}_S$ and all
	integers $k$ can be considered as the characters $x_p^k : y \longmapsto y^k$.
\end{frame}
\begin{frame}[shrink=5]
%	\frametitle{$p$-adic Mellin transform}
	Let us consider a bounded $\C_p$-analytic function $F$ on ${\Yc}_S$. The above
	statement about zeroes of bounded $\C_p$-analytic functions implies now that
	the function $F$ is uniquely determined by its values $F(\chi_0 \chi)$, where
	$\chi_0$ is a fixed character and $\chi$ runs through all elements
	$\chi \in {\Yc}_S^{\tors}$ with possible exclusion of a finite number of
	characters in each analyticity component of the decomposition
	(\ref{EQdecdeXS}). This condition is satisfied, for example, by the set of
	characters $\chi \in {\Yc}_S^{\tors}$ with the $S$-complete conductor
	(i.e. such that $S(\chi) = S$), and even for a smaller set of characters,
	for example for the set obtained by imposing the additional
	assumption that the character $\chi^2$ is not trivial (see \cite{Ma73}
	%,\cite{Man76}, \cite{Vis76}
	).
%\end{frame}
%[shrink=2]
%\iffalse
%\subsection{The analytic structure of ${\Yc}_S$}
%\begin{frame}[shrink=5]
	%[shrink=2]
	%\iffalse
%	\frametitle
%	{\blue \large $p$-adic Mellin transform}
	Let $\mu$ be a (bounded) $\C_p$-valued measure on $\ZZ_S^{\times}$. Then the
	{\it non-Archimedean Mellin transform}\index{non-Archimedean Mellin transform}
	of the measure $\mu$ is defined by
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.51)  
	\begin{equation}  \label{EQMeltrans}
	L_{\mu}(x) = \mu(x) = \int_{\ZZ_S^{\times}} x \dd \mu, ~~~~ (x \in {\Yc}_S), 
	\end{equation}
	which represents a bounded $\C_p$-analytic function
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.52)  
	\begin{equation}
	L_{\mu} : {\Yc}_S \longrightarrow \C_p.
	\end{equation}
	Indeed, the boundedness of the function $L_{\mu}$ is obvious since all
	characters $x \in {\Yc}_S$ take values in $\Or_p$ and $\mu$ also is bounded. The
	analyticity of this function expresses a general property of the integral
	(\ref{EQMeltrans}), namely that it depends analytically on the parameter
	$x \in {\Yc}_S$. However, we give below a pure algebraic proof of this fact
	which is based on a description of the Iwasawa algebra. This description
	will also imply that every bounded $\C_p$-analytic function on ${\Yc}_S$ is
	the Mellin transform of a certain measure $\mu$.
	
	\
	
\end{frame}
%\iffalse
%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%----------------------------PARAGRAPHE 1.4.4.-------------------------------
%\subsection{The Iwasawa algebra}
\begin{frame}
	\frametitle{The Iwasawa algebra}
	
	Let ${\Oc}$ be a closed subring in
	${\Oc}_p = \{ z \in \C_p ~~\mid~~ |z|_p \leq 1 \}$,
	$$
	G = \lim_{\stackrel{\longleftarrow}{i}} G_i, ~~~~ (i \in I),  
	$$
	a profinite group. 
	Then the canonical homomorphism
	$G_i \stackrel{\pi_{ij}} {\longleftarrow} G_j$
	induces a homomorphism of the corresponding group rings
	$$ 
	{\Oc}[G_i] \longleftarrow {\Oc}[G_j].  
	$$
	Then the {\it completed group ring}\index{completed group ring} ${\Oc}[[G]]$
	is defined as the projective limit
	$$
	{\Oc}[[G]] = \lim_{\stackrel{\longleftarrow}{i}} {\Oc}[[G_i]],
	~~~~(i \in I).  
	$$
	Let us consider also the set ${\rm Dist}(G,{\Oc})$ of all ${\Oc}$-valued distributions
	on $G$ which itself is an ${\Oc}$-module and a ring with respect to
	multiplication given by the
	{\it convolution of distributions}\index{convolution of distributions},
	which is defined in terms of families of functions
	$$ \mu_1^{(i)}, \mu_2^{(i)} : G_i \longrightarrow {\Oc},  $$
	%$
	(see the previous section) as follows:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.53)  
	\begin{equation}
	(\mu_1 \star \mu_2)^{(i)}(y) = \sum_{y=y_1y_2} \mu_1^{(i)}(y_1)
	\mu_2^{(i)}(y_2), ~~~~ (y_1,y_2 \in G_i)
	\end{equation}
	Recall also that the ${\Oc}$-valued distributions are identified with
	${\Oc}$-valued measures. Now we describe an isomorphism of ${\Oc}$-algebras
	${\Oc}[[G]]$ and ${\rm Dist}(G,{\Oc})$. In this case when $G=\Z_p$ the algebra
	${\Oc}[[G]]$ is called the {\it Iwasawa algebra}\index{Iwasawa algebra}.
	
	%THEOREM 1.10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	\begin{theorem}  \label{THEOisomIsa}
		(a) Under the same notation as above there is the canonical isomorphism of
		${\Oc}$-algebras
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.54)  
		\begin{equation}
		;{\rm Dist}(G,{\Oc}) \stackrel{\sim}{\longrightarrow} {\Oc}[[G]].
		\end{equation}
		
		(b) If $G = \Z_p$ then there is an isomorphism
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.55)  
		\begin{equation}  \label{EQisomOGOX}
		{\Oc}[[G]] \stackrel{\sim}{\longrightarrow} {\Oc}[[X]],
		\end{equation}
		where ${\Oc}[[X]]$ is the ring of formal power series in $X$ over ${\Oc}$. 
		The
		isomorphism (\ref{EQisomOGOX}) depends on a choice of the topological
		generator of the group $G = {\ZZ_p}$.
	\end{theorem}
\end{frame}
%\iffalse
\begin{frame}[shrink=5]
	
	%'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
	%---------------------------PARAGRAPHE 1.4.5.---------------------------------
	
	We noticed above that the theorem \ref{THEOisomIsa} would imply a description
	of $\C_p$-analytic bounded functions on ${\Yc}_S$ in terms of measures. Indeed,
	these functions are defined on analyticity components of the decomposition
	(\ref{EQdecdeXS}) as certain power series with $p$-adically bounded
	coefficients, that is, power series, whose coefficients belong to ${\Oc}_p$ after
	multiplication by some constant from ${\CC_p}^{\times}$. Formulas for
	coefficients of these series can be also deduced from the proof of the theorem.
	However, we give a more direct computation of these coefficients in terms of
	the corresponding measures. Let us consider the component $aU_p$ of the set
	$\ZZ_S^{\times}$ where
	$$ a \in (\ZZ/p^{\nu}\ZZ)^{\times} \times \prod_{q\neq} \ZZ_q^{\times},  $$
	and let $\mu_a(x) = \mu(ax)$ be the corresponding measure on $U_p$ defined by
	restriction of $\mu$ to the subset $aU_p \subset \ZZ_S ^{\times}$. 
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\iffalse
\begin{frame}[shrink=5]
	Consider the
	isomorphism $U_p \cong {\Z_p}$ given by:
	$$ y = \ga^x ~~(x \in {\Z_p}, y \in U_p),  $$
	with some choice of the generator $\ga$ of $U_p$ (for example, we can take
	$\ga = 1+p^{\nu}$). Let $\mu'_a$ be the corresponding measure on ${\Z_p}$. 
	Then
	this measure is uniquely determined by values of the integrals
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.56)  
	\begin{equation}  \label{EQintmu}
	\int_{{\Z_p}} \binom{x}{i} d\mu'_a(x) = a_i,
	\end{equation}
	with the interpolation polynomials $\binom{x}{i}$, since the ${\C_p}$-span of the
	family
	$$ \left\{ \binom{x}{i} \right\} ~~~~ (i \in \ZZ, i \geq 0)  $$
	is dense in $\Cr({\Z_p}, {\Oc}_p)$ according to Mahler's interpolation theorem
	for continuous functions on ${\Z_p}$).
	Indeed, from the basic properties of the interpolation polynomials
	it follows that
	$$ \sum_i b_i \binom{x}{i} \equiv 0 \pmod{p^n} ~~~~ (\mbox{for all }
	x \in {\Z_p}) \Longrightarrow b_i \equiv 0 \pmod{p^n}.  $$
	We can now apply the abstract Kummer congruences (see proposition
	\ref{PROPabKum}), which imply that for arbitrary choice of numbers
	$a_i \in {\Oc}_p$ there exists a measure with the property (\ref{EQintmu}).


\
	
\end{frame}
%\iffalse
\begin{frame}[shrink]
	\frametitle{Coefficients of power series and the Iwasawa isomorphism}
	We state that the Mellin transform $L_{\mu_a}$ of the measure
	$\mu_a$ is given by the power series $F_a(t)$ with coefficients as in 
	(\ref{EQintmu}), that is
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.57)  
	\begin{equation}  \label{EQchitsum}
	\int_{U_p} \chi_{(t)}(y) \dd \mu(ay) = \sum_{i=0}^{\infty} \left(
	\int_{{\Z_p}} \binom{x}{i} \dd \mu_a'(x) \right) (t-1)^i
	\end{equation}
	for all wild characters of the form $\chi_{(t)}$, $\chi_{(t)}(\ga) = t$,
	$|t - 1|_p<1$. It suffices to show that (\ref{EQchitsum}) is valid for all
	characters of the type $y \longmapsto y^m$, where $m$ is a positive integer.
	In order to do this we use the binomial expansion
	$$ \ga^{mx} = \left( 1 + (\ga^m - 1) \right)^x = \sum_{i=0}^{\infty}
	\binom{x}{i} (\ga^m - 1)^i,  $$
	which implies that
	$$ \int_{u_p} y^m \dd \mu(ay) = \int_{{\Z_p}} \ga^{mx} \dd \mu_a'(x) =
	\sum_{i=0}^{\infty} \left( \int_{{\Z_p}} \binom{x}{i} \dd \mu_a'(x) \right)
	(\ga^m-1)^i,  $$
	establishing (\ref{EQchitsum}).
	
	\
	
\end{frame}
%\iffalse

%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%-------------------------------PARAGRAPHE 1.4.6.----------------------------
\iffalse
\subsection{Example: Mazur's measure and the non-Archimedean
	Kubota-Leopoldt zeta function}
\begin{frame}[shrink=5]
	\frametitle{Example: Mazur's measure and the non-Archimedean
		Kubota-Leopoldt zeta function}
	%(see \cite{Lan76}, \cite{Kub-Leo64}, \cite{Leo75}, \cite{Was82}). 
	Let us first
	consider a positive integer $c \in \ZZ_S^{\times} \cap \ZZ$, $c>1$ coprime to
	all primes in $S$. Then for each complex number $s \in \CC$ there exists a
	complex distribution $\mu_s^c$ on $G_S = \ZZ_S^{\times}$ which is uniquely
	determined by the following condition
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.58)  
	\begin{equation}  \label{EQmusc}
	\mu_s^c(\chi) = (1 - \chi^{-1}(c) c^{-1-s}) L_{M_0}(-s,\chi),
	\end{equation}
	where $M_0 = \prod_{q \in S} q$.
	% (see \ref{SSECdist}). 
	Moreover, the right hand
	side of (\ref{EQmusc}) is holomorphic for all $s \in \CC$ including $s = -1$.
	If $s$ is an integer and $s \geq 0$ then according to criterion of proposition
	\ref{PROPabKum} the right hand side of (\ref{EQmusc}) belongs to the field
	$$ \QQ(\chi) \subset \QQ^{\rm ab} \subset \Qb  $$
	generated by values of the character $\chi$.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}%[shrink=3]
	Thus we get a distribution with
	values in $\QQ^{\rm ab}$. 
	If we now apply to (\ref{EQmusc}) the fixed embedding
	$i_p : \Qb \hookrightarrow {\C_p}$ we get a ${\C_p}$-valued distribution
	$\mu^{(c)} = i_p(\mu_0^c)$ which turns out to be an ${\Oc}_p$-measure in
	view of proposition \ref{PROPabKum}, and the following equality holds
	$$ \mu^{(c)} (\chi x_p^r) = i_p (\mu_r^c(\chi)).  $$
	This identity relates the special values of the Dirichlet $L$-functions at
	different non-positive points. 
	The function
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.59)  
	\begin{equation}
	L(x) = \left(1 - c^{-1} x(c)^{-1} \right)^{-1} L_{\mu^{(c)}}(x)
	~~~~ (x \in X_S)
	\end{equation}
	is well defined and it is holomorphic on $X_S$ with the exception of a simple
	pole at the point $x = x_p \in X_S$. This function is called the
	{\it non-Archimedean zeta-function of Kubota-Leopoldt}\index{non-Archimedean
		zeta function of Kubota-Leopoldt}. The corresponding measure
	$\mu^{(c)}$ will be called the
	{\it $S$-adic Mazur measure}\index{$S$-adic Mazur measure}.
\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{frame}[shrink]
	\frametitle{} 
	\
	
\end{frame} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{} 
	
	\
	
\end{frame} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{} 
	
	\
	
\end{frame} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lecture \Numero 4. $p$-adic $L$-functions on classical groups. Ordinary case. Admissible measures, special values.}
\begin{frame}[shrink]
	\frametitle{\ncours $p$-adic $L$-functions on classical groups.}
	\vskip -0.5cm  
	Ordinary case. Admissible measures, special values.
	("Fonctions $L$ $p$-adiques sur les groupes classiques : cas ordinaire, mesures admissibles, valeurs spéciales").\\
	\hskip-10pt{\Large \blue Admissible measures: Definition.} 
	Let $M$ be a $\Oc$-module of finite rank where $\Oc\subset \C_p$.
	For $h\ge 1$, consider the following $\C_p$-vector spaces of functions on $\Z_p^*$ : $\Cc^h\subset \Cc^{loc-an}\subset \Cc$.
	Then
	
	- a continuous homomorphism $\mu: \Cc\to M$  is called a {\magenta (bounded) measure} $M$-valued measure on $\Z_p^*$.
	
	- $\mu: \Cc^h\to M $ is called an {\magenta $h$ admissible measure} $M$-valued measure on $\Z_p^*$ measure
	if the following growth condition  is satisfied	
	$$
	\left |\int_{a+(p^v)}(x-a)^j d\mu\right|_p \le p^{-v(h-j)}
	$$	 
	for $j=0,1,\cdots,h-1$, and et  $\Yc_p=Hom_{cont}(\Z_p^*, \cb_p^*)$ be the space
	of definition of  {\magenta $p$-adic Mellin transform.}
	
	\hskip-10pt{\Large\blue  Theorem} (\cite{Am-V}, \cite{MTT}) For an $h$-admissible measure
	$\mu$, the Mellin transform $\Lr_\mu: \Yc_p\to\C_p$ exists and has growth $o(\log^h)$ (with infinitely many zeros).	

 	\
 	
 \end{frame}		
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 		
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Complex and $p$-adic $L$-functions on classical groups.}


	\begin{itemize}
	
	
\item	{\magenta Automorphic forms and their weights. Complex analytic weight space. Motivic weights, introduction \cite{EHLS}}.
	
 

\item{Main result stated with Hodge/Newton polygons of $\Dr(s)$}

\item	{Main Theorem in the Hermitian case}	


\item Appendix 1. Zeta Functions, $L$-Functions and Motives.
\cite{MaPa}	

\item Appendix 2. Automorphic $L$-functions (A.Borel, \cite{Bor79})



	\end{itemize}		
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=3]
	\frametitle{Modular forms as a  tool in arithmetic} 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{tabular}{ll}
		%\hskip-0.3cm
		\pause
		\begin{tabular}{l}  
			We view modular forms as:\\
			\ 
			
			1) {\blue $q$-power series}
			\\ 
			$\ds f=\sum_{n=0}^\infty a_n q^n\in {\mathbb C}[[ q]]$
			and as
			\\ \pause
			\begin{tabular}{l}
				2) {\blue holomorphic functions}
				\\  
				on the
				upper half plane
				\\
				$ \HH = \{ z\in {\mathbb C}\ |\ \Im\ z>0\}$\\
			\end{tabular}
		\end{tabular}
		\hskip0.3cm
		\pause
		\begin{tabular}{|l} 
			where 
			$q=\exp(2\pi i z)$, \\
			$z\in \HH$, 
			and define 
			\\
			$L$-function \\ 
			\blue
			$\ds L(f, s,\chi)=\sum_{n=1}^\infty \chi(n)a_n n^{-s}$ \\
			for a
			Dirichlet character 
			\\
			$\chi:(\Z/N\Z)^*\to {\mathbb C}^*$
			\black
			%\\
			(its Mellin transform)
		\end{tabular}
	\end{tabular}
	%\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\pause
	
	\
	
	{\large\blue A famous example: the Ramanujan function 
		$\tau (n)$}
	%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
	\begin{tabular}{ll} 
		\begin{tabular}{l|}  
			The function 
			$\Delta$ (of the variable $z$)
			\\
			is defined by the formal expansion 
			\\
			$\Delta=\sum_{n=1}^\infty  \tau (n) q^n$
			\\ 
			$=
			q
			\prod_{m=1}^\infty (1-q^m)^{24}$
			\\ 
			$= q - 24q^2 + 252q^3 + \cdots$
			\\
			is a  cusp form of weight $k=12$
			\\
			for the group $\Ga = \SL_2(\Z)$).
			%%%%%%%%%%%%%%%%%%%%%%%%%
		\end{tabular} 
		% \hskip-0.3cm 
		&\begin{tabular}{l} 
			$\tau (1)=1,  \tau (2)=-24$,\\
			$  \tau (3) =252, \tau(4)=-1472$\\
			$\tau (m) \tau (n) =\tau (mn)$ \\  for $(n,m)=1$, \\
			$|\tau(p)|\le 2p^{11/2}$  \\ 
			for all primes $p$ .
		\end{tabular}
	\end{tabular}
\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Analytic $p$-adic theory:
		zeta values vs. coefficients } 
	It was much developed in the 60th
	%	following the cours "Zeta Function" by I.R.Shafarevich in Moscow State University \cite{Sha69} on the work of Iwasawa, 
	in \cite{Iwa72}, \cite{Se73} and 
	\cite{Was82}.
	
	\smallskip\pause				
	{\magenta Modular methods are applicable to the $p$-adic analytic continuation of $\zeta(s)$ itself through the normalized Eisenstein series:  }
	$$
	\frac{(k-1)!}{2(2\pi i)^k}G_k(z)= \frac{\zeta(1-k)}{2}+\sum_{n=1}^{\infty}\sum_{d|n}d^{k-1}q^n
	= -\frac{B_k}{2k}+\sum_{d\ge 1}\frac {d^{k-1}q^d}{1-q^d},  
	$$  
	% \vskip-5pt
	modular forms  of even weight  $k \ge 4$ for $\SL_2(\Z)$ as follows:
	%,$B_k$  are Bernoulli numbers.
	
	\smallskip\pause 
	{\magenta J.-P.Serre noticed \cite{Se73}, p.206}, that the constant term
	%$\zeta^*_p(1-k)=$
	$$
	\frac{\zeta(1-k)}{2}(1-p^{k-1})\text{ expresses by }
	\si^*_{k-1}(n)=\sum_{ d|n}d^{k-1} \ (p\nd d, n\ge 1),
	$$
	the higher coefficients of the normalized Eisenstein series $\bmod p^r$.
	%$\di \si^*_{k-1}(n)=\sum_{ d|n\atop p\not | d}d^{k-1}$, 
	In this way $\zeta^*_p(1-k)$ can be continually extended to  $s\in {\mathbb Z}_p$ with a single simple pole at $s=1$ starting from $s=1-k$ (see \cite{Se73}).	
	
	
	\smallskip\pause
	The {\magenta Hurwitz numbers} naturally appear as the critical values of the  Hecke $L$-function  of ideal character $\di L(s,\psi)	=\sum_{\a} \psi(\a) N{\a^{-s}}$,  $\psi((\al))=\al^m, \al\equiv 1 \bmod(2+2i)$,
	also defined  for any imaginary quadratic field $K$, and 
	$g_\psi=
	\sum_{\a} \psi(\a) q^{N{\a}}$ is a modular form of weight $m+1$.
	Its {\magenta$p$-adic analytic continuation over $m$ and $s$} was constructed 
	by  Yu.I.Manin and M.M.Vishik (1974, \cite{Ma-Vi}){\tiny }. 
	
	\
	
	
\end{frame}

\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}%[shrink=3]
	\frametitle{Classical modular forms}%\label{ii43.2.} 
	are introduced
	as certain holomorphic functions on the
	upper half plane\index{Upper half plane}
	$ \mathbb H = \{ z\in \mathbb C\ |\ \Im\ z>0\}$,
	%\end{frame}
	%\end{document} 
	which can be regarded as a homogeneous space
	for the group $G(\R) = \GL_2(\R)$:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.83)
	\begin{align}\label{ii43.1}
	{\mathbb H}
	=
	\GL_2(\mathbb R) / \Or (2)\cdot Z,
	%\eqno (3.1)
	\end{align}
	where $Z = \{ \mat x,0,0,x,\ | x\in \R^\times\}$
	is the center of $G(\R)$ and $\Or (2)$ is the orthogonal group.
	The group $\GL_2^+(\R)$
	of matrices $\ga = \mat a_\ga, b_\ga, c_\ga, d_\ga,$ with positive 
	determinant
	acts on ${{\mathbb H}}$ by
	fractional linear
	transformations;\index{Fractional linear transformation}
	on cosets %(\ref{ii43.1}) 
	this action transforms
	into the natural action by group shifts. 
	Let $\Gamma$ be a
	subgroup of finite index in the modular group 
	$\SL_2(\Z)$.
\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=3]
	\frametitle{Definition of a modular form}
	A holomorphic function $f : {{\mathbb H}} \to \C$
	is called a
	modular form\index{Modular form}
	of (integral) weight $k$ with respect to $\Ga$
	iff the conditions a) and b) are satisfied:
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.84)
	%\medskip
	\begin{itemize}
		\item{a)} {\it Automorphy condition}
		\begin{align}\label{ii43.2}
		f((a_\ga z + b_\ga)/(c_\ga z + d_\ga))
		=
		(c_\ga z + d_\ga)^k f(z)
		% \eqno(3.2)
		\end{align}
		for all elements $\ga \in \Ga$;
		
		\item{b)} {\it Regularity at cusps}:
		$f$ is regular at
		cusps\index{Cusp}
		$z \in \Q \cup i\infty$
		(the cusps can be viewed as fixed points of parabolic elements of 
		$\Ga$);
		this means that for each element $\si = \mat a, b, c, d, \in 
		\SL_2(\Z)$
		the function $(cz + d)^{-k}f\left(\frac {az + b} {cz + d} \right )$
		admits a Fourier expansion
		over non--negative powers of $q^{1/N} = e(z/N)$
		for a natural number $N$.
		One writes traditionally 
		$$
		q = e(z) = \exp (2\pi i z).
		$$
		A modular form
		$$
		f(z) = \sum _{n=0}^\infty a(n) e(nz/N)
		$$
		is called a
		cusp form\index{Cusp!-- form}
		if $f$ vanishes at all cusps
		(i.e. if the above Fourier expansion
		contains only positive powers of $q^{1/N}$),
		see 
		\cite{Lang S. (1976)}, 
		\cite{Ma-Pa05}
	\end{itemize}
\end{frame}

\begin{frame}[shrink=3]
	
	The complex vector space of all modular (resp. cusp) forms
	of weight $k$ with respect to $\Ga$
	is denoted by $\Mc_k(\Ga)$ (resp.  $\Sc_k(\Ga)$).
	
	A basic fact from the theory of modular forms
	is that the spaces of modular forms are finite dimensional.
	Also, one has $\Mc_k(\Ga)\Mc_l(\Ga) \subset \Mc_{k + l}(\Ga)$.
	The direct sum
	$$
	\Mc (\Ga)
	=
	\bigoplus_{k=0}^\infty \Mc _k(\Ga)
	$$
	turns out to be a graded algebra over $\C$
	with a finite number of generators.
	
	An example of a modular form with respect to $\SL_2(\Z)$ of weight $k 
	\ge 4$
	is given by the
	{\it Eisenstein series}\index{Eisenstein series}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.85)
	\begin{align}\label{ii43.3}
	G_k(z)
	=
	\sum_{m_1, m_2 \in \Z}\!\!\!\!{}^{\displaystyle {}'} 
	(m_1+ m_2 z)^{-k}
	% \eqno(3.3)
	\end{align}
	(prime denoting
	$(m_1, m_2) \not= (0, 0)$).
	For these series the automorphy condition (\ref{ii43.2})
	can be deduced straight from the definition.
	One has $G_k(z) \equiv 0$ for odd $k$ and 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    (6.86)
	\begin{align}\label{ii43.4}
	G_k(z)
	=
	\frac{ 2(2\pi i)^k}{ (k-1)!}
	\left[ - \frac {B_k} {2k}  + \sum_{n=1}^\infty \si_{k-1}(n) e(nz) 
	\right ],
	% \eqno(3.4)
	\end{align}
	where $\si_{k-1} (n) = \sum_{d|n}d^{k-1}$
	and $B_k$ is the $k^{\rm th}$
	{\it Bernoulli number}.\index{Bernoulli number}
	
	The graded algebra $\Mc (\SL_2(\Z))$
	is isomorphic to the polynomial ring
	of the (independent) variables $G_4$ and $G_6$.
	
	\
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=3]
	\frametitle{Examples} 
	%Eisenstein series and Congruences for the Ramanujan  function
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	Recall that $B_k$ denote the Bernoullli numbers defined by the development
	$$
	\frac x {e^x-1} =\sum_{k=0}^\infty B_k\frac {x^k} {k!}, 
	$$
	One has 
	$$
	\z(k)=-\frac {(2\pi i)^k} 2 \ , 
	G_k(z)= \frac {(2\pi i)^k} {(k-1)!}, 
	\left[ -\frac {B_k} {2 k} + \sum_{n=1}^\infty
	\si_{k-1}(n)q^n\right].
	$$ 
	%\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{align*}
	&
	E_4(z)=1+240\sum_{n=1}^\infty\si_{3}(n)q^n \in \Mc_4(\SL(2, \Z)),\cr &
	E_6(z)=1-504\sum_{n=1}^\infty\si_{5}(n)q^n \in \Mc_6(\SL(2, \Z)),\cr &
	E_8(z)=1+480\sum_{n=1}^\infty\si_{7}(n)q^n\in \Mc_8(\SL(2, \Z)),\cr &
	E_{10}(z)=1-264\sum_{n=1}^\infty\si_{9}(n)q^n\in \Mc_{10}(\SL(2, \Z)),\cr &
	E_{12}(z)=1+\frac {65520} {691} \sum_{n=1}^\infty\si_{11}(n)q^n
	\in \Mc_{12}(\SL(2, \Z)),\cr &
	E_{14}(z)=1 - 24\sum_{n=1}^\infty\si_{13}(n)q^n\in \Mc_{14}(\SL(2, \Z)).
	\end{align*}
	\medskip
	\noindent
	{\sl Proof } see in  \cite{Se70}.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile=singleslide,shrink=20]
	\frametitle{Fast computation of the Ramanujan function:}
	Put
	$ \ds
	h_k:=\sum_{n=1}^\infty \sum_{d|n}d^{k-1}q^n
	=\sum_{d=1}^\infty \frac {d^{k-1}q^d}{1-q^d}.
	$ 
	The classical fact is that 
	$\De=(E_4^3-E_6^2)/1728$ where
	$E_4= 1+240h_4$ and
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	$ E_6= 1 - 504h_6$. 
	%\end{frame}
	%\end{document} 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\begin{frame}[fragile=singleslide,shrink=20]
	%\frametitle
	
	\
	
	{\large\blue Computing with PARI-GP} 
	see
	\rm \cite{BBBCO}, 
	The PARI/GP number theory system),
	{\tt http://pari.math.u-bordeaux.fr}
	
	$ \ds
	h_k:=\sum_{n=1}^\infty \sum_{d|n}d^{k-1}q^n
	=\sum_{d=1}^\infty \frac {d^{k-1}q^d}{1-q^d}
	\Longrightarrow
	$ 
	%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%{}
	\begin{verbatim}
	gp > h6=sum(d=1,20,d^5*q^d/(1-q^d)+O(q^20))
	gp > h4=sum(d=1,20,d^3*q^d/(1-q^d)+O(q^20)
	gp > Delta=((1+240*h4)^3-(1-504*h6)^2)/1728
	\end{verbatim}
	
	%\end{document} 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\small
	
	\begin{verbatim}
	q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 
	+ 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 
	- 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 
	+ 987136*q^16 - 6905934*q^17+ 2727432*q^18 + 10661420*q^19
	+ O(q^20)
	\end{verbatim}
\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile=singleslide,shrink=5]
	\frametitle{Congruence of Ramanujan}
	$\ds
	\tau (n) \equiv \sum_{d|n}d^{11} \ \bmod \ 691:
	$
	
	\small
	\begin{verbatim}
	gp > (Delta-h12)/691
	%10 = -3*q^2 - 256*q^3 - 6075*q^4 - 70656*q^5 - 525300*q^6
	- 2861568*q^7 - 12437115*q^8 - 45414400*q^9
	- 144788634*q^10 - 412896000*q^11 - 1075797268*q^12
	- 2593575936*q^13 - 5863302600*q^14 - 12517805568*q^15
	- 25471460475*q^16 - 49597544448*q^17
	- 93053764671*q^18 - 168582124800*q^19 + O(q^20)
	\end{verbatim}
	
	{\large \blue More programs of computing  $\tau(n)$ (see \cite{Sl})}
	\begin{verbatim}
	PROGRAM   	
	
	(MAGMA) M12:=ModularForms(Gamma0(1), 12); t1:=Basis(M12)[2]; 
	PowerSeries(t1[1], 100); Coefficients($1);
	
	(PARI) a(n)=if(n<1, 0, polcoeff(x*eta(x+x*O(x^n))^24, n))
	
	(PARI) {tau(n)=if(n<1, 0, polcoeff(x*(sum(i=1, (sqrtint(8*n-7)+1)\2,
	(-1)^i*(2*i-1)*x^((i^2-i)/2), O(x^n)))^8, n));} 
	gp > tau(6911)
	%3 = -615012709514736031488
	gp > ##
	***   last result computed in 3,735 ms.
	\end{verbatim}
	%\end{tabular}
	
\end{frame}
\fi
			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
			\begin{frame}[shrink]
				\frametitle{Recall: 
					Hermitian modular group $\Gamma_{n,K}$ and the standard zeta
					function $\Zc(s; \f)$}
				Let $\theta=\theta_K$ be the quadratic character attached to $K, n'=\left[\frac n 2 \right]$.
				\begin{align*}
				& \nonumber
				\Gamma_{{n},K}=
				\left \{  M=\mat A,B,C,D,\in  \GL_{2{{n}}}(\Oc_K) | M\eta_{n}M^*=\eta_{n} \right\}, \eta_{n}=\mat 0_{{n}},-I_{{n}},I_{{n}},0_{{n}},
				\pause
				\\ &
				\nonumber
				\Zc(s,  {\bf f} ) =\left(\prod_{i=1}^{2n}
				L(2s -i + 1, \theta^{i-1})\right)
				\sum_{\a}
				\lambda(\a)N(\a)^{-s}, 
				\\ & \nonumber
				(\text{via Hecke's eigenvalues: }{\f}|T(\a) = \lambda(\a)\f, \a\subset {\Oc_{\it K}})	
				\\ &
				=\prod_\q \Zc_\q(N(\q)^{-s})^{-1}
				(\text{an Euler product over primes } \q \subset\Oc_K,
				\\ & \text{ with }
				\deg \Zc_\q(X)= 2n, 
				\text{ the Satake parameters }  
				t_{i,\q} , i=1, \cdots, n), 
				\pause
				\\ &
				\Dr(s,{\f})=\Zc(s-\frac \ell 2+\frac{1}{2},{\f})\ \ 
				(\text{Motivically normalized standard zeta function} \\
				& 	
				\text{ with a functional equation }s \mapsto \ell -s; \rk = 4n)
				\end{align*}
				\pause
				{\Large \red Main result:} $p$-adic interpolation of all critical values $\Dr(s,{\f}, \chi)$,  
				$n \le s \le \ell - n , \chi \bmod p^r$ .
				
				
				
				\
				
			\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]
	\frametitle{Automorphic forms, $p$-adic theory of weights. }	
$p$-adic analytic weight space. Motivic and arithmetical weights, introduction to \cite{EHLS}	
	
[Lan13] , Arithmetic compactifications of PEL-type shimura varieties, London Mathematical
Society Monographs, vol. 36, Princeton University Press, 2013.

For the purposes of subsequently defining p-adic modular forms for unitary groups we
assume that the PEL data considered also satisfy:

$\bullet$ $B$ has no type $D$ factor;

$\bullet$ $\langle\cdot, \cdot \rangle$: $L \otimes \Z_p \times L \otimes \Z_p \to \Z_p(1)$ is a perfect pairing;

$\bullet$ p $\nd$ Disc$({\Oc}_B)$, where Disc$(\Oc_B)$ is the discriminant of $(\Oc_B)$ over $\Z$ defined in
[Lan13, Def. 1.1.1.6]; this condition implies that 
$(\Oc_B)\otimes (\Oc_B)$ is a maximal 
$(\Oc_B)$-order in $B$ and that $\Oc_B \otimes \Z_p$ is a product of matrix algebras.

\

Associate a group scheme $G = G_P$ over $\Z$ with such a PEL datum $P$: for any $\Z$-algebra $R$
$$
G(R) = \{(g, \nu) \in \GL_{\Oc_B\otimes R}(L \otimes R)\times R^\times : \langle gx, gy\rangle = \nu \langle x, y\rangle \forall x, y \in L \otimes R\}.
$$
Then $G_{/\Q}$ is a reductive group, and by our hypotheses with respect to $p$, $G_{/\Q}$ is smooth
and $G(\Z_p)$ is a hyperspecial maximal compact of $G(\Q_p)$.


Definitions of $\Xc_p$ $\Yc_p$ through $B$, $T$.
	
	\
	
\end{frame}		
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{frame}[shrink]
		\frametitle{Main result stated with Hodge/Newton polygons of $\Dr(s)$}  The Hodge polygon $P_{H}(t): [0,d]\to {\mathbb R}$ of the function $\Dr(s)$ and  \\ the Newton polygon $P_{N,p}(t): [0,d]\to {\mathbb R}$ at $p$ are piecewise linear: \\ 
		
		\smallskip\pause
		{\magenta The Hodge polygon } of pure weight $w$ has the slopes  ${j}$ of $length_{j}= {h^{{j}, w-{j}}}$ given by Serre's Gamma factors  of the functional equation of the form $s\mapsto w+1-s$, relating $\di \Lambda_\Dr(s,\chi) =\Gamma_\Dr(s) \Dr(s,\chi)$ and 
		$\Lambda_{\Dr^\rho}(w+1-s,\bar\chi)$, where  $\rho$ is the complex conjugation of $a_n$, and $\Gamma_\Dr(s) = \Gamma_{\Dr^\rho}(s)$ equals to the product  $\Gamma_\Dr(s)= 
		\prod_{{j} \le \frac w 2}\ds \Gamma_{{j}, w-{j}}(s)$, where
		$$
		\Gamma_{{j}, w-{j}}(s) =
		\begin{cases}\Gamma_\CC(s-{j})^{h^{{j}, w-{j}}}, &\text{ if } {j}< w,  \\ \Gamma_\R(s-{j})^{h_+^{{j}, {j}}}
		\Gamma_\R(s-{j}+1)^{h_-^{{j}, {j}}}
		, &\text{ if } 2{j}= w, \text{ where }
		\end{cases}
		$$
		$\ds
		\Gamma_\R(s)=\pi^{
			-\frac {s}{2}}\Gamma\left(\frac {s}{2}\right), \Gamma_\CC(s)=\Gamma_\R(s)\Gamma_\R(s+1)=
		2(2\pi)^{-s}\Gamma(s)$,
		$h^{{j}, {j}}=h_+^{{j}, {j}}+h_-^{{j}, {j}}$, 
		$\ds\sum_{{j}} h^{{j}, w-{j}}=d$.
		
		\smallskip\pause
		{\magenta The Newton polygon at $p$} is the convex hull of points
		$(i,\ord_p(a_i))\ \  (i=0,\dots,d)$; its slopes $\lambda$ are
		the   $p$-adic valuations $\ord_p(\al_i)$ of the inverse roots $\al_i$ of $\Dr_p(X)\in \bar \Q[X]\subset \CC_p[X]$:  
		%	under  $incl_p:\bar{\mathbb Q}\hookrightarrow {\mathbb C}_p$ :
		$length_\lambda= \sharp\{i\ |\ \ord_p(\al_i)=\lambda \}$.
		
		\smallskip\pause 
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{frame}[shrink]
		\frametitle{The Hodge polygon of the Hermitian  zeta function}
		Starting from the Gamma factors 
		%of the functional equation 
		of the standard Hermitian $L$-function $\Dr(s,{\f},  \chi)$ 
		%normalized with respect to the symmetry $s\mapsto 1-s$,
		let us describe the Hodge polygon for $F=\Q$.
		
		The explicit form of the Gamma factors of the standard Hermitian $L$-function
		$\Zc(s, {\f})$  was studied 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		in (cf. \cite{Shi00}, p.179,  \cite{Ha97},  \cite{Ha14},
		%and formulas (22.4a), (16.47)), 
		\cite{Kl},  \cite{Bou16}, \cite{Ge16}), and that of $\Dr(s,{\f},  \chi)$ follows with the Gamma factor 
		\iffalse
		Galois equivariance of critical values of L-functions for unitary groups
		Lucio Guerberoff, Jie Lin
		"\dots The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic L-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of $\GL_{n}\times\GL_{1}$. We show that our results are compatible with Deligne's conjecture".  
		Subjects: 	Number Theory (math.NT)
		MSC classes: 	11F67 (Primary) 11F70, 11G18, 11R39, 22E55 (Secondary)
		Cite as: 	arXiv:1612.09590 [math.NT]
		(or arXiv:1612.09590v1 [math.NT] for this version)
		\fi
		\begin{align*}&
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\Gamma_\Dr(s)=L_{\infty }(s,{\f},  \chi) = 
		\prod_{j = 0}^{n-1} \Gamma _{{\CC}}(s-j)^2, 
		\\ & \text{ with the symmetry } s \mapsto \ell-s. 
		\end{align*}
		
		
		\smallskip
		These factors suggest the following form of the Hodge polygon of $\Dr(s, {\f},\chi)$ of rank $d=4{n}$  
		as that of the Hodge numbers $h^{{j}, w-{j}}$ below (in the increasing
		order of slopes ${j}$, with weight $w=\ell-1$):
		\begin{align*} 
		&2\cdot (0, \ell-1), \dots , 2\cdot(n-1,  \ell-n), 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%\\ 
		%\nonumber  &
		%(-k+1+{m}, k-1-{m}), \cdots, ( -k+{n}+{m}, k-{n}-{m}) , 
		%\\ 
		%\nonumber  &
		%(2n'+1-n+k, -k+{n}), \cdots  , (2n'+k-1, 1),
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\ 
		\nonumber  &
		2\cdot(\ell-n, n-1),\dots ,2\cdot( \ell-1,0) , 
		\end{align*}
		following Serre's recipe \cite{Se70}, p.11.	 
		%$ \bar H^{rs;w}(W_\mu^\nabla)$ \cite{Ha97},p. 105
		
		%This comes from a Rankin-Selberg product of CM-type.
		
		%\smallskip
		%{\red A striking programm by M.Harris realizes general  Delignes %$\Dr(s)$-functions as factors of such products!}
		
		
		\
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
	\begin{frame}[shrink]
		\frametitle{Main Theorem (the Hermitian case)}
			Let $\Omega_{\f}=\langle{\f}, {\f} \rangle$ be the period attached to  a Hermitian cusp eigenform ${\f}$, $\magenta \Dr(s,{\f})=\Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})$ the standard zeta function, and
			$$
			\alpha _{{\f}}=\alpha _{{\f},p}=\left(\prod_{\q|p}\prod_{i=1}^n t_{\q,i}\right) p^{-n(n+1)}, \ \
			h 
			= \ord_{p}(\alpha _{{\f},p}),
			$$
			The number $\alpha_{\f}$	turns out to be an eigenvalue of Atkin's type
			operator $U_p : \sum_{H} A_H q^H\mapsto \sum_{H} A_{pH} q^H$ on some ${\f}_0$,
			and $h=P_{N}(\frac {d}2)-P_{H}(\frac {d}2)$.
			
		Let  ${\f} $ be a Hermitian cusp eigenform of 
		degree $n\ge 2$ and of weight $\ell > 4n+2$.
		There exist  distributions  
		$\mu_{\Dr, s}$ for $s=n,\cdots, \ell-n$ 
		%for $s=1-k+n,\cdots,2n'+1-n+k$
		% a ${\CC}_{p}$--analytic function
		%$L^{(\ep  _{0})}_{(p)}$ on ${\cal Y}_{p}$ 
		with the properties:
		\par
		\medskip
		i) 
		for all pairs $(s, \chi)$ such that $s \in  {\Z}$ with
		$n\le s\le \ell-n$,  
		%and  for $s=1$ the character $\chi ^{2}$ is non--trivial,
		$$
		\int_{\Z_p^*}\chi  d\mu_{\Dr,s} = 
		%G(\chi )^{-{n}-1}
		A_{p}(s,\chi)% = \prod_{\q | p}A_{\q}(\c,s, \chi) 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\frac{{\cal D}^{*}(s,  {\f},\overline \chi )}{\Omega_{\f}}
		$$
		(under the inclusion $i_p$), with elementary factors $A_{p}(s,\chi) = \prod_{\q | p}A_{\q}(s, \chi)$
		including a finite Euler product, gaussian sums, the conductor 
		of $\chi$; the integral is a finite sum. 
		
		\
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%			
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
	\begin{frame}[shrink]
		\frametitle{}
		%{Main theorem  (continued) for 
		%$ \Dr(s,{\f})=\Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})$ }	
		%\begin{theo}
		(ii) if				
		$\ord_{p}
		\left((\prod_{\q|p}\prod_{i=1}^n t_{\q,i}) 
		p^{-n(n+1)}\right)=0$
		then the above distributions $\mu_{\Dr,s}$ are bounded measures, we set  $\mu_\Dr=\mu_{\Dr,s^*}$ and the  integral is defined for all continuous characters 
		$y\in\Hom(\Z_p^*, \C_p^*) =:\Yc_p$.
		%%%%%%%%%%%%%%%%%%%%%%%%%%%
		
		\smallskip
		Their {\magenta  Mellin transforms} 
		$\Lr_{\mu_\Dr}(y)=
		\int_{\Z_p^*}yd \mu_\Dr$, 
		$\Lr_{\mu_\Dr}: \Yc_p\to \CC_p$, \\
		% $\Lr_{\mu_\Dr^-}(y)=$
		% $\int_{\Z_p^*}yd \mu_\Dr^-$, 
		% $\Lr_{\mu_\Dr^-}: \Yc_p\to \CC_p$,\\
		give bounded $p$-adic analytic interpolation   of the above $L$-values
		to %bounded ${\CC}_{p} $--analytic function 
		on the $\CC_p$-analytic group $\Yc_p$;
		and these distributions are related by: 
		$\ds \int_X \chi d\mu_{\Dr,s}=\int_X \chi x^{s^*-s}d\mu_\Dr^*$, $X=\Z_p^*$, 
		where $s^*=\ell-n$, $s_*=n$. 
		
		\smallskip\pause
		(iii) in the  {\magenta admissible}  case assume that $0<h \le 
		\ds \frac { s^*-s_*+1} 2= \frac{\ell+1-2n}2$, where
		$h=\ord_{p}\left((\prod_{\q|p}\prod_{i=1}^n t_{\q,i}) 
		p^{-n(n+1)}\right)>0$,
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		Then there exist $h$--admissible measures 
		$\mu_\Dr$ whose integrals 
		$\int_{\Z_p^*} \chi x_p^s d\mu_\Dr$ are
		given by $ \di i_p\left(	A_{p}(s,\chi)% = \prod_{\q | p}A_{\q}(\c,s, \chi) 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\frac{{\cal D}^{*}(s,  {\f},\overline \chi )}{\Omega_{\f}}\right)\in \CC_p$ with $A_{p}(s,\chi)$ as in (i);
		their Mellin transforms
		$\Lr_{\Dr}(y)=\int_{\Z_p^*} y d\mu_\Dr$,
		%$\Lr_{\Dr^\pm}:\Yc_p\to\CC_p$
		belong to the type
		$o(\log  x^{h}_{p})$.
		\iffalse 
		with 
		and it is given by the Mellin transform of an $h$--admissible measure;
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%Their  {\magenta Mellin transforms} 
		$\Lr_{\Dr^\pm}(y)=\int_{\Z_p^*} y d\mu^\pm_\Dr$,
		$\Lr_{\Dr^\pm}:\Yc_p\to\CC_p$
		give $p$-adic analytic interpolation  of growth  $\log_p^h(\cdot)$ of the $L$-values:   
		the values $\Lr_{\Dr^\pm}(\chi x_p^m)$ are integrals given by $ \di i_p\left(\frac
		{{\cal D}^{*}(m,  {\f},\chi )}{\Omega_{\f}}\right)\in \CC_p$.
		\fi
		\\
		\smallskip\pause
		(iv) %if $h \le  (s^{*} - s_{*} + 1)/2 = k - {n}$ then 
		the functions $\Lr_{\Dr}$ are determined by
		% the above conditions
		(i)-(iii).
		%\end{theo}
		
		\hskip-10pt{\blue\Large Remarks.} \\
		(a) Interpretation of  $s^{*}$: 
		the smallest of the "big slopes" of
		$P_{H}$ 
		
		(b) Interpretation of  $s_{*}-1$:
		the biggest of the "small slopes" of
		$P_{H}$.
		%	(c) We have  $\Dr^*(s_0,{\bf f}, \chi)=
		%	\Lambda_\Dr(s_0,{\bf f},\chi)$, 
		%	because in this case  $h^{\ell,\ell}=0$, and all the critical points $s_0$ %are	$s_{*} + 1 \le s_0\le  s^{*}$. %(no parity condition).
		
		\
		
	\end{frame}
\begin{frame}[shrink]
	\frametitle{Eisenstein series and congruences \red  (KEY POINT!)} The (Siegel-Hermite)Eisenstein series $E_{2\ell}^{({n})}(Z)$ of weight $2\ell$, character $\det^{-\ell}$,  is defined by
	$\di
	E_{2\ell}^{({n})}(Z)=
	\sum_{g\in\Gamma_{K,\infty}^{({n})}\bs \Gamma_K^{({n})}}
	(\det g)^{\ell}j(g,Z)^{-2\ell} .
	$
	The series  converges absolutely for $\ell >{n}$. 
	Define the normalized Eisenstein series ${\cal E}_{2\ell}^{({n})}(Z)$ by
	$
	{\cal E}_{2\ell}^{({n})}(Z)=$
	$2^{-{n}}\prod_{i=1}^{{n}}L(i-2\ell, \theta^{i-1})\cdot  E_{2\ell}^{({n})}(Z)
	$
	If $H\in\Lambda_{n}(\Oc)^+$, then the $H$-th Fourier coefficient of ${\cal E}_{2\ell}^{({n})}(Z)$ is polynomial over $\Z$ in $\{p^{\ell-({n}/2)}\}_p$, and equals 
	$$
	|\gamma(H)|^{\ell-({n}/2)}
	\prod_{p|\gamma(H)}\tilde F_p(H,p^{-\ell+({n}/2)}),
	\gamma(H) = (-D_K)^{[{n}/2]} \det H.
	$$	
	Here, $\tilde F_p(H,X)$ is a certain Laurent polynomial in the variables $\{X_p=p^{-s}, X_p^{-1}\}_p$ over $\Z$. % arising from the Siegel series for $H$.
	This polynomial is a {\red key point in proving congruences} for
	the modular forms in a {\red Rankin-Selberg integral}. 
	
	\smallskip
	Also, we set , for $s\in\CC$ and a Hecke ideal character $\psi \bmod \c$,
	$$ \magenta
	E(Z,s,\ell,\psi)=
	\sum_{g\in C_\infty\bs C}
	\psi(g) (\det g)^{\ell}j(g,Z)^{-2\ell}|(\det g)j(g,Z)|^{-s} .
	$$
	
	\	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Recall :Hermitian  Rankin-Selberg type  integral  } 
	%\iffalse	
	%Let us specialize to $F=\Q$
	%	{\magenta Include reading in the simplest elliptic modular case of $\GL_2$}
	
	%	The integral representation of Rankin-Selberg type in the Hermitian modular case: % is used
	% (\cite{Bou16}, p.13):
	%"The following theorem (see also [25, Theorem 7.8]) is obtained by combining results of
	%Shimura [31] and Klosin [25]. For details we refer to [6, Section 4], 
	
	{\blue \Large Theorem 4.1 (Shimura, Klosin)}, see \cite{Bou16}, p.13.\\
	Let $0 \ne {\f} \in\Mc_\ell(C,\psi ))$ of scalar weight $\ell$, $\psi
	\bmod \c$, such that $\forall \a, {\f} |T(\a) = \lambda(\a){\f}$, and assume that $2\ell\ge n$, then there exists ${\cal T} \in S_+\cap \GL_n(K)$ and ${\cal R} \in \GL_n(K)$ such that	
	\begin{align}\nonumber
	& \Gamma ((s)) \psi(\det({\cal T}))%c_{\f} (\tau, r)
	\Zc(s + 3n/2, {\f} , \chi) =	\\ & \nonumber
	\Lambda_{\c}(s + 3n/2, \theta\psi\chi)\cdot
	%\\ \nonumber
	%&  \left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2} C_0
	%\Lambda_\c(s + 3n/2, \theta(\psi\chi)_1)\cdot
	%\\ & \nonumber
	%\left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}
	C_0 \langle {\f} , \theta_{\cal T}(\chi)
	%\tilde E_\A(x, 
	{\Ec}(\bar s + n, \ell - \ell_\theta, \chi^\rho\psi) \rangle_{C\sec},
	\end{align}
	where ${\Ec}(Z, s, \ell-\ell_\theta, \psi)_{C\sec}$ is a
	normalized
	group theoretic %adelic
	Eisenstein series with components as above of level $\c\sec$ divisible by $\c$, and weight $\ell-\ell_\theta$. 	
	Here $\langle \cdot, \cdot \rangle_{C\sec}$ is the  
	%group theoretic %adelic
	normalized
	Petersson inner product associated to the congruence subgroup 
	$C\sec$ of level $\c\sec$.
	%$D[{\e}, {\e}{\h}]$,
	% as defined for example in [30, Equation 10.9.6],	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\
	$\ds \Gamma((s))=
	%\prod_{v\in {\bf a}}
	(4\pi)^{-n(s+h)}\Gamma^\iota_n(s + h), 
	{\Gamma}^\iota_n(s) = \pi^{\frac{n(n-1)}2} \prod_{j=0}^{n-1} {\Gamma}(s-j)$,\\ 
	where $h=0	\text{ or } 1, \ 
	C_0$
	%:=\frac{[\Gamma_0(\c\sec):\Gamma]}{\sharp X}.	
	\ a subgroup index.
	%\fi{}
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Zeta values  and Bernoulli Numbers}	
A key result in number theory  is the expansion of the
Riemann zeta-function $\zeta(s)$ into the Euler product:
$$
\zeta(s) = \prod_p  (1 - p^{-s})^{-1}
= \sum_{n = 1}^{\infty}
n^{-s} \hspace{1cm} (\text{defined for }\Re(s) > 1 ).  
$$
\pause
The set of arguments $s$ for which $\zeta(s)$ is defined was extended by Riemann
to all $s \in \CC$, $s \neq 1$.
%, and we may view $\CC$ as the group of all
%continuous homomorphisms
%$$
%\CC = \Hom_{\rm cont}(\RR_+^{\times}, \CC^{\times}), ~~~~ y \longmapsto y^s  
%$$
%of $\RR_+^{\times}$. 
The special values $\zeta(1 - k)$ at negative integers are
rational numbers:
$ 
\di\zeta(1 - k) = -\frac{B_k}{k},  
$
satifying certain Kummer congruences $\bmod p^m$, 
where $B_k$ are Bernoulli  numbers, defined by the 
\begin{align*}&
%"e^{Bt}" = 
\sum_{n = 0}^{\infty } \frac{B_n t^n}{n!}
= \frac{t e^t}{e^t - 1} ;  
B_0=1, \ B_1=-\frac 1 2 ,  \ B_2=\frac 1 6 ,  \ B_3=B_5=\cdots=0, B_4=-\frac 1 {30},\\ & 
 \ B_6= \frac 1 {42} , 
\ B_8=-\frac 1 {30} ,B_{10}=\frac {5} {66}, 
B_{12}=\frac {691} {2730},\ B_{14} =-\frac 7 6 ,
%\ B_{16}=\frac {3617} {510}, \cdots, 
%B_{18} = -\frac {43867} {798} , \  \dots.
{\displaystyle\z(2k)=- \frac {(2\pi i)^{2k}B_{2k}} {2(2k)!}}, 
\end{align*}
\pause
Their denominators are small by the Sylvester-Lipschitz theorem
$$ 
\forall c \in \ZZ \mbox{ implies } c^k (c^k - 1) \frac{B_k}{k} \in \ZZ\ \  (\text{see in }\cite{Mil-Sta74}),  
$$
using the known formula for the sum of $k$-th powers via Bernoulli polynomials $B_{k}(x)=\sum_{i=0}^k{k \choose
	i}B_ix^{k-i} ="(x+B)^k"$
$$ 
S_k(N) = \sum_{n=1}^{N-1} n^k = \frac{1}{k+1} \left[ B_{k+1}(N) -
B_{k+1} \right], \  B_{1}(x)=x-\frac 1 2,  B_{2}(x)=x^2-x+\frac 1 6, %B_{3}(x)=x^3-\frac 3 2 x^2+\frac 1 2 x,
\cdots 
$$
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{Kummer congruences and $p$-adic integration}
Kubota and Leopoldt constructed \cite{KuLe64} a $p$-adic interpolation of these special values, explained by Mazur via a $p$-adic measure $\mu_c$ on $\Z_p$  and Kummer congruences for the Bernoulli numbers, see \cite{Ka78} 
($p$ is a prime number,
$c > 1$ an integer prime to $p$). Writing the normalized values
$$ 
\zeta_{(p)}^{(c)}(-k) = (1 - p^k)(1 - c^{k+1}) \zeta(-k) = \int_{\Z_p^*}x^{k}d\mu_c(x)
$$
produces the {\magenta Kummer congruences} in the form: for any polynomial 
$h(x) = \sum_{i = 0}^{n}\al_i x^i$ over $\Z$,
$$ 
\forall x\in \ZZ_p, \sum_{i = 0}^{n} \al_i x^i  \in p^m \ZZ_p\Longrightarrow \sum_{i=0}^{n} \al_i \zeta_{(p)}^{(c)}(-i) \in p^m \ZZ_p,  
$$
\pause
Indeed, integrating the above polynomial $h(x)$ over 
$\mu_c$ produces the congruences.
%\pause
%This property expresses the fact that the numbers $\zeta_{(p)}^{(c)}(-k)$
%depend continuously (and even analytically) on $k$ in the $p$-adic sense; 
The existence of $\mu_c$  is deduced from the above formula for the sum of $k$-th powers $S_k(p^r)$ for $r\to \infty$, 
%$$ S_k(N) = \sum_{n=1}^{N-1} n^k = \frac{1}{k+1} \left[ B_{k+1}(N) -
%B_{k+1} \right]  $$
%in which  $B_{k}(x) = (x+B)^k = \sum_{i=0}^k \binom{k}{i} B_ix^{k-i}$
%denotes the Bernoulli polynomial. Indeed, all summands in $S_k(N)$
%depend $p$-adic analytically on $k$, 
restricted to numbers $n$,
prime to $p$.
%The desired congruence follows if we express the numbers
%$\zeta_{(p)}^{(c)}(-k)$ in terms of Bernoulli numbers. % (see \cite{Ka78}).

%\end{frame}
%\begin{frame}[shrink]
%	\frametitle
%	{\magenta Mazur's $p$-adic integral} : 
%	there exists a $p$-adic measure 
%	$\mu_c$ on $\Z_p^*$ , such that the special values
%	$$
%	\zeta(1-k)(1-p^{k-1})=%(1-\chi(c)c^{k})^{-1}
%	{\int_{\Z_p^*}x^{k-1}d\mu_c(x) \over 1-c^{k}}
%	\in \Q, k \ge 2\mbox{ even},  c=1+p.	$$
\pause
In order to define such a measure $\mu_c$  it suffices for any continuous function $\phi : \Z_p \to \Z_p$ to define its integral
$\int_{\Z_p} \phi(x)d\mu_c$.
Approximating $\phi (x)$ by a polynomial (when the integral is already defined),  pass to the limit (which is well defined due to Kummer congruences).

	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle	{Kubota-Leopoldt $p$-adic zeta-function} 
	The domain of definition of $p$-adic zeta functions is the $p$-adic
	analytic group  $\Yc_p=Hom_{cont}(\Z_p^*, \cb_p^*)$
	of all continuous $p$-adic characters of the profinite group $\ZZ_p^{\times}$,
	where $\CC_p = \widehat{\overline{\QQ}}_p$ denotes the Tate field
	(completion of an algebraic closure of the $p$-adic field $\QQ_p)$
	(over complex numbers $\C=Hom_{cont}(\R_+^*,\C^*)$, 
	$y$ run the characters $t\mapsto t^s$ . 
	\pause
	
	Define
	$
	\zeta_p: \Yc_p \to \cb_p
	$ 
	on the space
	as the  {\magenta $p$-adic Mellin transform} 
	%of  $\mu_c$: for any integer $c>1$ not divisible by $p$, 
	$$
	\zeta_p(y)=%(1-cy(c))^{-1}
	{\int_{\Z_p^*}y(x)d\mu_c (x)\over 1-cy(c)}= {\Lr_{\mu_c}(y)\over 1-cy(c)},
	$$ 
	%(notice that
	%$\int(1+t)^z d\mu(z)=\sum_{n=0}^\infty\int{z\choose n}d\mu(z)$).
	\\	with a single simple pole at 
	$y=y_p^{-1}\in \Yc_p$, 
	where  
	$y_p(x)=x$ the inclusion character $\Z_p^* \hookrightarrow \C_p^*$ and $y(x)=\chi(x)x^{k-1}$  is a typical  arithmetical character ($y=y_p^{-1}$ becomes $k=0$, $s=1-k=1$). 
	
	
	\smallskip\pause
	{\large \magenta Explicitly}: Mazur's measure is given by $\mu_c(a+p^v\Z_p) = $\\
	$
	\frac {1}c\left[ \frac {ca}{p^v}\right] + 
	\frac {1-c}{2c}=\frac{1}c B_1(\{\frac {ca}{p^v}\})-B_1(\frac {a}{p^v})$,~$B_1(x)=x-\frac 1 2$,
	(\cite{LangMF}, Ch.XIII), we see the zeta distribution $\mu_s|_{s=0} (a+(N))=-B_1(\frac{a}{N})$.
	{\red \large Then the binomial formula 
		$\int_Z(1+t)^z d\mu_c= \sum_{n=0}^\infty t^n \int_Z{z\choose n}d\mu_c$, 
		gives the analyticity of $\zeta_p(y)$ on $t=y(1+p)-1$ in the unit disc
		$\{t\in \C_p\| \ |t|_p<1\}$.}
	
	%\fi
	%and given by the {\red constant term of Eisenstein series}.	
	
	\
	
\end{frame}	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   2
\begin{frame}[shrink]
	\frametitle{$p$-adic 
		%methods and zeta functions
		zeta functions of  modular forms
		}
From the $p$-adic zeta function of Kubota-Leopoldt, one extends
$p$-adic zeta functions of various modular forms constructed,
such as $p$-adic interpolation of the special values  
$$
L_\Delta(s, \chi) =
\sum_{n=1}^{\infty}\chi(n)\tau(n)n^{-s}, \ (s=1,2,\cdots, 11)
$$
for the Ramanujan function  $\tau(n)$ defined by the expansion
$$
q\prod_{m\ge 1}{(1-q^m)}^{24}=\sum_{n\ge 1}\tau
(n)q^n=q-24q^2+252q^3-1472q^4+\cdots,
$$
twisted by Dirichlet characters $\chi : (\Z/p^r\Z)^* \to \C^*$; it was done in the elliptic and Hilbert modular cases by Yu.I.Manin and B.Mazur, via modular symbols and $p$-adic integration, see \cite{Ma73}, \cite{Ma76}).
\pause

In the Siegel modular case the $p$-adic  standard zeta functions of Siegel modular forms were constructed in \cite{Pa88}, \cite{Pa91} via Andrianov's identity (of Rankin-Selberg type).
		
	


\pause	
	{\red  PRESENT GOAL: } To describe analytic $p$-adic continuation of  the standard zeta function $L_F(s)$ of a 
	Hermitian modular form  $F=\sum_H A(H) q^H$   on  the Hermitian upper half plane $\mathcal H_{n}$ of degree ${n}$, where $q^H=\exp(2\pi i {\rm Tr}(HZ))$, 
	$H$ runs through all semi-integral positive definite Hermitian matrices of degree ${n}$, i.e. $H\in \Lambda_{n}({\mathcal O})$, in the integers ${\mathcal O}_K$ of an imaginary quadratic field $K=\Q(\sqrt{-D_K})$.
	
	\pause
	
	Analytic $p$-adic continuation of their {\magenta standard zeta functions} is constructed via   $p$-adic measures, bounded or growing. %Previously this problem was solved in  the symplectic Siegel modular %case.		
			
			\
			
		\end{frame}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  3
		
		\begin{frame}[shrink]
			\frametitle{Modular forms,  
zeta functions, $L$-functions}% (complex and $p$-adic)}
%{\large \red Examples}
%\begin{itemize} 
%	\item
{\Large Eisenstein series}
	$\displaystyle E_k=1+\frac 2{\zeta(1-k)}\sum_{n= 1}^\infty\sum_{d|n}d^{k-1}q^n \in \Mc_k$, a modular forms for even weight $k\ge 4$ for $\SL_2(\Z), q=e^{2\pi i z}$), and $E_2\in \Qc\Mc $ {\magenta a quasimodular form}. 
	The ring of quasimodular forms, closed under differential operator ${\displaystyle D=q\frac{d}{dq}
		=\frac{1}{2\pi i}\frac{d}{dz}}$, used  in arithmetic, $\zeta(s)$ is the Riemann zeta function, $\displaystyle \zeta(-1)=-\frac 1{12}$, $E_2=1-24\sum_{n= 1}^\infty\sum_{d|n} d q^n$ is also  a {\magenta $p$-adic modular form} (due to J.-P.Serre,\cite{Se73}, p.211)
		%\item 
		
		\
		
{\Large		Elliptic curves} $E: y^2=x^3+ax+b$ , $a,b\in \Z$,  {\red A.Wiles's modular forms} $\displaystyle f_E=\sum_{n=1}^\infty a_nq^n$ with $a_p=p-Card E(\F_p)$ $(p\nd 4a^3+27b^2)$,
	and the $L$-function $L(E,s)\displaystyle =\sum_{n=1}^\infty a_nn^{-s}$.
	%\end{itemize}
	%belong traditionally to the world of arithmetic, but also to the worlds of geometry, algebra and analysis.
			
	\end{frame}
\fi	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  4
	
	\begin{frame}[shrink]
		\frametitle{Zeta-functions or $L$-functions} They are attached to various mathematical objects as certain Euler products.

\begin{itemize}
\item $L$-functions link such objects to each other (a general form of functoriality); 

\pause
\item Special $L$-values answer fundamental questions about these objects in the form of a 
{\red number (complex or $p$-adic)}.
\end{itemize}
\pause
Computing these numbers use integration theory
of Dirichlet-Hecke characters along  $p$-adic and complex valued measures.


\pause
This approach originates in the {\red Dirichlet class number formula} using the $L$-values in order  to compute class numbers of algebraic number fields through Dirichlet's $L$-series $L(s, \chi)$: for an imaginary quadratic field $K$ 
of discriminant $-D<-4$, $\chi_D(n)={-D \choose n}$
$$
h_D
%= \sharp \text{class group} (\Q(\sqrt{-D})
=\frac { \sqrt{D}L(1,\chi_D)}{2\pi}= L(0,\chi)= - \frac {1} {D}
\sum_{a=1}^{D-1}\chi_D(a)a.
$$
%and the generalized Bernoulli numbers 
%$B_{k,\chi}=N^{k-1} \sum_{a=0}^{N-1} \chi(a)N^{-1}B_k(a/N)$. 
\pause
(Example: disc$(\Q(\sqrt{-5})))=-20$, $h_{20}=2$; in PARI/GP $\chi_{20}(n)=$  kronecker(-20,n), 
{\tt gp > -sum(x=1,19,x*kronecker(-20,x))/20
 \% 29 = 2 
}

\pause
Another famous example: the Millenium
{\red BSD Conjecture} gives the rank of an elliptic curve $E$ as the order of $L(E,s)$ at s=1
(i.e. the residue of its logarithmic derivative, see \cite{MaPa}, Ch.6).  


		\
		
	\end{frame}
\begin{frame}[shrink]
	\frametitle{A short story of critical values, see \cite{YS} } 	
	Euler  discovered  $\di\zeta(2) = \frac{\pi^2}{6}$,
	and $\displaystyle \frac{2\zeta(2n)}{(2\pi i)^{2n}}$ = $\displaystyle-\frac{B_{2n}}{(2n)!}\in \Q, (n\ge 1)$.  
	%in 1735-1742, the starting result about the zeta values.
	
	\pause
	These are examples of   {\magenta critical values} (in the sense of Deligne):  for a more general zeta function 
	$\Dr (s)$ the critical values are defined using	
	its gamma factor $\Gamma_\Dr(s)$ such that the product $\Gamma_\Dr(s)\Dr(s)$ satisfies a standard functional equation under the symmetry $s \mapsto v-s$.
	Then $\Dr(n)$, $n \in \Z$ is a critical value of $\Dr(s)$ if both $\Gamma_\Dr(n)$ and	$\Gamma_\Dr(v-n)$ are finite.
	
	\smallskip\pause
	{\magenta Hurwitz \cite{Hur1899} showed a striking analogy to Euler's theorem}: 
	%\begin{align}&\nonumber
	$$
	\frac{\sum'_{\al\in\Z[i]}\al^{-4m}}{\Omega^{4m}}=\frac{H_m}{(4m)!}\in\Q,   
	\Omega=2\int_0^1 \frac{dx}{\sqrt{1-x^4}}
	%\\ & \nonumber= 
	=2.6220575542\cdots
	$$
	%\end{align}
	for $1 \le m \in \Z$, where $\al = a + ib$, $a, b \in \Z$ 
	%extends to all 
	are non-zero Gaussian integers and $H_m$ are Hurwitz numbers (recursively computed, \cite{Sl}): 
	$\di H_1, H_2, \cdots= \frac{1}{10}, \frac{3}{10}, \frac{567}{130}, \frac{43659}{170}, \frac{392931}{10},\cdots$.	
	Recall the formula: 
	
	\pause
	Let $\wp$ be the Weierstrass $\wp$-function satisfying $\wp'^2 = 4\wp^3 - 4\wp$. Then
	 $\di\wp(z)
	= \frac 1{z^2}+ \sum_{n=1}^\infty\frac {2^{4n}H_nz^{4n-2}}{4n(4n-2)!}$.
	%$$ \sum_{ (r, s) \ne (0, 0) } 1/(r+si)^{4n} = (2w)^{4n}H_n/(4n)!
	%$$ where 
	%$w = 2 \int_0^1 dx/\sqrt{1-x^4}$.
		\pause
	{\red A rapid computation of these values}: take
	the Fourier expansion of the Eisenstein series at $z=i$, $q=e^{-2\pi}$:   %=0.0018674427: $ 
	$\di G_{4m}(z)=\sum^{}_{a,b}{{}'}(az+b)^{-4m}= 2\zeta(4m)+
	\frac{2(2\pi)^{4m}}{(4m-1)!}
	\sum_{d\ge 1}\frac{d^{4m-1}q^d}{(1-q^d)}$, 
	\pause
	{\red $\displaystyle\frac{G_{4m}(i)}{\Omega^{4m}}=\displaystyle \frac{H_m }{(4m)!}$}, $\pi,\Omega$ --  {\magenta periods} of $\zeta(s)$ and of
	$E:y^2=4x^3-4x$.
	%	$\zeta(s)$ and  %$\di\Dr(s)=
	%	$\sum_{\al}\bar\al^{m}N(\al)^{-s}$).
	
	\
	
	
	%\smallskip\pause	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\fi  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  6

\begin{frame}[shrink]
	\frametitle{Analytic $p$-adic theory:
		zeta values vs. coefficients } 
	It was much developed in the 60th
%	following the cours "Zeta Function" by I.R.Shafarevich in Moscow State University \cite{Sha69} on the work of Iwasawa, 
 in \cite{Iwa72}, \cite{Se73} and 
\cite{Was82}.
	
	\smallskip\pause				
	{\magenta Modular methods are applicable to the $p$-adic analytic continuation of $\zeta(s)$ itself through the normalized Eisenstein series:  }
	$$
	\frac{(k-1)!}{2(2\pi i)^k}G_k(z)= \frac{\zeta(1-k)}{2}+\sum_{n=1}^{\infty}\sum_{d|n}d^{k-1}q^n
	= -\frac{B_k}{2k}+\sum_{d\ge 1}\frac {d^{k-1}q^d}{1-q^d},  
	$$  
	% \vskip-5pt
	modular forms  of even weight  $k \ge 4$ for $\SL_2(\Z)$ as follows:
	%,$B_k$  are Bernoulli numbers.
	
	\smallskip\pause 
	{\magenta J.-P.Serre noticed \cite{Se73}, p.206}, that the constant term
	%$\zeta^*_p(1-k)=$
	$$
	\frac{\zeta(1-k)}{2}(1-p^{k-1})\text{ expresses by }
	\si^*_{k-1}(n)=\sum_{ d|n}d^{k-1} \ (p\nd d, n\ge 1),
	$$
	the higher coefficients of the normalized Eisenstein series $\bmod p^r$.
	%$\di \si^*_{k-1}(n)=\sum_{ d|n\atop p\not | d}d^{k-1}$, 
	In this way $\zeta^*_p(1-k)$ can be continually extended to  $s\in {\mathbb Z}_p$ with a single simple pole at $s=1$ starting from $s=1-k$ (see \cite{Se73}).	
	
	
	\smallskip\pause
	The {\magenta Hurwitz numbers} naturally appear as the critical values of the  Hecke $L$-function  of ideal character $\di L(s,\psi)	=\sum_{\a} \psi(\a) N{\a^{-s}}$,  $\psi((\al))=\al^m, \al\equiv 1 \bmod(2+2i)$,
	also defined  for any imaginary quadratic field $K$, and 
	$g_\psi=
	\sum_{\a} \psi(\a) q^{N{\a}}$ is a modular form of weight $m+1$.
	Its {\magenta$p$-adic analytic continuation over $m$ and $s$} was constructed 
	by  Yu.I.Manin and M.M.Vishik (1974, \cite{Ma-Vi}){\tiny }. 
\
	
	
\end{frame}



%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   7

	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  3
	%\fi
	\begin{frame}[shrink]
		\frametitle{Complex and $p$-adic analytic continuation}
A classical example of analytic continuation is given by the Riemenn zeta function with   
$$
\zeta (s)= \frac{(2\pi)^{s/2}}{2\Gamma(s/2)}\int_{0}^{\infty}(\theta (iy) -1)y^{(s/2) -1}dy \ \ ({\mathrm Re(s)}>1),
$$
through the theta function $\theta(z)=\sum_{n\in{\mathbb Z}}e^{2\pi i n^2 z}$ which is a modular form of weight  $1/2$ on the complex upper half plane ${\mathcal H}$.%=\{z=x+iy,y>0\}$.
%, as the first chapter in the book \cite{VoKar}.
% of S.M.Voronin and A.A.Karatsuba (1994), together with  $L(s,\chi)$. 

\medskip
\smallskip\pause
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For a Dirichlet $L$-function $L(s,\chi)$,   
an integral representation uses

%\smallskip\pause
I) {\magenta theta function with Dirichlet character} $\chi \bmod N$
$$
\theta(z, \chi)=\sum_{n\in{\mathbb Z}}\chi(n)n^\nu e^{2\pi i n^2 z},\ \ \chi(-1)=(-1)^\nu, \nu=0,1, \text{ or }
$$
\smallskip\pause
II)  {\magenta meromorphic zeta distributions} $\di{\mu_s(a+(N)):=\sum_{n\ge 1\atop n\equiv a \bmod N}n^{-s}}= N^{-s} \sum_{n\ge 1} (n+(\frac{a}{N}))^{-s}$: the integral $\di L(s,\chi)=\int_X \chi(x) d\mu_s(x)=\sum_{a\bmod N}
\chi(a)\mu_s (a+ (N))=: \mu_s(\chi)$ over $X=\hat \Z$ or $\Z_p$ is {\magenta a finite sum of partial series}, 
$\di = -N^{k-1}\frac{B_k(\frac a N )}
k$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Methods of  constructing $p$-adic $L$-functions}
Our long term purposes are 
to define and to use the $p$-adic  $L$-functions in a way similar to complex  $L$-functions
via the following methods:

(1) Tate, Godement-Jacquet;

(2) the method of Rankin-Selberg;

(3) the method of Euler subgroups of  Piatetski-Shapiro and the  doubling method of  Rallis-Böcherer (integral representations  on a subgroup of $G\times G$);

(4) Shimura's method (the convolution integral with theta series);

(5) Shahidi's method.

There exist already advances for (1) to (4), and we also tried to develop  (5),  see \cite{GMPS14}.

We used  the Eisenstein series  and a $p$-adic integral of Shahidi's type
for the reciprocal of a product of certain $L$-functions.

\

\end{frame}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{Happy Birthday Freydoon Shahidi!}
\begin{figure}[h]
	\begin{center}
		%\includegraphics[width=14cm,height=10cm]{IF-CG2-10-5}
		\includegraphics[width=14cm,height=10cm]{DSC05123}
		\caption{\label{23w1_2}
			Freydoon Shahidi in Grenoble, May 2010}
		 %\includegraphics[width=14cm,height=10cm]{IMG_1153}
		 %\caption{\label{IMG_1153}
		 %	Freydoon Shahidi in the Conference on October 21, 2017}
	\end{center}
\end{figure}
\

\end{frame}

%	\fi 
	\begin{frame}[shrink]
		\frametitle{Meeting Clermont-Grenoble, May 2010, Freydoon Shahidi with Steve Gelbart et al. }
		\begin{figure}[h]
			\begin{center}
				\includegraphics[width=16cm,height=11cm]{IF-CG2-2010-5}
				%\includegraphics[width=14cm,height=10cm]{DSC05123}
				%\caption{\label{23w1_2}
				%Summer School ``Serre's Modularity
				%Conjectures, Marseille-Luminy, July 2007} 
			\end{center}
		\end{figure}
		
		
		\
		
	\end{frame}  
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
	\begin{frame}[shrink]
		\frametitle{Conference at Purdue University on October 20-22, 2017}
		\begin{figure}[h]
			%\includegraphics[width=14cm,height=10cm]{DSC05123}
			%\caption{\label{23w1_2}
			%	Freydoon Shahidi in Grenoble, May 2010}
			\begin{center}	
				\includegraphics[width=14cm,height=10cm]{IMG_1153}
%				\caption{\label{IMG_1153}
%					Freydoon Shahidi in the Conference at Purdue University %on October 21, 2017}
			\end{center}
		\end{figure}
		
		
		\
		
	\end{frame}
\fi
			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\begin{frame}[shrink]
			\frametitle{
	Hermitian modular group $\Gamma_{n,K}$ and the standard zeta
	function $\Zc(s; \f)$ (definitions)}
Let $\theta=\theta_K$ be the quadratic character attached to $K, n'=\left[\frac n 2 \right]$.
\begin{align*}
& \nonumber
\Gamma_{{n},K}=
 \left \{  M=\mat A,B,C,D,\in  \GL_{2{{n}}}(\Oc_K) | M\eta_{n}M^*=\eta_{n} \right\}, \eta_{n}=\mat 0_{{n}},-I_{{n}},I_{{n}},0_{{n}},
 \pause
 \\ &
 	\nonumber
 	\Zc(s,  {\bf f} ) =\left(\prod_{i=1}^{2n}
 	L(2s -i + 1, \theta^{i-1})\right)
 	\sum_{\a}
 	\lambda(\a)N(\a)^{-s}, 
 	\\ & \nonumber
 	(\text{via Hecke's eigenvalues: }{\f}|T(\a) = \lambda(\a)\f, \a\subset {\Oc_{\it K}})	
 	\\ &
 	=\prod_\q \Zc_\q(N(\q)^{-s})^{-1}
 	(\text{an Euler product over primes } \q \subset\Oc_K,
 	\\ & \text{ with }
 	\deg \Zc_\q(X)= 2n, 
 	\text{ the Satake parameters }  
 	t_{i,\q} , i=1, \cdots, n), 
 	\pause
 	\\ &
 \Dr(s,{\f})=\Zc(s-\frac \ell 2+\frac{1}{2},{\f})\ \ 
 	(\text{Motivically normalized standard zeta function} \\
 	& 	
\text{ with a functional equation }s \mapsto \ell -s; \rk = 4n)
\end{align*}
\pause
{\Large \red Main result:} $p$-adic interpolation of all critical values $\Dr(s,{\f}, \chi)$,  
 $n \le s \le \ell - n , \chi \bmod p^r$ .
 
 

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{The idea of motivic normalization: Ikeda's lifting \cite{Ike08}} The Gamma factor of Ikeda's lifting, denoted by ${\f}$, of an elliptic modular form $f$ and used as a pattern,  extends to a general (not necessarily lifted) Hermitian modular form ${\f}$ of even weight $\ell$, which equals in the lifted case to 
$\ell=2k+2n'$, where 
$k=(\ell-2n')/2$ $=\ell /2-n'$=$\ell/2-n'$, when 
the Gamma factor of the standard zeta function with the symmetry $s\mapsto 1-s$ becomes (see p.\pageref{Sho}) 
\smallskip
$\prod_{i=1}^n 
\Gamma_\CC (s+\ell/2 -n'+n'-i+(1/2))^2
=\prod_{i=1}^n 
\Gamma_\CC (s+\ell/2-i+(1/2))^2
=\prod_{i=0}^{n-1} 
\Gamma_\CC (s+\ell/2-i-(1/2))^2$.

This Gamma factor suggests the following motivic normalization
$\Dr(s) = \Zc(s-(\ell/2)+(1/2))$  for which
$\Gamma_\Dr(s)=\Gamma_\Zc(s-(\ell/2)+(1/2))^2$, and the $L$-function becomes
$\Dr(s) = \Zc(s-(\ell/2)+(1/2))$ with symmetry
$s \mapsto 2(\ell/2) -1+1-s=\ell-s$ of motivic weight $\ell-1$
and 
$\ds \Gamma_\Dr (s)=
\prod_{i=0}^{n-1} \Gamma_\CC (s-i)^2,
$
with the slopes 
$2\cdot 0, 2\cdot 1, \dots 2\cdot (n-1)$,\\ 
$2\cdot (\ell -n), \cdots, 2\cdot(\ell-1)$,
so that Deligne's critical values are at $s=n, \dots, s=\ell-n$. 


\

	\end{frame}

	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
	
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   ***** 8

\begin{frame}[shrink]
	\frametitle{General zeta functions: critical values and coefficients}
	More general zeta functions are Euler products of degree $d$ 
	\begin{align} & \nonumber
	\Dr(s,\chi)=\sum_{n=1}^{\infty}\chi(n)a_nn^{-s} = \prod_{p }\frac 1 {\Dr_p(\chi(p)p^{-s})},\  \Lambda_\Dr(s,\chi)=\Gamma_\Dr(s) 
	\Dr(s,\chi),
	\end{align}
where $\deg\Dr_p (X)=d$ for all but finitely many $p$, and $\Dr_p(0)=1$.

\medskip
	%$\Dr(s)$ is determined by the generating function %$f=\sum_{n=1}^{\infty}a_nq^{n}$.


\smallskip\pause	
	In  many cases algebraicity of the zeta values was proven as
	%numbers
$$
	\frac{\Dr^*(s_0,\chi)}{\Omega^\pm_\Dr}\in \Q(\{\chi(n), a_n\}_n),   
	\text{ where } \Dr^*(s,\chi) \text{ is normalized by } \Gamma_\Dr,   
 $$ 
 %normalized zeta values  $\Dr^*(s_0, \chi)$ 
 at critical points $s_0\in{\mathbb Z}_{crit}$   %can be expressed as 
 as linear combinations of {\magenta coefficients} $a_n$ dividing out {\magenta periods} 
 $\Omega^\pm_\Dr$, where $\Dr^*(s_0,\chi)=\Lambda_\Dr(s_0,\chi)$ if $h^{\ell,\ell}=0$. 
	
	\iffalse
	if the functions $\Dr$ belong to a finite dimensional space  ${\mathcal M}$, $\dim_{\bar{\mathbb Q}}{\mathcal M}<\infty$, and $a_n\in {\bar{\mathbb Q}}$, because the linear form
	$f\mapsto \Dr^*(s_0),  {\mathcal M}\to {\bar{\mathbb Q}}$ 
	can be expanded on a basis of coefficients    
	$$
	f\mapsto a_n=a_n(f),  {\mathcal M}\to {\bar{\mathbb Q}}.
	$$
	\fi
	
	\smallskip 
	In  $p$-adic analysis,  the Tate field is used     
	${\mathbb C}_p=\hat {\bar {\mathbb Q}}_p$, 
	the completion of an algebraic closure ${\bar {\mathbb Q}}_p$, in place of $\C$.
	Let us fix embeddings  
	$
	\begin{cases} &
	i_p : \bar{\mathbb Q}\hookrightarrow {\mathbb C}_p  \\ &
	i_\infty : \bar{\mathbb Q}\hookrightarrow 
	{\mathbb C},
	\end{cases}
	$
	and try to continue analytically these zeta values to $s\in \Z_p$,  $\chi\bmod p^r$. 	
	
	\
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  *****  8
		\begin{frame}[shrink]
			\frametitle{Main result stated with Hodge/Newton polygons of $\Dr(s)$}  The Hodge polygon $P_{H}(t): [0,d]\to {\mathbb R}$ of the function $\Dr(s)$ and  \\ the Newton polygon $P_{N,p}(t): [0,d]\to {\mathbb R}$ at $p$ are piecewise linear: \\ 
			
		\smallskip\pause
			{\magenta The Hodge polygon } of pure weight $w$ has the slopes  ${j}$ of $length_{j}= {h^{{j}, w-{j}}}$ given by Serre's Gamma factors  of the functional equation of the form $s\mapsto w+1-s$, relating $\di \Lambda_\Dr(s,\chi) =\Gamma_\Dr(s) \Dr(s,\chi)$ and 
			$\Lambda_{\Dr^\rho}(w+1-s,\bar\chi)$, where  $\rho$ is the complex conjugation of $a_n$, and $\Gamma_\Dr(s) = \Gamma_{\Dr^\rho}(s)$ equals to the product  $\Gamma_\Dr(s)= 
			\prod_{{j} \le \frac w 2}\ds \Gamma_{{j}, w-{j}}(s)$, where
			$$
			\Gamma_{{j}, w-{j}}(s) =
			\begin{cases}\Gamma_\CC(s-{j})^{h^{{j}, w-{j}}}, &\text{ if } {j}< w,  \\ \Gamma_\R(s-{j})^{h_+^{{j}, {j}}}
			\Gamma_\R(s-{j}+1)^{h_-^{{j}, {j}}}
			, &\text{ if } 2{j}= w, \text{ where }
			\end{cases}
			$$
			$\ds
			\Gamma_\R(s)=\pi^{
				-\frac {s}{2}}\Gamma\left(\frac {s}{2}\right), \Gamma_\CC(s)=\Gamma_\R(s)\Gamma_\R(s+1)=
			2(2\pi)^{-s}\Gamma(s)$,
			 $h^{{j}, {j}}=h_+^{{j}, {j}}+h_-^{{j}, {j}}$, 
			 $\ds\sum_{{j}} h^{{j}, w-{j}}=d$.
			
		\smallskip\pause
			{\magenta The Newton polygon at $p$} is the convex hull of points
			$(i,\ord_p(a_i))\ \  (i=0,\dots,d)$; its slopes $\lambda$ are
			the   $p$-adic valuations $\ord_p(\al_i)$ of the inverse roots $\al_i$ of $\Dr_p(X)\in \bar \Q[X]\subset \CC_p[X]$:  
			%	under  $incl_p:\bar{\mathbb Q}\hookrightarrow {\mathbb C}_p$ :
			$length_\lambda= \sharp\{i\ |\ \ord_p(\al_i)=\lambda \}$.
			
		\smallskip\pause 
				
		\end{frame}
%	\end{document}
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\iffalse		
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  8
	\begin{frame}[shrink=1]
		\frametitle{Automorphic  zeta functions and their $p$-adic analogues} 
		%	\subsection*
	%	{Our main objects  in this talk are automorphic $L$-functions and their $p$-adic analogues.}
		
	For an algebraic group $G$ over a number field $F$,
	 such zeta functions
		%(or simply zeta-functions in Shimura's terminology) 
		are defined as
		certain Euler products over prime ideals $\p$ of the ring of integers $\Oc_F$ of $F$. 
		More precisely, we apply our constructions
		for the standard
		zeta functions of unitary groups studied in Shimura's book \cite{Shi00}.
		Recall first a simpler example of Yu.I.Manin-B.Mazur:
		\begin{exam}
			[$G = \GL(2), F =\Q, L_f (s) =
			\sum_{n\ge 1}a_nn^{-s} , s \in \C$.]
			Here $f(z) =
			\sum_{n\ge 1}a_nq^n$ is a modular form on the upper-half plane
			$$
			H =\{ z \in \C, \Im(z) > 0\}= \SL(2)/SO(2),  q = e^{2\pi iz}.
			$$ 
			The Euler product has the form
			$$
			L_f (s) =
			\prod_{p \ primes}
			(1- a_pp^{-s} + p^{k-1-2s} )^{-1}=\prod_{p \ primes}
			F_p(p^{-s})^{-1}, d=2,
			$$
			where $k$ is the weight of $f$. 
			%and $\psi_f$ the Dirichlet character of $f$. 
			%It is
			%defined iff the automorphic representation $\pi_f$ attached %to $f$ is irreducible, and
			%$\pi_f$ is generated by the lift $\tilde f$ of $f$ to the %group $G(\A)$. 
		\end{exam}
		\iffalse
		
		%, where $\A$
		%is the ring of adeles 
		%\begin{align*}
		%\A = &\{ x=(x_\infty, x_p)_{p } \ | \ x_\infty \in %\R, x_p\in \Q_p, \\ &\text{ such that }
		%x_p\in \Z_p\ 
		%\text{ for all but a finite number of }p\}.
		%\end{align*}
		%\fi
		
		\
		
	\end{frame}
	\fi
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 10
	%\iffalse
	\begin{frame}[shrink]
		\frametitle{A $p$-adic analogue of $L_f (s)$ (Manin-Mazur)}
		It is a $p$-adic analytic function $L_{f ,p}(s, \chi)$ of $p$-adic arguments
		$s \in\Z_p$,  $\chi \bmod p^r$ which interpolates algebraic numbers
		$$
		L^*_f (s,  \chi)/\omega^\pm\in\bar\Q\hookrightarrow\C_p = \hat{\bar\Q}_p \text{ (the Tate field)}
		$$
		for $1\le s\le k -1$, $\omega^\pm$ are {\magenta periods} of $f$
		(for example, in the Hurwitz case $f=g_\psi=f_m$, and $\omega^\pm=\Omega^m$ ). The complex analytic $L$ function of $f$ is defined for all $s\in\C$; in the absolutely
		convergent case $\Re(s) > (k + 1)/2$,
		$\di
		L_f^* (s,  \chi) = \Gamma_\CC(s)
		\sum_{n\ge 1}
		\chi(n)a_nn^{-s}, %\text{ where } \Gamma_\CC(s)=2\cdot (2\pi)^{-s}\Gamma(s),
		$
		is a holomorphic function with a functional equation $s\mapsto k-s$.
		According to Manin and Shimura, the ratio is algebraic if the {\magenta period} $\omega^\pm$ is chosen according to the parity $\chi(-1)(-1)^{-s} = \pm 1$.
		
		\medskip
		{\magenta Application}: if an elliptic curve $E/\Q$ with complex multiplication by $K$ has infinitly many rational points then 
		$L(E,1)=0$  because infinitely many primes $p$ divide 
		$L^*_{f_K} (1,  \chi)/\Omega$, see \cite{CoWi77}, \cite{RG78}. 
		
		
	\end{frame}
	\fi	
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ***** 		
		
		\begin{frame}[shrink]
			\frametitle{$p$-adic analytic interpolation of  %the Hermitian standard zeta function 
				$\Dr (s,  {\f} , \chi)$} The result expresses the zeta values as integrals with respect to $p$-adic Mazur-type measures.
			These measures are constructed from the Fourier coefficients of Hermitian modular forms, and from eigenvalues of Hecke operators on the unitary group. 
			
		\smallskip\pause
			{\bf \red Pre-ordinary case: $P_{H}(t)=P_{N,p}(t)$ at $t=\frac d 2$} 
			The integrality of  measures is proven 
			%by T.Bouganis \cite{Bou16}, 
			representing $\Dr^*(s,\chi)=\Gamma_\Dr(s) \Dr(s, \chi)$ as a Rankin-Selberg type 
			integral at critical points $s=m$. 
			Coefficients of modular forms in this integral satisfy Kummer-type congruences and produce certain bounded measures
			$\mu_\Dr$ from integral representations and Petersson product, \cite{CourPa}.  
			For the case of $p$ inert in $K$, see \cite{Bou16}. 
			
		\smallskip\pause
			{\bf \red Admissible case: $h=P_{N}(\frac {d}2)-P_{H}(\frac {d}2)>0$} 
			The zeta distributions are unbounded, but their sequence produce  $h$-admissible (growing) measures of
			 Amice-Vélu-type, allowing to integrate any continuous characters 
			$y\in\Hom(\Z_p^*, \C_p^*) =\Yc_p$.
			A general result is used on the existence of $h$-admissible (growing) measures from binomial congruences for the coefficients of Hermitian modular forms.
			 Their  $p$-adic {\magenta Mellin transforms} 
			  $\Lr_{\Dr}(y)=\int_{\Z_p^*} y(x) d\mu_\Dr(x)$,
			  $\Lr_{\Dr}:\Yc_p\to\CC_p$
			   give 
			 $p$-adic analytic interpolation  of growth  $\log_p^h(\cdot)$ of the $L$-values:   
			 the values $\Lr_{\Dr}(\chi x_p^m)$ are integrals given by $ \di i_p\left(\frac
			 	{{\cal D}^{*}(m,  {\f},\chi )}{\Omega_{\f}}\right)\in \CC_p$.
			 	
				\
				
			
			
			
			\end{frame}
			\iffalse		
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
		\begin{frame}[shrink]
			\frametitle{Hermitian modular groups: notations and definitions}
	Consider an imaginary quadratic field  $K={\mathbb Q}(\sqrt{-D_K})$ of the discriminant  $-D_K$, and its ring of integers
	${\mathcal O}={\mathcal O_K}$,  the quadratic character 
	$\theta_K$
	and Hermitian modular group   	of degree  ${{n}}$:
	$$
	\Gamma_{{n},K}=U(\eta_{n})({\mathbb Q})\cap \GL_{2{{n}}}({\mathcal O})
	$$
   
	
	Here  the notation is used $\eta_{n}=\mat 0_{{n}},-I_{{n}},I_{{n}},0_{{n}},$ 
	and the unitary group %of degree  ${{n}}$,
	 $$ U(\eta_{n})(R)=
	  \left \{  M=\mat A,B,C,D,\in  \GL_{2{{n}}}(R\otimes K) | M\eta_{n}M^*=\eta_{n} \right\}, \             
	$$
	for any commutative ring  $R$, 
	%$J_{{n}}=\mat 0_{{n}},-I_{{n}},I_{{n}},0_{{n}}, $ 
	 it is 
	a ${\mathbb Q}$-algebraic subgroup in 
	 the group of unitary similitudes 
	$$
	GU(\eta_{n})(R)= \left\{  M\in  \GL_{2{{n}}}(R\otimes K) \ |\   M\eta_{n} M^* =\nu \eta_{n}, \nu\in R^* \right\},          
	$$
where $M^*=	{}^t M^\rho$, $\rho$ is given by the complex conjugation
in $M_n(R\otimes K)$, and
$S = S_n = \{ s\in M_n(K)\ |\  s^*=s\}$, the set of Hermitian matrices with entries in $K$.

			
			\
			
			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
\fi
%\end{document}
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
	\begin{frame}[shrink]
		\frametitle{A Hermitian modular form of weight $\ell$ with
			character $\sigma$} is  a  holomorphic function $F$ on $\Hr_{n}$ $({n}\ge 2)$ such that 
		$F(g\langle Z\rangle) = \sigma(g)F(Z)j(g,Z)^\ell$ for any 
		$g \in \Gamma_{n,K}$. 
		Here $\sigma$ be a character of $\Gamma_K^{({n})}$, trivial on
		$
		\left\{\mat 1_{n},B,0,1_{n},
		\right\},
		$
		and for $Z \in \Hr_{n}$, put
		$g\langle Z\rangle = (AZ + B)(CZ + D)^{-1}$, 
		$j(g,Z) = \det(CZ +D)$.
		
	\smallskip\pause
		{\magenta Fourier expansions}:  a semi-integral Hermitian matrix is a Hermitian matrix $H \in (\sqrt{-D_K})^{-1}M_{n}(\Oc)$
		whose diagonal entries are integral.
		
		Denote the set of semi-integral Hermitian matrices by $\Lambda_{n}(\Oc)$,\\
		the subset of its positive definite elements  is  $\Lambda_{n}(\Oc)^+$.
		%For $ H \in \Lambda_{n}(\Oc)$, put
		%$\gamma(H) = (-D_K)^{[{n}/2]} \det H$, $\gamma(H)\in\Z$. 
		
		\smallskip
		{\magenta A Hermitian modular form $F$ is called a cusp form}  
		if it has a Fourier expansion of the form
		$\di
		F(Z)=\sum_{H\in\Lambda_{n}(\Oc)^+}A(H) q^H.
		$
		Denote the space of cusp forms of weight $\ell$ with character $\sigma$ by $\Sr_\ell(\Gamma_{n,K},\sigma)$.
		
		\
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8		
\begin{frame}[shrink]
	\frametitle{The standard zeta function of  a Hermitian modular form}
	
	Fix  an integral ideal ${\mathfrak c}$ of $\Oc_K$. 
	Denote by $C\subset \Gamma_{n,K}$ the congruence subgroup of level $\c$; the group is essentially a principal congruence subgroup; it is an analogue of the group $\Gamma_0(N)$ in the elliptic modular case.  
	Write $T(\a)$ for the Hecke operator associated to it as it is defined in \cite{Shi00}, page 162, using the action of double cosets $C\xi C$ with
	$\xi=\diag(\hat D,  D)$, $(\det (D))=(\al)$, $\hat D=(D^*)^{-1}$.
	
	\smallskip\pause
	Consider a non-zero Hermitian modular form ${\f} \in \Mc_k(C,\psi)$ and assume ${\f}|T(\a) = \lambda(\a)\f$ with $\lambda(\a) \in \CC$ for all integral ideals $\a \in\Oc$. Then
	%For  a Hecke ideal character $\chi$ of $K$ of conductor ${\mathfrak f}$, Consider the Dirichlet series: for  $Re(s) \gg 0$,
	\begin{align}
	\nonumber
	\Zc(s,  {\f} ) =\left(\prod_{i=1}^{2n}
	L_\c(2s -i + 1, \theta^{i-1})\right)
	\sum_{\a}
	\lambda(\a)N(\a)^{-s}, 
	\end{align}
	the sum is over all integral ideals of $\Oc_K$. 
	%It is shown in [\cite{Shi00}, page 171] that 
	
	\smallskip\pause
	This series has an Euler product representation
	%, which we write as 
	$\Zc(s, {\f}) =
	\prod_{\q} (\Zc_\q (N(\q)^{-s})^{-1}$, where the product is over all prime ideals of $\Oc_K$,
	{\red  $\Zc_\q (X)$ is the numerator
	of the series 
	$\sum_{r\ge 0} \lambda(\q^r)X^r\in \CC(X)$, computed by Shimura as follows.}
	
	\
	
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{Euler factors of the standard zeta function, \cite{Shi00}, p. 171 }
	%	{\magenta Euler factors of the stanard zeta function.}
	%	Following %\cite{Bou16}, p.12, and 
	%	\cite{Shi00}, page 171, 
	The Euler factors $\Zc_{\q}(X)$ in the Hermitian modular case
	at the prime ideal $\q$ of $\Oc_K$ are
	\smallskip\pause
	\begin{align} \nonumber
	\text{ (i) } &\Zc_\q(X) =\prod_{i=1}^{n}
	{\left((1- N(\q)^{n-1}t_{\q,i}X)(1- N(\q)^{n}t_{\q,i}^{-1} X)\right)}^{-1},\\ & 
	\nonumber
	\text{ if } \q^{\rho} = \q \text{ and }\q \nd \c, \text{(\magenta the inert case outside level }\c),\\
	\smallskip\pause
	\text{(ii) } \nonumber & \Zc_{\q_1}(X_1) \Zc_{\q_2}(X_2) =
	\prod_{i=1}^{2n}
	\left((1- N(\q_1)^{2n}t_{\q_1\q_2,i}^{-1}X_1)
	(1- N(\q_2)^{-1}t_{\q_1\q_2,i}X_2)\right)^{-1},  \\ &  \nonumber
	\text{ if }
	\q_1 \ne \q_2, \q_1^{\rho}= \q_2 \text{ and } \q_i \nd \c \text{
		for } i = 1,2\text{ \magenta (the split case outside  level) }, \\
	\nonumber
	\smallskip\pause
	\text{(iii) } &\Zc_\q(X) = \prod_{i=1}^{n} 
	\left(1- N(\q)^{n-1}t_{q,i}X\right)^{-1}, \text{ if } \q^{\rho} = \q \text{ and } \q|\c 	\text{   \magenta  (inert level divisors )},\\  \nonumber
	\text{(iv) } & \Zc_{\q_1}(X_1) \Zc_{\q_2}(X_2) =
	\prod_{i=1}^{n}
	\left((1- N(\q_1)^{n-1}t_{\q_1\q_2,i}^{-1}X_1)
	(1- N(\q_2)^{n-1}t_{\q_1\q_2,i}X_2)\right)^{-1},  \\ &  \nonumber
	\text{ if } \q_{1} \ne \q_{2},  \q_i | \c \text{ for } i = 1,2
	\text{   \magenta (split level divisors).  } 
	\end{align}
	where the $t_{?,i}$ above for $? = \q, \q_{1}\q_{2}$, are the {\magenta Satake parameters } of the eigenform $\f$.

	\
	
	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[shrink]
	\frametitle{Notice the important dychotomy for the $L$-factors} 
in the Siegel modular case (that is, of symplectic type)  vs. the Hermite modular case (of unitary type). 
In these cases the correspopnding complex 
component of the Langlands $L$-group is either $GSpinO(2n+1)(\CC)$,
with the Euler factors of degree $2n+1$ (the standard representation of $GO(2n+1)$,  resp. of  degree $2^n$ (the spinor representation of the  $L$-group) (the symplectic case),
or, in the Hermite case,  the complex component of the $L$-group is
$GL_{2n}(\CC)\times GL_{2n}(\CC)$,  with the Euler factors of degree  $4n$  (the standard representation of the $L$-group),
	see also 16.16,  p.133, in particular, formula (16.16.2) at p.134 of \cite{Shi97a} 
	or \cite{Shi97b} for a concise exposition.
	
	\



\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[shrink]
	\frametitle{The standard motivic-normalized zeta
		 $\Dr (s,\f,\chi)$}
	The standard zeta	function of $\f$ is defined by means of the
	 $p$-parameters as the
	following Euler product:
	$$ 
	\Dr(s, {\bf f}, \chi ) = \prod_p \prod_{i=1}^{2n}
	\left\{  \left( 1 - \frac{\chi(p)\al_i(p)}
	{p^s} \right) 
	\left( 1 - \frac{\chi(p)\al_{4n-i}(p)}
	{p^s} \right) \right\}^{-1} ,  
	$$
	where $\chi$ is an arbitrary Dirichlet character.
	The  $p$--parameters  
	$\alpha _{1}(p),  \dots,  \alpha _{4{n}}(p)$ of $\Dr (s,\f,\chi)$ for $p$ not dividing the level $C$ of the form $\f$ are related to the
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%******
	the $4{n}$ characteristic numbers
	$$
	\alpha _{1}(p),\   \cdots,\   \alpha _{2{n}}(p),\  \alpha_{2{n}+1}(p),\  \cdots,\   \alpha _{4n}(p)
	$$
	of the product of all $\q$-factors $\Zc_\q( N\q^{(n'+\frac{1}{2}) }X)^{-1}$ for all $\q|p$, which is a polynomial of degree 4${n}$ of the variable  $X=p^{-s}$ (for almost all $p$) with coefficients in a number field $T=T(\f)$ .	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\smallskip
	There is a relation between the two normalizations 
	$\magenta \Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})=\Dr(s,{\f})$ 
	explained below, 
	see \cite{Ha97} for general zeta functions
	 $\Zc(s,\f)$ of type introduced in \cite{Shi00}, 
	 using reprsentation theory of unitary groups and 
	 Deligne's motivic $L$-functions.
	
	\
	
	\smallskip\pause
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
	\begin{frame}[shrink]
		\frametitle{Description of the Main theorem} 		
	Let $\Omega_{\f}$ be a period attached to  an Hermitian cusp eigenform ${\f}$, $\magenta \Dr(s,{\f})=\Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})$ the standard zeta function, and
	$$
			\alpha _{{\f}}=\alpha _{{\f},p}=\left(\prod_{\q|p}\prod_{i=1}^n t_{\q,i}\right) p^{-n(n+1)}, \ \
		h 
	= \ord_{p}(\alpha _{{\f},p}),
		$$
The number $\alpha_{\f}$	turns out to be an eigenvalue of Atkin's type
operator $U_p : \sum_{H} A_H q^H\mapsto \sum_{H} A_{pH} q^H$ on some $\f_0$,
and $h=P_{N}(\frac {d}2)-P_{H}(\frac {d}2)$.
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pause
	\hskip-10pt {\Large \blue Definition.} 
	Let $M$ be a $\Oc$-module of finite rank where $\Oc\subset \C_p$.
	For $h\ge 1$, consider the following $\C_p$-vector spaces of functions on $\Z_p^*$ : $\Cc^h\subset \Cc^{loc-an}\subset \Cc$.
	Then
	 
	- a continuous homomorphism $\mu: \Cc\to M$  is called a {\magenta (bounded) measure} $M$-valued measure on $\Z_p^*$.
	
	- $\mu: \Cc^h\to M $ is called an {\magenta $h$ admissible measure} $M$-valued measure on $\Z_p^*$ measure
if the following growth condition  is satisfied	
	 $$
	 \left |\int_{a+(p^v)}(x-a)^j d\mu\right|_p \le p^{-v(h-j)}
$$	 
for $j=0,1,...,h-1$, and et  $\Yc_p=Hom_{cont}(\Z_p^*, \cb_p^*)$ be the space
of definition of  {\magenta $p$-adic Mellin transform}

\hskip-10pt{\Large\blue  Theorem} (\cite{Am-V}, \cite{MTT}) For an $h$-admissible measure
$\mu$, the Mellin transform $\Lr_\mu: \Yc_p\to\C_p$ exists and has growth $o(\log^h)$ (with infinitely many zeros).	
		
		
	\
	
	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	
\begin{frame}[shrink]
	\frametitle{Main Theorem.}
		Let  ${\f} $ be a Hermitian cusp eigenform of 
		degree $n\ge 2$ and of weight $\ell > 4n+2$.
		 There exist  distributions  
		$\mu_{\Dr, s}$ for $s=n,\cdots, \ell-n$ 
		%for $s=1-k+n,\cdots,2n'+1-n+k$
		% a ${\CC}_{p}$--analytic function
		%$L^{(\ep  _{0})}_{(p)}$ on ${\cal Y}_{p}$ 
		with the properties:
		\par
		\medskip
		i) 
		for all pairs $(s, \chi)$ such that $s \in  {\Z}$ with
		 $n\le s\le \ell-n$,  
		%and  for $s=1$ the character $\chi ^{2}$ is non--trivial,
		$$
		\int_{\Z_p^*}\chi  d\mu_{\Dr,s} = 
		%G(\chi )^{-{n}-1}
		A_{p}(s,\chi)% = \prod_{\q | p}A_{\q}(\c,s, \chi) 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\frac{{\cal D}^{*}(s,  {\f},\overline \chi )}{\Omega_{\f}}
		$$
		(under the inclusion $i_p$), with elementary factors $A_{p}(s,\chi) = \prod_{\q | p}A_{\q}(s, \chi)$
		including a finite Euler product, gaussian sums, the conductor 
		of $\chi$; the integral is a finite sum. 
		
		\
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%			
			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
			\begin{frame}[shrink]
				\frametitle{}
				%{Main theorem  (continued) for 
				%$ \Dr(s,{\f})=\Zc(s-\frac{\ell}{2}+\frac{1}{2},{\f})$ }	
					%\begin{theo}
				(ii) if				
             $\ord_{p}
\left((\prod_{\q|p}\prod_{i=1}^n t_{\q,i}) 
				p^{-n(n+1)}\right)=0$
				 then the above distributions $\mu_{\Dr,s}$ are bounded measures, we set  $\mu_\Dr=\mu_{\Dr,s^*}$ and the  integral is defined for all continuous characters 
				 $y\in\Hom(\Z_p^*, \C_p^*) =:\Yc_p$.
				 %%%%%%%%%%%%%%%%%%%%%%%%%%%
				 
				 \smallskip
				 Their {\magenta  Mellin transforms} 
				 $\Lr_{\mu_\Dr}(y)=
				 \int_{\Z_p^*}yd \mu_\Dr$, 
				 $\Lr_{\mu_\Dr}: \Yc_p\to \CC_p$, \\
				 % $\Lr_{\mu_\Dr^-}(y)=$
				 % $\int_{\Z_p^*}yd \mu_\Dr^-$, 
				 % $\Lr_{\mu_\Dr^-}: \Yc_p\to \CC_p$,\\
				 give bounded $p$-adic analytic interpolation   of the above $L$-values
				to %bounded ${\CC}_{p} $--analytic function 
				on the $\CC_p$-analytic group $\Yc_p$;
				and these distributions are related by: 
				$\ds \int_X \chi d\mu_{\Dr,s}=\int_X \chi x^{s^*-s}d\mu_\Dr^*$, $X=\Z_p^*$, 
				where $s^*=\ell-n$, $s_*=n$. 
				
			\smallskip\pause
				(iii) in the  {\magenta admissible}  case assume that $0<h \le 
				\ds \frac { s^*-s_*+1} 2= \frac{\ell+1-2n}2$, where
					$h=\ord_{p}\left((\prod_{\q|p}\prod_{i=1}^n t_{\q,i}) 
					p^{-n(n+1)}\right)>0$,
					%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
					Then there exist $h$--admissible measures 
					$\mu_\Dr$ whose integrals 
				$\int_{\Z_p^*} \chi x_p^s d\mu_\Dr$ are
					given by $ \di i_p\left(	A_{p}(s,\chi)% = \prod_{\q | p}A_{\q}(\c,s, \chi) 
					%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
					\frac{{\cal D}^{*}(s,  {\f},\overline \chi )}{\Omega_{\f}}\right)\in \CC_p$ with $A_{p}(s,\chi)$ as in (i);
					 their Mellin transforms
					 $\Lr_{\Dr}(y)=\int_{\Z_p^*} y d\mu_\Dr$,
					 %$\Lr_{\Dr^\pm}:\Yc_p\to\CC_p$
					 belong to the type
				  $o(\log  x^{h}_{p})$.
				  \iffalse 
				  with 
			and it is given by the Mellin transform of an $h$--admissible measure;
				%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
				 %Their  {\magenta Mellin transforms} 
				 $\Lr_{\Dr^\pm}(y)=\int_{\Z_p^*} y d\mu^\pm_\Dr$,
				 $\Lr_{\Dr^\pm}:\Yc_p\to\CC_p$
				 give $p$-adic analytic interpolation  of growth  $\log_p^h(\cdot)$ of the $L$-values:   
				 the values $\Lr_{\Dr^\pm}(\chi x_p^m)$ are integrals given by $ \di i_p\left(\frac
				 {{\cal D}^{*}(m,  {\f},\chi )}{\Omega_{\f}}\right)\in \CC_p$.
\fi
			\\
			\smallskip\pause
				(iv) %if $h \le  (s^{*} - s_{*} + 1)/2 = k - {n}$ then 
				the functions $\Lr_{\Dr}$ are determined by
				% the above conditions
				 (i)-(iii).
			%\end{theo}
	
	\hskip-10pt{\blue\Large Remarks.} \\
	(a) Interpretation of  $s^{*}$: 
	the smallest of the "big slopes" of
	$P_{H}$ 
	
	(b) Interpretation of  $s_{*}-1$:
	the biggest of the "small slopes" of
	$P_{H}$.
%	(c) We have  $\Dr^*(s_0,{\bf f}, \chi)=
%	\Lambda_\Dr(s_0,{\bf f},\chi)$, 
%	because in this case  $h^{\ell,\ell}=0$, and all the critical points $s_0$ %are	$s_{*} + 1 \le s_0\le  s^{*}$. %(no parity condition).
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{Eisenstein series and congruences \red  (KEY POINT!)} The (Siegel-Hermite)Eisenstein series $E_{2\ell}^{({n})}(Z)$ of weight $2\ell$, character $\det^{-\ell}$,  is defined by
	$\di
	E_{2\ell}^{({n})}(Z)=
	\sum_{g\in\Gamma_{K,\infty}^{({n})}\bs \Gamma_K^{({n})}}
	(\det g)^{\ell}j(g,Z)^{-2\ell} .
	$
	The series  converges absolutely for $\ell >{n}$. 
	Define the normalized Eisenstein series ${\cal E}_{2\ell}^{({n})}(Z)$ by
	$
	{\cal E}_{2\ell}^{({n})}(Z)=$
	$2^{-{n}}\prod_{i=1}^{{n}}L(i-2\ell, \theta^{i-1})\cdot  E_{2\ell}^{({n})}(Z)
	$
	If $H\in\Lambda_{n}(\Oc)^+$, then the $H$-th Fourier coefficient of ${\cal E}_{2\ell}^{({n})}(Z)$ is polynomial over $\Z$ in $\{p^{\ell-({n}/2)}\}_p$, and equals 
	$$
	|\gamma(H)|^{\ell-({n}/2)}
	\prod_{p|\gamma(H)}\tilde F_p(H,p^{-\ell+({n}/2)}),
	\gamma(H) = (-D_K)^{[{n}/2]} \det H.
	$$	
	Here, $\tilde F_p(H,X)$ is a certain Laurent polynomial in the variables $\{X_p=p^{-s}, X_p^{-1}\}_p$ over $\Z$. % arising from the Siegel series for $H$.
	This polynomial is a {\red key point in proving congruences} for
	the modular forms in a {\red Rankin-Selberg integral}. 
	
	\smallskip
	Also, we set , for $s\in\CC$ and a Hecke ideal character $\psi \bmod \c$,
	$$ \magenta
	E(Z,s,\ell,\psi)=
	\sum_{g\in C_\infty\bs C}
	\psi(g) (\det g)^{\ell}j(g,Z)^{-2\ell}|(\det g)j(g,Z)|^{-s} .
	$$
	
\	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{An integral representation of Rankin-Selberg type} 
	%\iffalse	
	%Let us specialize to $F=\Q$
	 The integral representation of Rankin-Selberg type in the Hermitian modular case: % is used (\cite{Bou16}, p.13):
	%"The following theorem (see also [25, Theorem 7.8]) is obtained by combining results of
	%Shimura [31] and Klosin [25]. For details we refer to [6, Section 4], 
	
	{\blue \Large Theorem 4.1 (Shimura, Klosin)}, see \cite{Bou16}, p.13.\\
	Let $0 \ne {\f} \in\Mc_\ell(C,\psi ))$ of scalar weight $\ell$, $\psi
	\bmod \c$, such that $\forall \a, {\f} |T(\a) = \lambda(\a){\f}$, and assume that $2\ell\ge n$, then there exists ${\cal T} \in S_+\cap \GL_n(K)$ and ${\cal R} \in \GL_n(K)$ such that	
	\begin{align}\nonumber
	& \Gamma ((s)) \psi(\det({\cal T}))%c_{\f} (\tau, r)
	\Zc(s + 3n/2, {\f} , \chi) =	\\ & \nonumber
	\Lambda_{\c}(s + 3n/2, \theta\psi\chi)\cdot
	%\\ \nonumber
	%&  \left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2} C_0
	%\Lambda_\c(s + 3n/2, \theta(\psi\chi)_1)\cdot
	%\\ & \nonumber
	%\left( \prod_{v\in {\bf b}}
	%g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
	%\right)
	%\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}
    C_0 \langle {\f} , \theta_{\cal T}(\chi)
	%\tilde E_\A(x, 
	{\bf E}(\bar s + n, \ell - \ell_\theta, \chi^\rho\psi) \rangle_{C\sec},
	\end{align}
	where ${\bf E}(Z, s, \ell-\ell_\theta, \psi)_{C\sec}$ is a
	normalized
	 group theoretic %adelic
	Eisenstein series with components as above of level $\c\sec$ divisible by $\c$, and weight $\ell-\ell_\theta$. 	
	Here $\langle \cdot, \cdot \rangle_{C\sec}$ is the  
	%group theoretic %adelic
	normalized
	Petersson inner product associated to the congruence subgroup 
	$C\sec$ of level $\c\sec$.
	%$D[{\e}, {\e}{\h}]$,
	% as defined for example in [30, Equation 10.9.6],	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\
	$\ds \Gamma((s))=
	%\prod_{v\in {\bf a}}
	(4\pi)^{-n(s+h)}\Gamma^\iota_n(s + h), 
	{\Gamma}^\iota_n(s) = \pi^{\frac{n(n-1)}2} \prod_{j=0}^{n-1} {\Gamma}(s-j)$,\\ 
	where $h=0	\text{ or } 1, \ 
	C_0$
	%:=\frac{[\Gamma_0(\c\sec):\Gamma]}{\sharp X}.	
	\ a subgroup index.
	%\fi
	
	\
	
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8



	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{The Hodge polygon of the Hermitian  zeta function}
Starting from the Gamma factors 
%of the functional equation 
of the standard Hermitian $L$-function $\Dr(s,{\f},  \chi)$ 
%normalized with respect to the symmetry $s\mapsto 1-s$,
let us describe the Hodge polygon for $F=\Q$.

The explicit form of the Gamma factors of the standard Hermitian $L$-function
$\Zc(s, {\f})$  was studied 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
in (cf. \cite{Shi00}, p.179,  \cite{Ha97},  \cite{Ha14},
%and formulas (22.4a), (16.47)), 
 \cite{Kl},  \cite{Bou16}, \cite{Ge16}), and that of $\Dr(s,{\f},  \chi)$ follows with the Gamma factor 
 \iffalse
 Galois equivariance of critical values of L-functions for unitary groups
 Lucio Guerberoff, Jie Lin
"\dots The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic L-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of $\GL_{n}\times\GL_{1}$. We show that our results are compatible with Deligne's conjecture".  
 Subjects: 	Number Theory (math.NT)
 MSC classes: 	11F67 (Primary) 11F70, 11G18, 11R39, 22E55 (Secondary)
 Cite as: 	arXiv:1612.09590 [math.NT]
 (or arXiv:1612.09590v1 [math.NT] for this version)
 \fi
\begin{align*}&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Gamma_\Dr(s)=L_{\infty }(s,{\f},  \chi) = 
\prod_{j = 0}^{n-1} \Gamma _{{\CC}}(s-j)^2, 
\\ & \text{ with the symmetry } s \mapsto \ell-s. 
\end{align*}
 

\smallskip
These factors suggest the following form of the Hodge polygon of $\Dr(s, {\f},\chi)$ of rank $d=4{n}$  
as that of the Hodge numbers $h^{{j}, w-{j}}$ below (in the increasing
order of slopes ${j}$, with weight $w=\ell-1$):
\begin{align*} 
	&2\cdot (0, \ell-1), \dots , 2\cdot(n-1,  \ell-n), 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\\ 
	%\nonumber  &
	%(-k+1+{m}, k-1-{m}), \cdots, ( -k+{n}+{m}, k-{n}-{m}) , 
	%\\ 
	%\nonumber  &
	%(2n'+1-n+k, -k+{n}), \cdots  , (2n'+k-1, 1),
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\ 
		\nonumber  &
	 2\cdot(\ell-n, n-1),\dots ,2\cdot( \ell-1,0) , 
\end{align*}
following Serre's recipe \cite{Se70}, p.11.	 
%$ \bar H^{rs;w}(W_\mu^\nabla)$ \cite{Ha97},p. 105

%This comes from a Rankin-Selberg product of CM-type.

%\smallskip
%{\red A striking programm by M.Harris realizes general  Delignes %$\Dr(s)$-functions as factors of such products!}


\
		
	\end{frame}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
	\begin{frame}[shrink]
		\frametitle{Geometric study in the $p$-ordinary case } This case corresponds to the coincidence of the Hodge polygon and the Newton polygon, it was considered 
		in \cite{EHLS}
		%by Eischen Ellen E., Harris, Michael, Li, Jian-Shu, Skinner,Christopher M., $p$-adic $L$-functions for unitary groups, arXiv:1602.01776v3 [math.NT]) 
		using methods of algebraic geometry and the theory of algebraic modular forms, 
% These powerful methods also used and lead to the proof of the Iwasawa % Main Conjecture (2014) in
%	\cite{MC}
%	by E.Urban and Ch.Skinner. 
\smallskip\pause
	These methods use infinite dimensional towers of spaces over
	${\bar {\mathbb Q}}$ containing automorphic forms of all levels of type $Np^r$, and their specializations at CM-points on Shimra varieties.  
	
\smallskip\pause
	On the other hand, the case $p$ inert in $K$  was studied in \cite{Bou16}, based on methods in \cite{CourPa}. % with my student Michel Courtieu. 
	
\smallskip\pause
	The present  method treats all $p$ unramified in $K$ and coprime to the level $\c$ of ${\bf f}$; it is  based on a modular construction of {\magenta admissible measures} as sequences of zeta distributions via an integral representation of Rankin-Selberg type.
	This method allows to reduce consideration to congruences 
	between Hermitian modular forms
	 of fixed level $\c p$.
		
\

\end{frame} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{document}

\iffalse

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 11
\begin{frame}[shrink]
\frametitle{Constructions of $p$-adic analogues
	of complex $L$-functions }
An irreducible automorphic representation $\pi$ of
adelic group $G(\A_K)$:
$$
L(s,  \pi,  r, \chi) =
\prod_{
	{\mathfrak p}_v\, primes\, in\, K}
\prod_{j=1}^d
(1-\beta _{j,{\mathfrak p}_v} N {\mathfrak p}_v^{-s} )^{-1}, 
\text{ where }\pi=\pi(f).% \text{ is generated by } f.
$$
 is an Euler product giving an $L$-function,
where $v \in \Sigma_K$ (places in
$K$), ${\mathfrak p} ={\mathfrak p}_v$,
$\alpha_{i,{\mathfrak p}}$ the Satake parameters of $\pi=\bigotimes_v\pi_v$, 
$$
\prod_{j=1}^d
(1-\beta _{j,{\mathfrak p}} X )
= \det(1_d-r(\diag(\alpha_{i,{\mathfrak p}})X)),
$$
$h_v =\diag(\alpha_{i,\mathfrak{p}})\in {}^LG(\C)$ (the Langlands group), $r : {}^LG(\C)\to \GL_d(\C)$ a finite dimensional representation, and
$\chi : \A^*_K/K^*\to \C^*$ is a character of finite order.

{\red Constructions extend to general automorphic representations on Shimura varieties} via the following tools: 


\begin{itemize}
	\item Modular symbols and their higher analogues (linear forms on cohomology spaces related to automorphic forms)
	\item Petersson products with a fixed automorphic form, or
	\item linear forms coming from the Fourier coefficients (or Whittaker
	functions), or throught the
	\item CM-values (special points on Shimura varieties),
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\

\end{frame}
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9
\begin{frame}[shrink]
	\frametitle{A general setting of the theory of $p$-adic $L$-functions} Following \cite{EHLS}, p.4 one can consider "a $p$-adic analytic
space $Y$ and a subset $Y^{alg}$ of points such that, for each $y \in Y^{alg}$ there is a motive $M_y$,
such that 0 is a critical value of the
$L$-function $L(s,M_y)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In our situation, we start with a CM field $K$ over $\Q$, a quadratic extension of a totally
real field $K^+$, and an $n$-dimensional hermitian vector space $V/K$. Then $Y$ is the space
of pairs $(\lambda,\chi)$, where $\lambda$ runs through the set of ordinary $p$-adic modular forms on the
Shimura variety $Sh(V )$ attached to $U(V )$ and $\chi$ runs through $p$-adic Hecke characters
of $K$; both $\lambda$ and $\chi$ are assumed to be unramified outside a finite set $S$ of primes of $K$,
including those dividing $p$, and of bounded level at primes not dividing $p$". 

\

\end{frame}
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 11
\begin{frame}[shrink]
\frametitle{Accessible cases: symplectic and unitary groups}
	\begin{itemize}
		\item
		$G = \GL_1$ over $\Q$ (Kubota-Leopoldt-Mazur) for the Dirichlet
		$L$-function $L(s, \chi)$.
		\item
		$G = \GL_1$ over a totally real field $F$ (Deligne-Ribet, using algebraicity result by Klingen).
		
		\item
		$G = \GL_1$ over a CM-field $K$, i.e. a totally imaginary extension of a totally real field $F$ (N.Katz, Manin-Vishik).
		\item {the Siegel modular case}
		$G = \GSp_n$ (the Siegel modular case, $F=\Q$). 
		\item{General symplectic and unitary groups  over a CM-field $K$}
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		Certain Euler products in Chapter 5 of \cite{Shi00}, 
		with critical values computed in Chapter 7, Theorem 28.8 using general nearly holomorphic arithmetical automorphic  forms for the group
		$$
		G= G(\varphi) =\{ \al\in \GL_{n+m}(K)\ |\  \al\varphi\al^*=\nu(\alpha)\varphi
		\},\nu(\al)\in F^*,
		$$
		where
		$
		\varphi=
		\begin{pmatrix}
		1_n&0\\
		0&1_m
		\end{pmatrix}
		$, or
		$\varphi=\eta_n =
		\begin{pmatrix}
		0&1_n\\
		-1_n& 0
		\end{pmatrix}, n=m 
		$, see also
		Ch.Skinner and E.Urban \cite{MC} and Shimura G., \cite{Shi00}.
	\end{itemize}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
\

\end{frame}
%\fi
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13
\begin{frame}[shrink]
	\frametitle{Nearly holomorphic modular forms for $\GL(2)$}N.M.Katz \cite{Ka76} 
		 used real-analytic and quasimodular forms coming
	from derivatives of holomorphic forms, and $p$-adic modular forms. 
	
	A relation real-analytic $\leftrightarrow p$-adic modular forms comes from the
	notion of $p$-adic modular forms invented by J.-P.Serre \cite{Se73} as
	$p$-adic limits of $q$-expansions of modular forms with rational
	coefficients for $\Gamma= \SL_2(\Z)$, using the ring $M_p$ of $p$-adic modular forms contains $\Mc = \oplus_{k\ge 0}\Mc_k(\Gamma, \Z) = \Z[E_4,  E_6]$, and it contains $E_2$ as an element with the $q$-expansion
	$E_2 = 1- 24
	\sum_{n\ge 1}\sigma_1(n)q^n$. 
	On the other hand, the function of $z=x+iy$,  
	$$
	\tilde E_2 = -\frac {3}{\pi y}
	+ E_2 =-12S + E_2,  \text{ where }
	S =
	\frac {1}{4\pi y},
	$$
	is a nearly holomorphic modular form (that is,  its coefficients are
	polynomials of $S$ over $\Q$). 
Then ${\mathcal N}=\Z[\tilde E_2, E_4, E_6]$ is the ring of such forms, 	
	$\tilde E_2|_{S=0} = E_2$
	and $E_2$ is a $p$-adic modular form. 
	
	Elements of the ring $\Qc{\Mc}={\mathcal N}|_{S=0}
	=\Z[E_2, E_4, E_6]$ are called quasimodular forms hence such forms are all $p$-adic modular forms. 
	These
	phenomena are quite general and used in computations and
	proofs.
	S.Boecherer extended these
	results to the Siegel modular case.	
	
	\
	
\end{frame}
%\fi
%\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\iffalse
\begin{frame}[shrink]
	\frametitle{Automorphic $L$-functions attached to symplectic and unitary groups}
	Let us briefly describe the $L$-functions attached to symplectic and unitary groups as certain Euler products in Chapter 5 of \cite{Shi00}, 
	with critical values computed in Chapter 7, Theorem 28.8 using general nearly holomorphic arithmetical automorphic  forms for the group
	$$
	G= G(\varphi) =\{ \al\in \GL_{n+m}(K)| \al\varphi\al^*=\nu(\alpha)\varphi
	\},\nu(\al)\in F^*,
	$$
	where
	$
	\varphi=
	\begin{pmatrix}
	1_n&0\\
	0&1_m
	\end{pmatrix}
	$, or
	$\varphi=\eta_n =
	\begin{pmatrix}
	0&1_n\\
	-1_n& 0
	\end{pmatrix}, n=m 
	$, see also
	Ch.Skinner and E.Urban \cite{MC} and Shimura G., \cite{Shi00}.
\end{frame}
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13
\begin{frame}[shrink]
\frametitle{The groups and automorphic forms in \cite{Shi00}}
Let $F$ be a totally real algebraic number field, $K$ be a totally
imaginary quadratic extension of $F$ and $\rho$ be the generator of
$\Gal(K/F)$. 
Take $\eta_n =
\begin{pmatrix}
0&1_n\\
-1_n& 0
\end{pmatrix}
$
and define
\begin{align*} &G = \Sp(n, F) &&\text{(Case Sp)}
\\ & G = \{\al\in \GL_{2n}(K)| \al\eta_n\al^*= \eta_n\} &&\text{(Case UT = unitary tube)}
\\ &G = \{\al\in \GL_{2n}(K)| \al T\al^* = T\} &&\text{(Case UB =
	unitary ball)}
\end{align*}
according to three cases.
Assume $F=\Q$ for a while. The group of the real points $G_\infty$ acts on the associated domain
\begin{align*}
{\mathcal H} =
\begin{cases}\{ z \in M(n, n,\C)\  | \ {}^tz = z, \Im(z) > 0\} & \text{(Case Sp)}\\
\{ z \in M(n, n,\C)\  | \  i(z^*-z) > 0\} & \text{(Case UT)}\\
\{ z \in M(p, q,\C)\  | \  1_q-z^*z > 0\} & \text{(Case UB).}
\end{cases}
\end{align*}
$(p, q), p + q = n$, being the signature of $iT$ . Here $z^* = {}^t\bar z$ and >
means that a hermitian matrix is positive definite.
In Case UB, there is the standard automorphic factor
$M(g,  z)$, $g\in G_\infty$,  $z \in \Hr$ taking values in 
$\GL_p(\C) \times\GL_q(\C)$.

\

\end{frame}
%\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}[shrink]
	\frametitle{Shimura's arithmeticity  \cite{Shi00}, $p$-adic zeta functions
		and nearly-holomorphic forms on classical groups}
%	{\it Automorphic $L$-functions via general quasi-modular forms.}
	Automorphic $L$-functions and their $p$-adic analogues can be obtained
	for quite general automorphic representations on Shimura varieties by constructing $p$-adic distributions out of algebraic numbers attached to automorphic forms. 
	This means that these numbers satisfy certain Kummer-type congruences established 
	in different  ways: via
	\begin{itemize}
		\item Normalized Petersson products with a fixed automorphic form, or
		\item linear forms coming from the Fourier coefficients (or Whittaker
		functions), or through the
		\item CM-values (special points on Shimura varieties),
		see The Iwasawa Main Conjecture for GL(2) by C. Skinner and E.
		Urban, [MC], Shimura G., Arithmeticity in the theory of automorphic forms \cite{Shi00}.
		
	
		
	\end{itemize}
	
	The combinatorial structure of the Fourier coefficients of the
	holomorphic forms used in these constructions is quite complicated.
	
	In order to prove the congruences needed for the $p$-adic constructions, we use a simplification due to nearly-holomorphic and general
	quasi-modular forms, related to algebraic automorphic forms.
	In this paper, a new method of constructing $p$-adic zeta-functios is
	presented using general quasi-modular forms and their Fourier	coefficients.

\	

\end{frame}
\iffalse
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 16
\begin{frame}[shrink]
	\frametitle{Algebraicity and congruences of the critical values of the zeta functions} of automorphic forms on unitary and
	symplectic groups, we follow the review by H.Yoshida \cite{YS} of Shimura's book \cite{Shi00}.
	
	Shimura's mathematics developed by stages:
	
	(A) Complex multiplication of abelian varieties =>
	
	(B) The theory of canonical models = Shimura varieties =>
	
	(C) Critical values of zeta functions and periods of automorphic forms.
	
	(B) includes (A) as 0-dimensional special case of canonical models.
	The relation of (B) and (C) is more involved, but (B) provides a solid foundation of the notion of the arithmetic automorphic forms.
	Also unitary Shimura varieties have recently attracted much interest
	(in particular by Ch. Skinner and E. Urban), see \cite{MC}, in relation
	with the proof of the The Iwasawa Main Conjecture for $\GL(2)$.
	
	\
	
\end{frame}
%\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14
\begin{frame}[shrink]
	\frametitle{Integral representations and critical values of the zeta functions}
	%	Automorphic forms are assumed scalar valued for simplicity. 
	In	Cases Sp and UT, Eisenstein series $E(z,  s)$ associated to the	maximal parabolic subgroup of $G$ of Siegel type is introduced. 
		Its analytic behavior and those values of $\si\in 2^{-1}\Z$ at which $E(z,  \sigma)$ is nearly
		holomorphic and arithmetic are studied in \cite{Shi00}. 
		This is achieved by proving
		a relation giving passage from $s$ to $s - 1$ for $E(z, s)$, involving a
		differential operator, then examining Fourier coefficients of
		Eisenstein series using the theory of confluent hypergeometric functions on
		tube domains.
		
		For a Hecke eigenform $f$ on $G_\A$ and an algebraic Hecke character $\chi$
		on the idele group of $K$ (in Case Sp, $K = F$), the zeta function
		{\red ${\mathcal Z}(s,  f,  \chi)$,
			 an Euler product extended over prime ideals of $F$, the degree of the Euler factor is $2n + 1$ in Case
		Sp, $4n$ in Case UT, and $2n$ in Case UB}, except for finitely many prime ideals, see  Theorem 19.8 on Euler products of Chapter 5, \cite{Shi00}.
		
		This zeta function is almost the same as the so called standard $L$-function attached to $f$ twisted by $\chi$ but it turns out to be more general in the unitary case, see also \cite{EHLS}.
		
		Main results on critical values of 
		the $L$-functions studied in Shimura's book \cite{Shi00}
		%Shimura's zeta functions
		is stated in Theorem 28.5, 28.8 (Cases Sp, UT), and in Theorem 29.5 in Case UB.
		
		\
		
	\end{frame}
%	\end{document}
	\fi
	
	
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Integral representation of Rankin-Selberg type over $F=\Q$} \ 
	
	
%\iffalse	
Let us specialize to $F=\Q$ the integral representation of Rankin-Selberg type in the Hermitian modular case the following % is used (\cite{Bou16}, p.13):
%"The following theorem (see also [25, Theorem 7.8]) is obtained by combining results of
%Shimura [31] and Klosin [25]. For details we refer to [6, Section 4], 
 \\
{\bf Theorem 4.1 (Shimura, Klosin)}
Let $0 \ne {\f} \in\Mc_k(C,\psi ))$ of scalar weight $k$, $\psi\bmod \c$, such that $\forall \a, {\f} |T(\a) = \lambda(\a){\f}$, and assume that $2k\ge n$, then there exists $\tau \in S_+\cap \GL_n(K)$ and $r \in \GL_n(K)_{\bf h}$ such that	
\begin{align}\nonumber
& \Gamma ((s)) \psi_\c(\det(r))c_{\f} (\tau, r)L(s + 3n/2, {\f} , \chi) =
 \Lambda_{\c}(s + 3n/2, \theta(\psi\chi)_1)\cdot
\\ \nonumber
 &  \left( \prod_{v\in {\bf b}}
g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
\right)
\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}
C_0\Lambda_\c(s + 3n/2, \theta(\psi\chi)_1)\cdot
\\ & \nonumber
  \left( \prod_{v\in {\bf b}}
g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
\right)
\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}
\\ & \nonumber
\times C_0 \langle {\f} (x), \theta_{\A, \chi}(x) \tilde E_\A(x, \bar s + n; k - l, ( \psi'/\psi)^c; \c\sec \rangle_{\c\sec},
\end{align}
	where $\tilde E_\A(x, \bar s + n; k - l, ( \psi'/\psi)^c; \c\sec >_{\c\sec}$ is the group theoretic %adelic
	Eisenstein series with $q$-component	
	and $\langle \cdot, \cdot \rangle_{\c\sec}$ is the  
	%group theoretic %adelic
	normalized
	Petersson inner product associated to the congruence subgroup 
	$D[{\e}, {\e}{\h}]$,
	% as defined for example in [30, Equation 10.9.6],	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\Gamma((s)) :=
\prod_{v\in {\bf a}}
(4\pi)^{-n(s+h_v)}\Gamma_n(s + h_v),\text{ and } 
C_0:=
\frac{[\Gamma_0(\c\sec):\Gamma]A}{\sharp X}.	
$$
%\fi

\

\end{frame}
%\fi
%\end{document}
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19
\begin{frame}[shrink]
	\frametitle{Integral representation of Rankin-Selberg type over $F=\Q$} 
	\	
%\end{frame}
%\end{document}	
%	\iffalse
	In the Hermitian modular case the following comes
	"The following theorem (see also [25, Theorem 7.8]) is obtained by combining results of
	Shimura [31] and Klosin [25]. For details we refer to [6, Section 4], 
	see Theorem 4.1 (Shimura, Klosin), in \cite{Bou16}, p.13.
	%\begin{prop}[integral representation  of the Hermitian standard zeta function of degree $4n$]
%\end{frame}
%\end{document}
	%\iffalse
		Let $0 \ne \f 
		\in\Mr_k(C,\psi  ))$ such that $\f |T(\a) = \lambda(\a)\f$ for	
		every $\a$, and assume that $k_v+k{_v\rho} \ge n$ for some $v \in {\bf a}$, then there exists $\tau \in S_+\cap \GL_n(K)$ and $r \in \GL_n(K)_{\bf h}$ such that	
%\end{frame}
%\end{document}
			%\iffalse
		\begin{align} & \nonumber
		\Gamma ((s)) \psi_\c(\det(r)) 
		c_{\f} (\tau, r)L(s + 3n/2, {\f} , \chi) =
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\Lambda_\c(s + 3n/2, \theta(\psi\chi)_1)\cdot 
		\\ & \nonumber \left( \prod_{v\in {\bf b}}
		g_v(\chi(\pi_\p)N(\p)^{-2s-3n})
		\right)
		\det(\tau )^{s{\bf a}+h} |\det(r)|_K^{-s-n/2}\\ & \nonumber
		C_0\times
		 \langle \f (x); \theta_{\A, \chi}(x) \tilde E_\A(x, \bar s + n; k - l, ( \psi'/\psi)^c; \c\sec \rangle_{\c\sec},
		%\iffalse
		\end{align}
%\end{frame}
%\end{document}
%\iffalse	
		where $\tilde E_\A(x, \bar s + n; k - l, ( \psi'/\psi)^c; \c\sec >_{\c\sec}$ is the group theoretic %adelic
		Eisenstein series with $q$-component	
		and $\langle \cdot, \cdot \rangle_{\c\sec}$ is the  group theoretic %adelic
		%\iffalse
		Petersson inner product associated to the congruence subgroup 
		$D[{\e}, {\e}{\h}]$ as defined for example in [30, Equation 10.9.6],	
		\begin{align} \nonumber
		\Gamma((s)) :=
		\prod_{v\in {\bf a}}
		(4\pi)^{-n(s+h_v)}\Gamma_n(s + h_v),\text{ and } 
		C_0:=
		\frac{[\Gamma_0(\c\sec):\Gamma]A}{\sharp X}.	
		\end{align}	
%\fi

\
		
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  copie
\fi

%\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *****
\begin{frame}[shrink]
	\frametitle{Proof of the Main Theorem (ii): Kummer congruences}
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		Let us se the notation
		$\di {{\cal D}_{p}^{alg}(m,  {\f},\chi )}=
		A_{p}(s,\chi)\frac
		{{\cal D}^{*}(m,  {\f},\chi )}{\Omega_{\f}}$
			\\
			The integrality of  measures is proven 
			%by T.Bouganis \cite{Bou16}, 
			representing $\Dr_p^{alg}(m,\chi)$ as Rankin-Selberg type 
			integral at critical points $s=m$. 
			Coefficients of modular forms in this integral satisfy Kummer-type congruences and produce bounded measures
			$\mu_\Dr$ whose construction reduces to congruences of Kummer type between the Fourier coefficients of modular forms, see also \cite{Bou16}.
			\smallskip\pause
			Suppose that we are given 
			%some $L$-function ${\Dr}_p^\pm(s, {\bf f}, \chi)$ attached to a Hermitian modular form ${\bf f}$ using 
			infinitely many "critical pairs" $(s_j , \chi_j )$ 
			at which one has an integral representation
		$ \di
		{\Dr}_p^{alg}(s_j, {\bf f}, \chi_j)
		=A_{p}(s,\chi) \frac{\langle {\bf f} , h_j \rangle}
		{\Omega_{\bf f}}
		$
		with all $h_j =
		\sum_\Tc b_{j, \Tc} q^\Tc \in \Mc$
		in a certain
		finite-dimensional space $\Mc$ containing ${\bf f}$ and defined over $\bar{\Q}$.
		We prove the following Kummer-type congruences:
		$$
		\forall x\in\Z^*_p, \  \sum_j\beta_j\chi_jx^{k_j}\equiv 0 \mod p^N
		\Longrightarrow
		\sum_j\beta_j
		 {\Dr}_p^{alg}(s_j, {\bf f}, \chi)
		\equiv 0 \mod p^N
		$$
		%for any choice of 
		$$
		\beta_j \in \bar \Q, k_j =
		s^*-s_j, \text{ where } s^* = \ell-n \text{ in our case.}
		$$ 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\smallskip\pause
		{\blue \Large Computing the Petersson products}
		of a given modular form
		${\bf f} (Z) =\sum_H a_H q^H\in \Mc_*(\bar{\Q})$
		by another modular form
		$h (Z) =\sum_H b_H q^H\in \Mc_{*}(\bar{\Q})$ uses a linear form
		$\di
		\ell_{\bf f} : h \mapsto \frac{\langle {\bf f} , h \rangle}{\langle {\bf f} , {\bf f} \rangle}
		$ defined over a subfield $k \subset \bar\Q.
$		
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

		\
		
	\end{frame}
%	\end{document}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
	\begin{frame}[shrink]
		\frametitle{Admissible Hermitian case} 
		%p.33 of \cite{Bou16} " Definition 7.2. 
		Let ${\f} \in \Sc_k(C; \psi )$ be a Hecke eigenform for 
		the congruence subgroup $C$ of level $\c$. 
		Let $\p$ be a prime of $K$ prime to $\c$, which is inert over $F$.
		Then we say that $\f$ is {\magenta pre-ordinary} at $\p$ if
		there exists an eigenform $0\ne {\f}_0 \in \Mc_{\{p\}} \subset \Sc_k(Cp,%D\b^{-1}, \b \c\p\p^\rho],
		\psi)$ with Satake parameters
		$t_{\p,i}$ such that 
		\begin{align}\label{ordinary}\nonumber
		\left \| 
		\left( 
		\prod_{i=1}^n t_{\p,i}
		\right) 
		N(\p)^{-\frac{n(n+1)} 2}
		\right \|_p=1,
		\end{align}
		where $\| \|_p$ the normalized absolute value at $p$. 
		
	\smallskip\pause
		The {\magenta admissible case} corresponds to
		
		\begin{align*}%\label{admissible} 
		\left \| 
		\left(\prod_{\q|p} 
		\prod_{i=1}^n t_{\q,i}
		 \right)
		p^{-n(n+1)}
		\right \|_p=p^{-h}\text{ for a positive  } h>0.
		\end{align*}
	{\magenta  An interpretation of $h$ as the difference
		$h=P_{N,p}(d/2)-P_{H}(d/2)$ comes from the above explicit 
	relations.}
		
		\
		
	\end{frame}
	%\end{document}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
	\begin{frame}[shrink]
		\frametitle{Existence of $h$-admissible  measures} of Amice-Vélu-type gives an unbounded  $p$-adic analytic interpolation of the $L$-values of growth  $\log_p^h(\cdot)$, using the Mellin transform of the constructed measures.
		This condition says that the product $\prod_{i=1}^n t_{\p,i}$ is nonzero  and divisible by a certain power of $p$ in $\Oc$:
		$$
		\ord_p\left(\prod_{\q|p}\left( 
		\prod_{i=1}^n t_{\q,i}
		\right) 
		p^{-n(n+1)}\right)=h.
		$$
		
		We use an {\magenta easy condition of admissibility} of a sequence of modular distributions $\Phi_{j}$ on $X=\Oc_K \otimes \Z_p$ with values in $\Oc[[q]]$ as in Theorem 4.8 of \cite{CourPa}
		% with $A=\Or$
		and check {\magenta congruences} of the type
		$$
		U^{\vkp {v}}
		\Bigl(\sum_{j^\prime =0}^j{j\choose j^\prime}
		(-a_{p}^0)^{j-j^\prime}
		\Phi_{j^\prime} ({a+(p^{v})}\Bigr)\in
		Cp^{{v} j}\Oc[[q]]%\eqno (congruences)
		$$
		for all $j = 0, 1, \dots,  \vkp h-1$.
		Here $s=j'+s_*$, $\Phi_{j^\prime} ({a+(p^{v})})$
		a certain convolution, i.e.
		$$
		\Phi_{j^\prime} (\chi)=\theta(\chi)\cdot {\bf E}(s, \chi)
		$$
		of a Hermitian theta series $\theta(\chi)$ and an Eisenstein series ${\bf E}(s, \chi)$ with any Dirichlet character $\chi \bmod p^r$.  
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		We use a general sufficient condition of admissibility of a sequence of modular distributions $\Phi_{j}$ on $X=\Z_p$  with values in $\Oc[[q]]$ as in Theorem 4.8 of \cite{CourPa}.% with $A=\Oc$		
		
\
	
		
		
\end{frame}
%\end{document}


\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 23
\begin{frame}[shrink]
	\frametitle{Using algebraic and $p$-adic modular forms}
		There are several methods to compute various $L$-values starting
		from the constant term of the Eisenstein series in \cite{Se73},
		$$
		G_k (z) =
		\frac{\zeta(1-k)} 2
		+
		\sum_{
			n=1}^\infty
		\si _{k-1}(n)q^n =
		\frac{\Gamma(k)}
		{(2\pi i )^k}
		\sum_{(c,d)} {}'
		(cz + d)^{-k} \ \ (\text{for} k\ge 4),
		$$
		and using Petersson products of nearly-holomorphic Siegel modular
		forms and arithmetical automorphic forms as in \cite{Shi00}:
		
		the Rankin-Selberg method,
		
		the doubling method (pull-back method).
		
		A known example is the standard zeta function 
		$D(s,  f, \chi)$ of a
		Siegel cusp eigenform   $f\in \Sc_n^k (\Gamma)$ of genus $n$ (with local factors of
		degree 2$n$ + 1) and $\chi$ a Dirichlet character.
		
		{\Large\blue Theorem} (the case of even genus $n$ (\cite{Pa91}, \cite{CourPa}), via the
		Rankin-Selberg method) gives a $p$-adic interpolation of the
		normailzed critical values $D^*(s, f, \chi)$ using Andrianov-Kalinin integral representation of these values $1 + n -k \le  s \le  k - n$
		through the Petersson product $\langle f,  \theta_{T_0}\delta^rE\rangle$ where $\delta^r$ is a certain
		composition of Maass-Shimura differential operators, $\theta_{T_0}$ a theta-series of weight $n/2$, attached to a fixed $n \times n$ matrix $T_0$.
		
		
\

\end{frame}
%\fi
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24
\begin{frame}[shrink]
	\frametitle{Using $p$-adic doubling method}		
		%\begin{theo}[
		{\Large\blue Theorem}
			($p$-adic interpolation of
			$D(s, f, \chi)$)\\
			 (1) (the case of odd genus (Boecherer-Schmidt, \cite{BS00}) 
			\ \\
		{\em	Assume that $n$ is arbitrary genus, and a prime $p$ ordinary then there exists a $p$-adic interpolation of
			$D(s, f, \chi)$ 
			
			{\rm (2)} (Anh-Tuan Do (non-ordinary case, PhD Thesis, 2014)), via
				the doubling method)}
			\ \\  Assume that $n$ is arbitrary genus, and  $p$ an arbitrary prime not dividing level of $f$ 
			then there exists a $p$-adic interpolation of
			$D(s, f, \chi)$. 	
			
		%\end{theo}
		
		Proof
		uses the following Boecherer-Garrett-Shimura identity
		(a pull-back formula, related to the Basic Identity of {P}iatetski-Shapiro and Rallis, \cite{GPSR}),
		%$\theta_{T_0}$
		which allows to compute the critical values through certain double
		Petersson product by integrating over 
		$z\in \HH_n$ the identity:
		$$
		\Lambda(l + 2s, \chi)D(l + 2s-n,  f, \chi) f =
		\langle f (w),  E^{2n}_{l, \nu, \chi, s} (\diag[z, w])_w, 
		$$
		Here $k = l +\nu, \nu \ge 0, \Lambda(l + 2s, \chi)$ is a product of special values of
		Dirichlet $L$-functions and $\Gamma$-functions,
		$E^{2n}_{l, \nu, \chi, s}$ a higher twist of a Siegel-Eisenstein series on $(z,w)\in \Hb_n\times \Hb_n$ (see \cite{Boe85},
		\cite{BS00}).
		

\

\end{frame}
%\end{document}
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24
\begin{frame}[shrink]
	\frametitle{Using differential operators
		 in $p$-adic constructions} is via congruences for the $L$-values, expressed
		through the Fourier coefficients of the Siegel modular forms and
		nearly-modular forms.
		Computing the Petersson products and
		$L$-values uses an injection of algebraic nearly holomorphic modular forms
		into $p$-adic modular forms.
		
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%		
		The essential point here is the use of differential operators, complex and $p$-adic, "Relationships via differential operators, ... play a role in the non-holomorphic case. 
		... the $p$-adic (not necessarily algebraic) automorphic form 
		$\theta^\kappa G_{k,\nu, \chi_{unitary}, \mu}$ 
	is related to the $C^\infty$ (not necessarily holomorphic) automorphic form associated to the data above via $\theta^\kappa G_{k,\nu,\chi_{unitary},\mu} =D^\kappa_kG_{k,\nu,\chi_{unitary}, \mu}$, where $D^\kappa_k$ is a $C^\infty$-differential operator analogous to $\theta^\kappa$, and generalizing the Maass-Shimura operators...",  in the notation of \cite{EHLS}, p.69.
	
	{\red On quasimodular forms such operators act very simply}, multiplying the  Fourier coefficient $a(\Tc)$ by a power of $\det(\Tc)$ of a non-negative Hermitian matrix $\Tc$, then the studied congruences follow naturally.     
		
		
\

\end{frame}
%\end{document}
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 25
\begin{frame}[shrink]
	\frametitle{Injecting nearly-holomorphic forms into $p$-adic modular forms}
	A recent discovery by Takashi Ichikawa (Saga University),
	[Ich12], J. reine angew. Math., [Ich13]
	allows to inject nearly-holomorphic arithmetical (vector valued)
	Siegel modular forms into $p$-adic modular forms.
	Via the Fourier expansions, the image of this injection is
	represented by certain quasi-modular holomorphic forms like
	$E_2 = 1- 24
	\sum_{n\ge 1}
	\si_1(n)q^n$, with algebraic Fourier expansions.
	This description provides many advantages, both computational
	and theoretical, in the study of algebraic parts of Petersson
	products and $L$-values, which we would like to develop here.
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	In fact,
	the realization of nearly holomorphic forms as $p$-adic modular forms has been
	studied by Eric Urban, who calls them "Nearly overconvergent modular forms" \cite{U14}, Chapter 10.
	
	%http://link.springer.com/chapter/10.1007/978-3-642-55245-8_14
	Urban only treats the elliptic modular case in that paper, but I believe
	he and Skinner are working on applications of a more general theory.
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	This work is related to a recent preprint \cite{BoeNa13} by S. Boecherer
	and Shoyu Nagaoka where it is shown that Siegel modular forms of
	level $\Gamma_0(p^m)$ are $p$-adic modular forms. Moreover they show that
	derivatives of such Siegel modular forms are $p$-adic. Parts of these
	results are also valid for vector-valued modular forms.
	\

\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 26
\begin{frame}[shrink]
	\frametitle{Arithmetical nearly-holomorphic Siegel modular forms}
	Nearly-holomorphic Siegel modular forms over a subfield $k$ of $\C$ are
	certain $\C^d$ -valued smooth functions $f$ of 
	$Z = X + \sqrt{-1}Y  \in \Hb_n$
	given by the following expression
	$
	f (Z) =
	\sum_T
	P_T (S)q^T 
	$
	where $T$ runs through the set $B_n$ of all half-integral semi-positive
	matricies, $S = (4\pi Y )^{-1}$ a symmetric matrix,
	$q^T = \exp(2\pi \sqrt{-1} \tr(TZ)), P_T (S)$ are vectors of degree d whose
	entries are polynomials over $k$ of the entries of $S$.
	
	{\magenta A geometric construction of such arithmetical forms uses the algebraic theory af moduli spaces of abelian varieties.}
	Following \cite{Ha81}, consider the columns $Z_1, Z_2, \dots Z_n$ of $Z\in \Hb_n$ and the
	$\Z$-lattice $L_Z$ in $\C^n$ generated by $\{
	E_1, E_2, \dots E_n, Z_1, Z_2, \dots Z_n \}, 
	$
	where
	$E_1, E_2, \dots E_n$ are the columns of the identity matrix $E$. 
	The torus $A_Z = \C^n/L_Z$ is an abelian variety, and there is an analytic family
	$\Ar \to \Hb_n$ whose fiber over the point $Z\in \Hb_n$ is $A_Z$ .
	Let us consider the quotient space $\Hb_n/\Gamma(N)$ of the Siegel upper half space $\Hb_n$ of degree $n$ by the integral symplectic group
	%\begin{align*}
	$$
	\Gamma(N) =
	\left\{
	\gamma=
	\begin{pmatrix}
	&A_\gamma 
	&B_\gamma
	\\
	&C_\gamma 
	&D_\gamma
	\end{pmatrix} 
	\ \Big| \
	\begin{matrix}
	& A_\gamma \equiv D_\gamma \equiv 1_n \\
	&B_\gamma \equiv C_\gamma \equiv 0_n
	\end{matrix}
	\right\}, 
	$$
	%\end{align*}
	\iffalse
	If $N > 3$, $\Gamma(N)$ acts without fixed points on $\Ac = \Ac_n$ and the
	quotient is a smooth algebraic family $\Ac_{n, N}$ of abelian varieties with
	level N structure over the quasi-projective variety
	$\Hr_{n, N}(\C) = \Hb_n/\Gamma(N)$ defined over $\Q(\zeta_N)$, where $\zeta_N$ is a primitive
	$N$-th root of 1.
	For positive integers $n$ and $N$, $\Hr_{n, N}$ is 
	\fi
	giving the moduli space classifying
	principally polarized abelian schemes of relative dimension $n$ with a symplectic level $N$ structure.	


\
	
\end{frame}
%\end{document}
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 27
\begin{frame}[shrink]
	\frametitle{Applications to constructions of $p$-adic $L$-functions}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%It is best to treat it as 
	We present here a survey
	of some methods of construction of $p$-adic $L$-functions. 
	Two important ideas
	that are not as well known as they should be are developed biefly in this section.
	%pp. 17-18, but
	%the treatment is very brief.
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	There exist two kinds of $L$-functions
	\begin{itemize}
		\item Complex $L$-functions (Euler products) on 
		$\C = \Hom(\R^*_+; \C^*)$.
		\item $p$-adic $L$-functions on the $\C_p$-analytic group 
		$\Hom_{cont}(\Z^*_p, \C_p^*)$
		(Mellin transforms $\Lr_{\mu}$ of $p$-adic measures $\mu$ on $\Z^*_p$).
	\end{itemize}
	
	Both are used in order to obtain a number ($L$-value) from an
	automorphic form. Such a number can be algebraic (after
	normalization) via the embeddings,
	$$
	\bar\Q \hookrightarrow \C, \bar\Q \hookrightarrow \C_p=\hat{\bar\Q}_p 
	$$  
	and we may compare the complex and $p$-adic $L$-values at many
	points.
	
	\
	
\end{frame}
%\end{document}
%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 28
\begin{frame}[shrink]
	\frametitle{ How to define and to compute $p$-adic $L$-functions?}
	The Mellin transform of a $p$-adic distribution $\mu$ on $\Z^*_p$ gives an
	analytic function on the group of $p$-adic characters
	$$
	y \mapsto \Lr_\mu(y) =
	\int_{\Z^*_p}
	y(x)d\mu(x), \  y \in \Hom_{cont}(\Z^*_p, \C^*_p)
	$$
	A general idea is to construct $p$-adic measures directly from Fourier
	coefficients of modular forms proving Kummer-type congruences for
	$L$-values. 
	Here we present a new method to construct $p$-adic
	$L$-functions via quasimodular forms.
	
	\
	
%\end{frame}
\fi
%\end{document}

%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 29
\begin{frame}[shrink]
	\frametitle
{How to prove Kummer-type congruences
		 using the Fourier coefficients?}

	
	
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%****
	Suppose that we are given some $L$-function $L^*_f (s, \chi)$ attached to a Siegel modular form $f$ and assume that for infinitely many "critical pairs" $(s_j ; \chi_j )$ one has an integral representation
	$
	\boxed{L^*_f (s,  \chi)=\langle f , h_j \rangle}
	$
	with all $h_j =
	\sum_\Tc b_{j, \Tc} q^\Tc \in \Mc$
	in a certain
	finite-dimensional space $\Mc$ containing $f$ and defined over $\bar{\Q}$.
	We want to prove the following Kummer-type congruences:
	$$
	\forall x\in\Z^*_p, \  \sum_j\beta_j\chi_jx^{k_j}\equiv 0 \mod p^N
	\Longrightarrow
	\sum_j\beta_j\frac{ L^*_f (s_j,  \chi)_j}{\langle f , f \rangle}\equiv 0 \mod p^N
	$$
	%for any choice of 
	$$
	\beta_j \in \bar \Q, k_j =
	\begin{cases}
	s_j -s_+ & \text{ if }s_+ = \min_j s_j \text{ or }\\
	s_--s_j&\text{ if }s_- =  \max_j s_j .
	\end{cases}
	$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{Computing the Petersson products}
of a given modular form
$f (Z) =\sum_\Tc a_\Tc q^\Tc\in \Mc_\rho(\bar{\Q})$
by another modular form
$h (Z) =\sum_\Tc b_\Tc q^\Tc\in \Mc_{\rho^*}(\bar{\Q})$ uses a linear form
$$
\ell_f : h \mapsto \frac{\langle f , h \rangle}{\langle f , f \rangle}
$$
defined over a subfield $k \subset \bar\Q$.
 
\

\end{frame}
%\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ***** 20
\begin{frame}[shrink]
	\frametitle{Proof of the Main Theorem (iii): (admissible case)}
Using a Rankin-Selberg integral representation 
for $\Dr^{alg}(s,{\f},\chi)$ and an eigenfunction ${\f}_0$ of Atkin's operator $U(p)$ of eigenvalue $\al_{\f}$ on ${\f}_0$ the Rankin-Selberg integral of
${\cal F}_{s,\chi}:=\theta( \chi)\cdot 	{\bf E}(s, \chi)$ gives
\begin{align} & \nonumber
\Dr^{alg}(s,{\f},\chi)=\frac{\langle {\bf f}_0, 
	\theta( \chi)\cdot 	{\bf E}(s, \chi)\rangle}{\langle {\bf f}, 
	{\bf f}\rangle} 
\text{    (the Petersson product on } G=GU(\eta_{n}) )
\\ & \nonumber
=
\al_{\f}^{-v}\frac{\langle {\bf f}_0, U(p^v)(\theta( \chi)\cdot
	{\bf E}(s, \chi))\rangle}{\langle {\bf f}, {\bf f}\rangle}=
\al_{\f}^{-v}
\frac{\langle f_0, U(p^v) ({\cal F}_{s,\chi})\rangle} {\langle {\bf f}, {\bf f}\rangle}.
\end{align}
\smallskip\pause
Modication in the admissible case: {\magenta instead of Kummer congruences},
to {\magenta estimate $p$-adically the integrals of test functions}: $M=p^v$: 
$\displaystyle  \int_{a+(M)} (x-a)^j d{\Dr^{alg}} := \sum_{j^\prime =0}^j{j\choose j^\prime}
(-a )^{j-j^\prime}\int_{a+(M)} x^{j'} d{\Dr^{alg}}$,
using the {\magenta orthogonality of characters} and the sequence of zeta distributions 
$\displaystyle \int_{a+(M)} x^{j} d{\Dr^{alg}}=\frac{1}{\sharp (\Oc / M\Oc)^\times}\sum_{\chi \bmod M}\chi^{-1}(a)\int_X\chi (x)x^{j} d{\Dr^{alg}},$
$ \int_X\chi  d{\Dr^{alg}_{s_-+j}}= 
\Dr^{alg}(s^*-j,f,\chi)=:\int_X\chi (x)x^{j} d{\Dr^{alg}}$.

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20
\begin{frame}[shrink]
	\frametitle{Congruences between the 
		coefficients of the Hermitian modular forms}
In order to integrate any locally-analytic function on $X$, it suffices to check the following binomial congruences for the 
coefficients of the Hermitian modular form 
$
 {\cal F}_{s^*-j,\chi}
= \sum_{\xi }v({\xi}, s^*-j, \chi)q^\xi :
$
for $v\gg 0$, and a constant $C$
\begin{align} & \nonumber
 \frac{1}{\sharp (\Oc / M\Oc)^\times}
  \sum_{j^\prime =0}^j{j\choose j^\prime}
(-a )^{j-j^\prime}
\sum_{\chi \bmod M} \chi^{-1}(a) v(p^v\xi, s^*-j', \chi)q^\xi
% \label{stde6.21H} &  \nonumber  
\\ & \nonumber \in Cp^{{v} j}\Oc [[q]] \ \ \text{    (This is a quasimodular form if } j'\ne s^*)
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\smallskip\pause
The resulting measure $\mu_\Dr$ allows
to integrate all continuous characters in 
$\Yc_p =\Hom_{cont}(X,\CC_p^*)$, including Hecke characters,  
as they are always locally analytic.

{\magenta Its $p$-adic Mellin transform} $\Lr_{\mu_\Dr}$ is an analytic function 
on $\Yc_p$ of the logarithmic growth $\Oc( \log^h)$,  $h=\ord_p(\alpha)$. 





	 
	 \
	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ***** 18
\begin{frame}[shrink]
	\frametitle{Proof of the main congruences}
Thus the Petersson product in $\ell_{\bf f}$ can be expressed through
the Fourier coeffcients of $h$ in the case when there is a finite basis
of the dual space consisting of certain Fourier coeffcients:
$\ell_{\Tc_i} : h \mapsto b_{\Tc_i} (i = 1, \dots, n)$.
It follows that 
$\ell_{\bf f} (h) =
\sum_i\gamma_i  b_{\Tc_i}$, where 
$\gamma_i\in k$.	
	
		Using the expression for $\ell_f (h_j ) =
	\sum_i \gamma_{i,j} b_{j,\Tc_i}$, the above
	congruences reduce to
	$$
	\sum_{i,j} \gamma_{i,j}\beta_j b_{j,\Tc_i} \equiv 0 \mod p^N.
	$$
	%\fi
	The last congruence is done by
	an elementary check on the Fourier coefficients $b_{j,\Tc_i}$.
	
	
	The abstract Kummer congruences are checked for a family of test elements.
	
	In the admissible case it suffices to check {\magenta binomial congruences} 
	for the Fourier coefficients
	as above in place of Kummer congruences.
	
	
\

\end{frame} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ***** 8
\begin{frame}[shrink]
	\frametitle{Appendix A. Rewriting the local factor at $p$ with character $\theta$ } 
	Notice that if $\theta$ is the quadratic character attached to $K/\Q$ then 
	%\iffalse
	%\end{frame}
	%\end{document}
	$$
	(1-\al_pX)(1-\al_p\theta(p)X)=
	\begin{cases}
	(1-\al_pX)^2& \text{ if } \theta(p)=1, p\r=\q_1\q_2, N(\q_i)=p,\\
	(1-\al_p^2X^2),&\text{ if } \theta(p)=-1, 
	p{\r}=\q, N(\q)=p^2,\\ 
	(1-\al_pX)&\text{ if } \theta(p)=0, p{\r}=\q^2, N(\q)=p.\\
	\end{cases}
	$$		
	Thus, if $X=p^{-s}$, $X^2=p^{-2s}$, $N(\q)=p$, 
	$\Zc_\q(X)^{-1}$ 
	$$=
	\begin{cases}
	\prod_{i=1}^{2n}
	(1- N(\q_1)^{2n}t_{\q_1\q_2,i}^{-1}X)
	(1- N(\q_2)^{-1}t_{\q_1\q_2,i}X),  
	& \text{ if } \theta(p)=1,\\
	\prod_{i=1}^{n}
	{(1- N(\q)^{n-1}t_{\q,i} X^2)(1- N(\q)^{n}t_{\q,i}^{-1}
		X^2)},& \text{ if }  \theta(p)=-1,\\
	\prod_{i=1}^{n}
	{(1- N(\q)^{n-1}t_{\q,i}X)(1- N(\q)^{n}t_{\q,i}^{-1} X)},	
	& \text{ if } \theta(p)=0.
	\end{cases}
	$$	
	$$	
	= \begin{cases}
	\prod_{i=1}^{n}(1-\ga_{p,i}X)^2
	\prod_{i=1}^{n}(1-\de_{p,i}X)^2
	& \text{ if } 
	%N(\q_1)^{2n}t_{\q_1\q_2,i}^{-1}=
	%N(\q_2)^{-1}t_{\q_1\q_2,i}
	\theta(p)=1, \text{ i.e. }p\r=\q_1\q_2,\\
	\prod_{i=1}^{n}(1-\al_{p,i}^2X^2)
	\prod_{i=1}^{n}(1-\beta_{p,i}^2X^2)
	,&\text{ if } \theta(p)=-1, \text{ i.e. } 
	p{\r}=\q,\\ 
	\prod_{i=1}^{n}(1-\al'_{p,i}X)
	\prod_{i=1}^{n}(1-\beta'_{p,i}X)
	&\text{ if } \theta(p)=0, 
	\text{ i.e. }p{\r}=\q^2,
	\end{cases}
	$$
	where $\al'_{p,i}=p^{n-1}t_{\q,i}$, $\beta'_{p,i}p^{n}t_{\q,i}^{-1}$,
	$\ga_{p,i}= p^{2n}t_{\q_1\q_2,i}^{-1}$,
	$p^{-1}t_{\q_1\q_2,i}$.
	It follows that $\prod_{\q|p}\Zc_\q (N(\q)^{-n-(1/2)}X)=X^{4n}+\cdots$ 
	
	
	\
	
\end{frame}
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Appendix A (continued).Relations between 
		$\al_i(p)$ and $t_{i,\q}$} were studied and explained by M.Harris \cite{Ha97} for general Hermitian zeta functions $\Zc(s,\f)$ of type introduced in \cite{Shi00}, using reprsentation theory of unitary groups and Deligne's approach to $L$-functions, see 
	\cite{De79}, in terms of  a $n$-dimensional Galois representations
	$\rho_\lambda: \Gal(\bar K/K) \longrightarrow
	\GL(M_{\f, \lambda})\cong\GL_n(E_\lambda)$
	over a completion $E_\lambda$ of a number field $E$
	containing $K$ and the Hecke eigenvalues of a vector-valued Hermitian
	modular form $\f$: 									
	\begin{align} \nonumber
		\Zc(s-n'-\frac{1}{2},{\f})=
		\Dr(s,{\f})
		= L(s,M_{\f, \lambda}\boxtimes M(\psi))
	\end{align}					
	for an algebraic Hecke ideal character $\psi$ as above of the infinity type $m_\psi$, see \cite{GH16}, p.20. 
	Here the symbol $L(s,M_{\f, \lambda}\boxtimes M(\psi))$ denotes the Rankin-Selberg type convolution (it corresponds to tensor product of Galois representations).
	Notice that $L(s,M_{\f, \lambda})$ is of degree $2n$, and 
	$L(s,M_{\f, \lambda}\boxtimes M(\psi))$ is of degree $4n$ because $L(s,\psi)=L(s,R(\psi))$ is of degree 2.
	
	Moreover, M.Harris suggested a general description of  $\Dr(s)$ with given Gamma factors and analytic properties as some  $\Dr(s,{\f})$ 
	some under natural conditions on Gamma factors, giving higher versions of Shimura-Taniyama-Weil conjecture (i.e. higher Wiles' modularity theorem).
	This can be stated also over a totally real field $F$ (instead of $\Q$),
	and its quadratic totally imaginary extension $K$, see \cite{GH16}, \cite{Pa94}. 
	
	\
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  20
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 15
\begin{frame}[shrink]
	\frametitle{Appendix B. Shimura's Theorem: algebraicity of critical values in Cases Sp and UT, p.234 of \cite{Shi00}}
	%\begin{theo}
	Let ${\bf f}\in {\mathcal V} (\bar\Q)$ be a non zero  arithmetical automorphic form of type Sp or UT. 
	Let $\chi$ be a Hecke character of $K$ such that 
	$\chi_{\bf a}(x)=x_{\bf a}^\ell |x_{\bf a}|^{-\ell}$ with $\ell\in\Z^{{\bf a}}$, \text{ and let }
	$\sigma_0\in 2^{-1}\Z$.
	Assume, in the notations of Chapter 7 of \cite{Shi00} on the weights $k_v, \mu_v, \ell_v$, that 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{align*}%{ll}
	&\text{\rm Case Sp} &&
	\  2n+1-k_v+\mu_v \le 2\si_0\le k_v-\mu_v,  
	\\&{}  && \  \text{ where }
	\mu_v=0  \text{ if } [k_v]-l_v \in 2\Z \\&{}  && \  \text{ and } \mu_v=1
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\text{ if } [k_v]-l_v \not\in 2\Z; \ 
	\si_0-k_v+\mu_v
	\\&{}  && \  \text{ for every } v \in  {\bf a}  \text{ if }\si_0>n
	\text{ and } 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\&{}  &&\ \si_0-1-k_v+\mu_v \in 2\Z\text{ for every } v \in  {\bf a}
	\text{ if }\si_0\le n. \\
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	&\text{\rm Case UT} &&\  4n-(2k_{v\rho}+\ell_v)\le 2\si_0\le 
	m_v-| k_v-k_{v\rho}-\ell_v |\\&{}  && \  \text{ and } 2\si_0-\ell_v\in 2\Z  \text{ for every  } v \in  {\bf a}.
	\end{align*}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19
\begin{frame}[shrink]
	\frametitle{Appendix B. Shimura's Theorem (continued)}
	Further exclude the following cases
	\begin{align*}%{ll}
	&\text{\rm (A) Case Sp} && \si_0=n+1,F=\Q  
	\text{ and }
	\chi^2=1;
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\ &\text{\rm (B) Case Sp} &&\si_0=n+(3/2),F=\Q; \chi^2=1  
	\text{ and }
	[k]-\ell\in 2\Z
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\ &\text{\rm (C) Case Sp} &&\si_0=0,
	{\mathfrak c}={\mathfrak g} 
	\text{ and }
	\chi=1;
	\\ &\text{\rm (D) Case Sp} &&0<\si_0\le n, {\mathfrak c}={\mathfrak g}, \chi^2=1
	\text{ and the conductor of } \chi \text{ is }{\mathfrak g};
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\&\text{\rm (E) Case UT} && 2\si_0=2n+1,F=\Q, \chi_1=\theta,  
	\text{ and }
	k_v-k_{v\rho}=\ell_v;
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\\&\text{\rm (F) Case UT} && 0\le 2\si_0 < 2n, {\mathfrak c}={\mathfrak g},
	\chi_1=\theta^{2\si_0} 	\text{ and the conductor of } \chi 	\text{ is } {\mathfrak r}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\end{align*}
	Then 
	$$
	{\mathcal Z}(\si_0, {\bf f}, \chi)/\langle  {\bf f},  {\bf f} \rangle\in\pi^{n|m|+d\varepsilon}\bar \Q, 
	$$
	where $d = [F:\Q]$, $|m|=\sum_{v\in {\bf a}}m_v$, and
	$$\varepsilon=
	\begin{cases}
	(n+1)\si_0-n^2-n,&\text{Case Sp}, k\in \Z^{{\bf a}}, 
	\text{ and } \si_0>n_0),
	\\
	n\si_0-n^2, &\text{Case Sp}, k \not\in \Z^{{\bf a}}, 
	\text{or} \si_0\le n_0),
	\\ 2n\si_0-2n^2+n
	&\text{Case UT}
	\end{cases}
	$$
	%\end{theo}		
	Notice that $\pi^{n|m|+d\varepsilon}\in\Z$ in all cases; if $k\not \in\Z^{\bf a}$,  the above parity condition on $\sigma_0$ shows that  $\sigma_0+k_v\in\Z$, so that $n|m|+d\varepsilon\in\Z$.
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Appendix C. Examples of Hermitian cusp forms} 
	{\blue\Large The Hermitian Ikeda lift, \cite{Ike08}}.
	Assume 
	${n} = 2n'$ even. Let
	$\di
	f(\tau) =
	\sum_{N=1}^\infty
	a(N)q^N \in  \Sr_{2k+1}(\Gamma_0(D_K), \chi)
	$
	be a primitive form, whose $L$-function is given by
	$$
	L(f, s) =
	\prod_{p \nd D_K}
	(1 - a(p)p^{-s} + {\theta}(p)p^{2k-2s})^{-1}
	\prod_{p|D_K} (1 - a(p)p^{-s})^{-1}.
	$$
	For each prime $p \nd D_K$, define the Satake parameter 
	$\{\al_p, \beta_p\} = \{\al_p, {\theta}(p)\al_p^{-1}\}$  by
	$$
	(1 - a(p)X + {\theta}(p)p^{2k}X^2)=
	(1-p^k\al_pX)(1-p^k\beta_pX)
	$$
	For $p | D_K$, we put $\al_p = p^{-k} a(p)$. 
	Put
	\begin{align}
	& \nonumber
	A(H)=|\gamma(H)|^{k}
	\prod_{p|\gamma(H)}\tilde F_p(H;\al_p), H \in \Lambda_{n}(\Oc)^+\\ & \nonumber
	F(H)=\sum_{H\in\Lambda_{n}(\Or)^+}A(H) q^H, Z\in\Hr_{2n}.
	\end{align}	
	
	\
	
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{Appendix C (continued).The first theorem (even case)} 
	{\Large \blue Theorem 5.1 (Case E) of \cite{Ike08}} Assume that ${n} = 2n'$ is even. Let $f(\tau )$, $A(H)$ and $F(Z)$ be as above. Then we have
	$F\in \Sr_{2k+2n'}(\Gamma_K^{(n)},\det^{-k-n'}).$
	
	
	\smallskip
	In the case when ${n}$ is odd, consider a similar lifting for a normalized Hecke eigenform ${n} = 2n'+1$ is odd. Let
	$\di
	f(\tau) =
	\sum_{N=1}^\infty
	a(N)q^N$ $\in  \Sr_{2k}(\SL_2(\Z))
	$
	be a primitive form, whose $L$-function is given by
	$$
	L(f, s) =
	\prod_{p}
	(1 - a(p)p^{-s} + p^{2k-1-2s})^{-1}.
	$$
	For each prime $p $, define the Satake parameter 
	$\{\al_p, \al_p^{-1}\}$  by
	$$
	(1 - a(p)X + p^{2k-1}X^2)=
	(1-p^{k-(1/2)}\al_pX)(1-p^{k-(1/2)}\al^{-1}X).
	$$
	Put
	\begin{align}
	& \nonumber
	A(H)=|\gamma(H)|^{k-(1/2)}
	\prod_{p|\gamma(H)}\tilde F_p(H;\al_p), H \in \Lambda_{n}(\Oc)^+\\ &
	\nonumber
	F(H)=\sum_{H\in\Lambda_{n}(\Or)^+}A(H) q^H, Z\in\Hr_{n}.
	\end{align}
	
	
	
	
	\
	
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
\begin{frame}[shrink]
	\frametitle{Appendix C (continued).
		The second theorem (odd case)}
	{\blue \Large Theorem 5.2 (Case O) of \cite{Ike08}}.
	Assume that ${n} = 2n'+1$ is odd. Let $f(\tau )$, $A(H)$ and $F(Z)$ be as above. Then we have
	$F\in \Sr_{2k+2n'}(\Gamma_K^{(n)},\det^{-k-n'}).$
	
	\smallskip
	The lift $Lift^{({n})}(f)$ of $f$ is a common Hecke eigenform of all
	Hecke operators of the unitary group, if it is not identically zero (Theorem 13.6). 
	
	\smallskip
	{\blue \Large Theorem 18.1 of \cite{Ike08}}. Let ${n}$, $n'$, and $f$ be as in Theorem 5.1 or as in Theorem 5.2. 
	Assume that $Lift^{({n})}(f)\ne 0$. 
	Let $L(s, Lift^{({n})}(f), st)$ be the $L$-function of $Lift^{({n})}(\f)$ associated to $st : {}^L{\cal G} \to \GL_{4{n}}(\CC)$. 
	Then up to bad Euler factors, $L(s, Lift^{({n})}(f), st)$ is equal to
	$$
	\prod_{i=1}^{n}
	L(s + k + n' - i + \frac 1 2, f)L(s + k + n' - i + \frac 1 2, f, {\theta}).
	$$
	Moreover, the %Satake parameters and
	 $4{n}$ charcteristic roots of
	$L(s, Lift^{({n})}(f), st)$ given
	as follows:  for $i=1, \cdots ,{n}$ 
	\begin{align} & \nonumber
	\al_p p^{-k-n'+i-\frac 1 2}, \al_p^{-1}p^{-k-n'+i-\frac 1 2},
	{\theta}(p) \al_pp^{-k-n'+i-\frac 1 2}, {\theta}(p)\al_p^{-1}p^{-k-n'+i-\frac 1 2}
	%\  \ (i=1, \cdots ,{n}) %\\ &
	%\{\al_p, \beta_p\} = \{\al_p, {\theta}(p)\al_p^{-1}\}
	\end{align}
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Sho}
\begin{frame}[shrink]
	\frametitle{Functional equation of the lift (thanks to Sho Takemori!)}	
There are two cases \cite{Ike08}: the even case (E) and the odd case (O): 
 $
 \begin{cases}
 f \in S_{2k+1}(\Gamma_0(D), \theta),  F=Lift^{(n)}(f) &(E) \\
(\text{the lift is of even degree }n = 2n'\text{ and of weight } 2k + 2n')&\\
f \in S_{2k}(\SL(\Z)), F=Lift^{(n)}(f) & (O)  \\
(\text{the lift is of odd degree }n = 2n'+1 \text{ and of weight } 2k + 2n').&
 \end{cases}
$ 
Then,
up to bad Euler factors, the standard $L$-function of $ F=Lift^{(n)}(f)$ is given by	
$
\prod_{i=1}^{n}
L(s + k + n' - i + \frac 1 2, f)L(s + k + n' - i + \frac 1 2, f, {\theta})
$
$=\begin{cases}
\prod_{i=1}^{2n'}
 L(s + k + n' - i + \frac 1 2, f)L(s + k + n' - i + \frac 1 2, f, {\theta})&  (E) \\
\prod_{i=1}^{n'}
L(t(s,i), f)L(t(s,2n'+1-i), f) & \\ 
L(t(s,i), f, {\theta})L(t(s,2n'+1-i), f, {\theta}) & \\
\prod_{i=1}^{2n'+1}
L(s + k + n' - i + \frac 1 2, f) &
\\
\times L(s + k + n' - i + \frac 1 2, f, {\theta})  & (O) \\
= L(s + k- \frac 1 2, f)L(s + k - \frac 1 2, f, {\theta})&\\ 
\prod_{i=1}^{n'}
L(t(s,i), f)L(t(s,2n'+2-i), f) & \\
L(t(s,i), f, {\theta})L(t(s,2n'+2-i), f, {\theta})&
\end{cases}
$
where $t(s,i)=s + k + n' - i + \frac 1 2$.	
 
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{Sho2}
\begin{frame}[shrink]
	\frametitle{The Gamma factor $\Gamma_\Zc(s)$ of Ikeda's lift}
In the even case since $(2k +1)-t(s, i)$ = $t(1-s, 2n'+1-i)$, using the Hecke functional equation in the symmetric terms of the product, gives the functional equation of the standard $L$ function of the form $s \mapsto 1 - s$, and the gamma factor is given by 
$$
\prod_{i=1}^{n}\Gamma_\CC(s + k + n' - i + 1/2)^2 =\Gamma_\Dr(s+n'+  \frac 1 2).
$$	
In the odd case when $f \in S_{2k}(SL_2(\Z))$, the lift is of degree $n = 2n' + 1$ and of weight $2k + 2n'$. By $2k - t(s, i)$ = $t(1 - s, 2n + 2 - i)$, the standard $L$ functions has functional equation of the
form $s \mapsto 1 - s$ and the gamma factor is the same.	

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hence the Gamma factor of Ikeda's lifting, denoted by ${\f}$, of an elliptic modular form $f$ and used as a pattern,  extends to a general (not necessarily lifted) Hermitian modular form ${\f}$ of even weight $\ell$, which equals in the lifted case to 
$\ell=2k+2n'$, where 
$k=(\ell-2n')/2$ $=\ell /2-n'$=$\ell/2-n'$, when 
the Gamma factor of the standard zeta function with the symmetry $s\mapsto 1-s$ becomes (see p.\pageref{Sho}) 
\smallskip
$\prod_{i=1}^n 
\Gamma_\CC (s+\ell/2 -n'+n'-i+(1/2))^2
=\prod_{i=1}^n 
\Gamma_\CC (s+\ell/2-i+(1/2))^2
=\prod_{i=0}^{n-1} 
\Gamma_\CC (s+\ell/2-i-(1/2))^2$.

	\
	
\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Proof of the Main Theorem (ii): Kummer congruences}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	Let us se the notation
	$\di {{\cal D}_{p}^{alg}(m,  {\f},\chi )}=
	A_{p}(s,\chi)\frac
	{{\cal D}^{*}(m,  {\f},\chi )}{\Omega_{\f}}$
	\\
	The integrality of  measures is proven 
	%by T.Bouganis \cite{Bou16}, 
	representing $\Dr_p^{alg}(m,\chi)$ as Rankin-Selberg type 
	integral at critical points $s=m$. 
	Coefficients of modular forms in this integral satisfy Kummer-type congruences and produce bounded measures
	$\mu_\Dr$ whose construction reduces to congruences of Kummer type between the Fourier coefficients of modular forms, see also \cite{Bou16}.
	\smallskip\pause
	Suppose that we are given 
	%some $L$-function ${\Dr}_p^\pm(s, {\bf f}, \chi)$ attached to a Hermitian modular form ${\bf f}$ using 
	infinitely many "critical pairs" $(s_j , \chi_j )$ 
	at which one has an integral representation
	$ \di
	{\Dr}_p^{alg}(s_j, {\bf f}, \chi_j)
	=A_{p}(s,\chi) \frac{\langle {\bf f} , h_j \rangle}
	{\Omega_{\bf f}}
	$
	with all $h_j =
	\sum_\Tc b_{j, \Tc} q^\Tc \in \Mc$
	in a certain
	finite-dimensional space $\Mc$ containing ${\bf f}$ and defined over $\bar{\Q}$.
	We prove the following Kummer-type congruences:
	$$
	\forall x\in\Z^*_p, \  \sum_j\beta_j\chi_jx^{k_j}\equiv 0 \mod p^N
	\Longrightarrow
	\sum_j\beta_j
	{\Dr}_p^{alg}(s_j, {\bf f}, \chi)
	\equiv 0 \mod p^N
	$$
	%for any choice of 
	$$
	\beta_j \in \bar \Q, k_j =
	s^*-s_j, \text{ where } s^* = \ell-n \text{ in our case.}
	$$ 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\smallskip\pause
	{\blue \Large Computing the Petersson products}
	of a given modular form
	${\bf f} (Z) =\sum_H a_H q^H\in \Mc_*(\bar{\Q})$
	by another modular form
	$h (Z) =\sum_H b_H q^H\in \Mc_{*}(\bar{\Q})$ uses a linear form
	$\di
	\ell_{\bf f} : h \mapsto \frac{\langle {\bf f} , h \rangle}{\langle {\bf f} , {\bf f} \rangle}
	$ defined over a subfield $k \subset \bar\Q.
	$		
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\
	
\end{frame}
%	\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}[shrink]
	\frametitle{Admissible Hermitian case} 
	%p.33 of \cite{Bou16} " Definition 7.2. 
	Let ${\f} \in \Sc_k(C; \psi )$ be a Hecke eigenform for 
	the congruence subgroup $C$ of level $\c$. 
	Let $\p$ be a prime of $K$ prime to $\c$, which is inert over $F$.
	Then we say that $\f$ is {\magenta pre-ordinary} at $\p$ if
	there exists an eigenform $0\ne {\f}_0 \in \Mc_{\{p\}} \subset \Sc_k(Cp,%D\b^{-1}, \b \c\p\p^\rho],
	\psi)$ with Satake parameters
	$t_{\p,i}$ such that 
	\begin{align}\label{ordinary}\nonumber
	\left \| 
	\left( 
	\prod_{i=1}^n t_{\p,i}
	\right) 
	N(\p)^{-\frac{n(n+1)} 2}
	\right \|_p=1,
	\end{align}
	where $\| \|_p$ the normalized absolute value at $p$. 
	
	\smallskip\pause
	The {\magenta admissible case} corresponds to
	
	\begin{align*}%\label{admissible} 
	\left \| 
	\left(\prod_{\q|p} 
	\prod_{i=1}^n t_{\q,i}
	\right)
	p^{-n(n+1)}
	\right \|_p=p^{-h}\text{ for a positive  } h>0.
	\end{align*}
	{\magenta  An interpretation of $h$ as the difference
		$h=P_{N,p}(d/2)-P_{H}(d/2)$ comes from the above explicit 
		relations.}
	
	\
	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}[shrink]
	\frametitle{Existence of $h$-admissible  measures} of Amice-Vélu-type gives an unbounded  $p$-adic analytic interpolation of the $L$-values of growth  $\log_p^h(\cdot)$, using the Mellin transform of the constructed measures.
	This condition says that the product $\prod_{i=1}^n t_{\p,i}$ is nonzero  and divisible by a certain power of $p$ in $\Oc$:
	$$
	\ord_p\left(\prod_{\q|p}\left( 
	\prod_{i=1}^n t_{\q,i}
	\right) 
	p^{-n(n+1)}\right)=h.
	$$
	
	We use an {\magenta easy condition of admissibility} of a sequence of modular distributions $\Phi_{j}$ on $X=\Oc_K \otimes \Z_p$ with values in $\Oc[[q]]$ as in Theorem 4.8 of \cite{CourPa}
	% with $A=\Or$
	and check {\magenta congruences} of the type
	$$
	U^{\vkp {v}}
	\Bigl(\sum_{j^\prime =0}^j{j\choose j^\prime}
	(-a_{p}^0)^{j-j^\prime}
	\Phi_{j^\prime} ({a+(p^{v})}\Bigr)\in
	Cp^{{v} j}\Oc[[q]]%\eqno (congruences)
	$$
	for all $j = 0, 1, \dots,  \vkp h-1$.
	Here $s=j'+s_*$, $\Phi_{j^\prime} ({a+(p^{v})})$
	a certain convolution, i.e.
	$$
	\Phi_{j^\prime} (\chi)=\theta(\chi)\cdot {\Ec}(s, \chi)
	$$
	of a Hermitian theta series $\theta(\chi)$ and an Eisenstein series ${\Ec}(s, \chi)$ with any Dirichlet character $\chi \bmod p^r$.  
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	We use a general sufficient condition of admissibility of a sequence of modular distributions $\Phi_{j}$ on $X=\Z_p$  with values in $\Oc[[q]]$ as in Theorem 4.8 of \cite{CourPa}.% with $A=\Oc$		
	
	\
	
	
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Proof of the Main Theorem (iii): (admissible case)}
	Using a Rankin-Selberg integral representation 
	for $\Dr^{alg}(s,{\f},\chi)$ and an eigenfunction ${\f}_0$ of Atkin's operator $U(p)$ of eigenvalue $\al_{\f}$ on ${\f}_0$ the Rankin-Selberg integral of
	${\cal F}_{s,\chi}:=\theta( \chi)\cdot 	{\Ec}(s, \chi)$ gives
	\begin{align} & \nonumber
	\Dr^{alg}(s,{\f},\chi)=\frac{\langle {\bf f}_0, 
		\theta( \chi)\cdot 	{\Ec}(s, \chi)\rangle}{\langle {\bf f}, 
		{\bf f}\rangle} 
	\text{    (the Petersson product on } G=GU(\eta_{n}) )
	\\ & \nonumber
	=
	\al_{\f}^{-v}\frac{\langle {\bf f}_0, U(p^v)(\theta( \chi)\cdot
		{\Ec}(s, \chi))\rangle}{\langle {\bf f}, {\bf f}\rangle}=
	\al_{\f}^{-v}
	\frac{\langle f_0, U(p^v) ({\cal F}_{s,\chi})\rangle} {\langle {\bf f}, {\bf f}\rangle}.
	\end{align}
	\smallskip\pause
	Modication in the admissible case: {\magenta instead of Kummer congruences},
	to {\magenta estimate $p$-adically the integrals of test functions}: $M=p^v$: 
	$\displaystyle  \int_{a+(M)} (x-a)^j d{\Dr^{alg}} := \sum_{j^\prime =0}^j{j\choose j^\prime}
	(-a )^{j-j^\prime}\int_{a+(M)} x^{j'} d{\Dr^{alg}}$,
	using the {\magenta orthogonality of characters} and the sequence of zeta distributions 
	$\displaystyle \int_{a+(M)} x^{j} d{\Dr^{alg}}=\frac{1}{\sharp (\Oc / M\Oc)^\times}\sum_{\chi \bmod M}\chi^{-1}(a)\int_X\chi (x)x^{j} d{\Dr^{alg}},$
	$ \int_X\chi  d{\Dr^{alg}_{s_-+j}}= 
	\Dr^{alg}(s^*-j,f,\chi)=:\int_X\chi (x)x^{j} d{\Dr^{alg}}$.
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20
\begin{frame}[shrink]
	\frametitle{Congruences between the 
		coefficients of the Hermitian modular forms}
	In order to integrate any locally-analytic function on $X$, it suffices to check the following binomial congruences for the 
	coefficients of the Hermitian modular form 
	$
	{\cal F}_{s^*-j,\chi}
	= \sum_{\xi }v({\xi}, s^*-j, \chi)q^\xi :
	$
	for $v\gg 0$, and a constant $C$
	\begin{align} & \nonumber
	\frac{1}{\sharp (\Oc / M\Oc)^\times}
	\sum_{j^\prime =0}^j{j\choose j^\prime}
	(-a )^{j-j^\prime}
	\sum_{\chi \bmod M} \chi^{-1}(a) v(p^v\xi, s^*-j', \chi)q^\xi
	% \label{stde6.21H} &  \nonumber  
	\\ & \nonumber \in Cp^{{v} j}\Oc [[q]] \ \ \text{    (This is a quasimodular form if } j'\ne s^*)
	\end{align}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\smallskip\pause
	The resulting measure $\mu_\Dr$ allows
	to integrate all continuous characters in 
	$\Yc_p =\Hom_{cont}(X,\CC_p^*)$, including Hecke characters,  
	as they are always locally analytic.
	
	{\magenta Its $p$-adic Mellin transform} $\Lr_{\mu_\Dr}$ is an analytic function 
	on $\Yc_p$ of the logarithmic growth $\Oc( \log^h)$,  $h=\ord_p(\alpha)$. 
	
	
	
	
	
	
	\
	
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ***** 18
\begin{frame}[shrink]
	\frametitle{Proof of the main congruences}
	Thus the Petersson product in $\ell_{\bf f}$ can be expressed through
	the Fourier coeffcients of $h$ in the case when there is a finite basis
	of the dual space consisting of certain Fourier coeffcients:
	$\ell_{\Tc_i} : h \mapsto b_{\Tc_i} (i = 1, \dots, n)$.
	It follows that 
	$\ell_{\bf f} (h) =
	\sum_i\gamma_i  b_{\Tc_i}$, where 
	$\gamma_i\in k$.	
	
	Using the expression for $\ell_f (h_j ) =
	\sum_i \gamma_{i,j} b_{j,\Tc_i}$, the above
	congruences reduce to
	$$
	\sum_{i,j} \gamma_{i,j}\beta_j b_{j,\Tc_i} \equiv 0 \mod p^N.
	$$
	%\fi
	The last congruence is done by
	an elementary check on the Fourier coefficients $b_{j,\Tc_i}$.
	
	
	The abstract Kummer congruences are checked for a family of test elements.
	
	In the admissible case it suffices to check {\magenta binomial congruences} 
	for the Fourier coefficients
	as above in place of Kummer congruences.
	
	
	\
	
\end{frame} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Appendix A. Rewriting the local factor at $p$ with character $\theta$ } 
	Notice that if $\theta$ is the quadratic character attached to $K/\Q$ then 
	%\iffalse
	%\end{frame}
	%\end{document}
	$$
	(1-\al_pX)(1-\al_p\theta(p)X)=
	\begin{cases}
	(1-\al_pX)^2& \text{ if } \theta(p)=1, p\r=\q_1\q_2, N(\q_i)=p,\\
	(1-\al_p^2X^2),&\text{ if } \theta(p)=-1, 
	p{\r}=\q, N(\q)=p^2,\\ 
	(1-\al_pX)&\text{ if } \theta(p)=0, p{\r}=\q^2, N(\q)=p.\\
	\end{cases}
	$$		
	Thus, if $X=p^{-s}$, $X^2=p^{-2s}$, $N(\q)=p$, 
	$\Zc_\q(X)^{-1}$ 
	$$=
	\begin{cases}
	\prod_{i=1}^{2n}
	(1- N(\q_1)^{2n}t_{\q_1\q_2,i}^{-1}X)
	(1- N(\q_2)^{-1}t_{\q_1\q_2,i}X),  
	& \text{ if } \theta(p)=1,\\
	\prod_{i=1}^{n}
	{(1- N(\q)^{n-1}t_{\q,i} X^2)(1- N(\q)^{n}t_{\q,i}^{-1}
		X^2)},& \text{ if }  \theta(p)=-1,\\
	\prod_{i=1}^{n}
	{(1- N(\q)^{n-1}t_{\q,i}X)(1- N(\q)^{n}t_{\q,i}^{-1} X)},	
	& \text{ if } \theta(p)=0.
	\end{cases}
	$$	
	$$	
	= \begin{cases}
	\prod_{i=1}^{n}(1-\ga_{p,i}X)^2
	\prod_{i=1}^{n}(1-\de_{p,i}X)^2
	& \text{ if } 
	%N(\q_1)^{2n}t_{\q_1\q_2,i}^{-1}=
	%N(\q_2)^{-1}t_{\q_1\q_2,i}
	\theta(p)=1, \text{ i.e. }p\r=\q_1\q_2,\\
	\prod_{i=1}^{n}(1-\al_{p,i}^2X^2)
	\prod_{i=1}^{n}(1-\beta_{p,i}^2X^2)
	,&\text{ if } \theta(p)=-1, \text{ i.e. } 
	p{\r}=\q,\\ 
	\prod_{i=1}^{n}(1-\al'_{p,i}X)
	\prod_{i=1}^{n}(1-\beta'_{p,i}X)
	&\text{ if } \theta(p)=0, 
	\text{ i.e. }p{\r}=\q^2,
	\end{cases}
	$$
	where $\al'_{p,i}=p^{n-1}t_{\q,i}$, $\beta'_{p,i}p^{n}t_{\q,i}^{-1}$,
	$\ga_{p,i}= p^{2n}t_{\q_1\q_2,i}^{-1}$,
	$p^{-1}t_{\q_1\q_2,i}$.
	It follows that $\prod_{\q|p}\Zc_\q (N(\q)^{-n-(1/2)}X)=X^{4n}+\cdots$ 
	
	
	\
	
\end{frame}
%\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{Appendix B. Shimura's Theorem: algebraicity of critical values in Cases Sp and UT, p.234 of \cite{Shi00}}
	%\begin{theo}
	Let ${\bf f}\in {\mathcal V} (\bar\Q)$ be a non zero  arithmetical automorphic form of type Sp or UT. 
	Let $\chi$ be a Hecke character of $K$ such that 
	$\chi_{\bf a}(x)=x_{\bf a}^\ell |x_{\bf a}|^{-\ell}$ with $\ell\in\Z^{{\bf a}}$, \text{ and let }
	$\sigma_0\in 2^{-1}\Z$.
	Assume, in the notations of Chapter 7 of \cite{Shi00} on the weights $k_v, \mu_v, \ell_v$, that 
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{align*}%{ll}
		&\text{\rm Case Sp} &&
		\  2n+1-k_v+\mu_v \le 2\si_0\le k_v-\mu_v,  
		\\&{}  && \  \text{ where }
		\mu_v=0  \text{ if } [k_v]-l_v \in 2\Z \\&{}  && \  \text{ and } \mu_v=1
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\text{ if } [k_v]-l_v \not\in 2\Z; \ 
		\si_0-k_v+\mu_v
		\\&{}  && \  \text{ for every } v \in  {\bf a}  \text{ if }\si_0>n
		\text{ and } 
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\&{}  &&\ \si_0-1-k_v+\mu_v \in 2\Z\text{ for every } v \in  {\bf a}
		\text{ if }\si_0\le n. \\
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		&\text{\rm Case UT} &&\  4n-(2k_{v\rho}+\ell_v)\le 2\si_0\le 
		m_v-| k_v-k_{v\rho}-\ell_v |\\&{}  && \  \text{ and } 2\si_0-\ell_v\in 2\Z  \text{ for every  } v \in  {\bf a}.
	\end{align*}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19
\begin{frame}[shrink]
	\frametitle{Appendix B. Shimura's Theorem (continued)}
	Further exclude the following cases
	\begin{align*}%{ll}
		&\text{\rm (A) Case Sp} && \si_0=n+1,F=\Q  
		\text{ and }
		\chi^2=1;
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\ &\text{\rm (B) Case Sp} &&\si_0=n+(3/2),F=\Q; \chi^2=1  
		\text{ and }
		[k]-\ell\in 2\Z
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\ &\text{\rm (C) Case Sp} &&\si_0=0,
		{\mathfrak c}={\mathfrak g} 
		\text{ and }
		\chi=1;
		\\ &\text{\rm (D) Case Sp} &&0<\si_0\le n, {\mathfrak c}={\mathfrak g}, \chi^2=1
		\text{ and the conductor of } \chi \text{ is }{\mathfrak g};
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\&\text{\rm (E) Case UT} && 2\si_0=2n+1,F=\Q, \chi_1=\theta,  
		\text{ and }
		k_v-k_{v\rho}=\ell_v;
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\\&\text{\rm (F) Case UT} && 0\le 2\si_0 < 2n, {\mathfrak c}={\mathfrak g},
		\chi_1=\theta^{2\si_0} 	\text{ and the conductor of } \chi 	\text{ is } {\mathfrak r}
		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\end{align*}
	Then 
	$$
	{\mathcal Z}(\si_0, {\bf f}, \chi)/\langle  {\bf f},  {\bf f} \rangle\in\pi^{n|m|+d\varepsilon}\bar \Q, 
	$$
	where $d = [F:\Q]$, $|m|=\sum_{v\in {\bf a}}m_v$, and
	$$\varepsilon=
	\begin{cases}
	(n+1)\si_0-n^2-n,&\text{Case Sp}, k\in \Z^{{\bf a}}, 
	\text{ and } \si_0>n_0),
	\\
	n\si_0-n^2, &\text{Case Sp}, k \not\in \Z^{{\bf a}}, 
	\text{or} \si_0\le n_0),
	\\ 2n\si_0-2n^2+n
	&\text{Case UT}
	\end{cases}
	$$
	%\end{theo}		
	Notice that $\pi^{n|m|+d\varepsilon}\in\Z$ in all cases; if $k\not \in\Z^{\bf a}$,  the above parity condition on $\sigma_0$ shows that  $\sigma_0+k_v\in\Z$, so that $n|m|+d\varepsilon\in\Z$.
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{}	
%%%%%%%%%%%%%%%%%%%%%
\vskip2.5cm 
%\hskip3.5cm
 {\huge \bf Thanks for your attention!}
\vskip1.5cm 

		
	{\large 	Many thanks to
		Jean-Louis  Verger-Gaugry
		for his invitation to this Summer School on Theory of Motives and Number Theory  
	at the crossroad of automorphic  $L$ functions (complex and $p$-adic), zeta functions, polyzeta functions and dynamical zeta function, held on the scientific campus of Le-Bourget-du-Lac,
		to Siegfried Boecherer (Mannheim), 
%		Sho Takemori (MPIM) and Emmanuel Royer (University Clermont Auvergne) 
		for valuable discussions and observations.}	
	
	\smallskip	
	My special thought go to the memory of Alexey Zykin , professor of the French Polynesia University, tragically dissapeared  in Tahiti in 2017.
	The present Summer School on Theory of Motives and Number Theory
was conceived as a continuation of the series of Conferences "Zeta-functions I-VI", held in J.-V.Poncelet Laboratory UMI 2615 du CNRS, and to commemorate Alexey.

	\
	
\end{frame} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{} 
	
	\	
	
\end{frame} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\begin{frame}[shrink]
		\frametitle{} 
		
			
	\
	
	
	
\end{frame} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{} 
	
	\
	
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Références}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\ifslide
{\large\blue References}
\fi


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20
%\newpage
\bibliographystyle{plain}
\begin{thebibliography}{10xxxx}
%\bibitem[Am75]{Am75}
%{\sc Amice,  Y.}, 
%{\em  Les nombres $p$-adiques}, 1975, PUF, Collection SUP, p.189.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Am-V]{Am-V}
{\sc Amice,  Y.} and {\sc Vélu,  J.}, 
{\em Distributions $p$-adiques associ\'ees aux
	s\'eries de Hecke}, 
Journ\'ees Arithm\'etiques de Bordeaux (Conf. Univ.
Bordeaux, 1974), Ast\'e\-risque no. 24/25, Soc. Math. France, Paris
1975, pp. 119-131
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[An74]{An74}
{\sc Andrianov, A.N.}, {\em 
Euler products corresponding to Siegel modular forms of genus  2}. Russian Math. Surveys 29:3 (1974), 45-116,
from Uspekhi Mat. Nauk 29:3 (1974) 43-110.

\bibitem[An87]{An87} 
{\sc Andrianov, A.N.}, {\em Quadratic Forms and Hecke Operators},
Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo,
1987.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}

\end{document}
