
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\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>
\mode<presentation>
{
  \usetheme{}%{Boadilla} %Madrid Singapore Montpellier
  \setbeamercovered{transparent}
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%\pagestyle{plain}
%\markboth{left head}{right head}
%%%%%%%%%%%%%%%%%%%%%%%%%
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%\uncover<+(1)->{
%\hskip5pt
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%%%%%%%%%%%%%%%%%%%%%%
%\mode<article>{\usepackage{fullpage}}
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%  \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{\Er}{{\mathcal E}}
\newcommand{\Hr}{{\mathcal H}}
\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} \\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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\newcommand{\vkp}{\varkappa}
\newcommand{\chiba}{\bar{\chi}}
\newcommand{\chibar}{\bar{\chi}}
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\newcommand{\lb}{\lambda}
\newcommand{\vr}{}
\def\dbQbar{\overline {\dbQ}}
\def\Lb{\Lambda}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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
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\newfam\Brus
\textfont\Brus=\tenbrus
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\font\tenirus=wncyi10
\font\eightirus=wncyi8
\newfam\Irus
\textfont\Irus=\tenirus
\scriptfont\Irus=\eightirus
\def\iru{\fam\Irus\tenirus}
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\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}
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\def\l1{l{}p1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\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
]
{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
Workshop "Arithmetic of automorphic forms and special $L$-values"\\ \vskip0.5cm \large\blue		
%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[March 26-27, 2018 ] % (optional)
{March 26-27, 2018 \\ 
	Durham University,
	UK
	%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[]
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{
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\addtocounter{framenumber}{-1}
  \end{frame}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\FF{\mathbb F}
\def\ZZ{\mathbb Z}

\def\f{\bf f}
\def\p{\mathfrak p}
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%\def\Numero{${\rm N\sp\circ}$}
%\newcounter{ncours}{\setcounter{ncours} {1}}
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%\addtocounter{ncours}{1}}

\begin{document}
	
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%\setbeamertemplate{footline}{}


\maketitle 

%[shrink=5]
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  INTRODUCTION
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setbeamertemplate{footline}
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		\vskip-10pt
		%%%%%%%%%%%%%%%%%%%
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		}
		\ {}
	\end{beamercolorbox}%
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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, 
	{\Er}(\bar s + n, \ell - \ell_\theta, \chi^\rho\psi) \rangle_{C\sec},
	\end{align}
	where ${\Er}(Z, s, \ell-\ell_\theta, \psi)_{C\sec}$ 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.
		$$
		\Phi_{j^\prime} (\chi)=\theta(\chi)\cdot {\Er}(s, \chi)
		$$
		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}
\iffalse
\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
${\cal F}_{s,\chi}:=\theta( \chi)\cdot 	{\Er}(s, \chi)$ gives
\begin{align} & \nonumber
\Dr^{alg}(s,{\f},\chi)=\frac{\langle {\bf f}_0, 
	\theta( \chi)\cdot 	{\Er}(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
	{\Er}(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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
	\frametitle{}	
%%%%%%%%%%%%%%%%%%%%%
\vskip2.5cm 
%\hskip3.5cm
 {\huge \bf Thanks for your attention!}
\vskip1.5cm 


\


		
	{\large 	Many thanks to Athanasios BOUGANIS 
		%Edray H. Goins and David Goldberg
		for his invitation to  a two days workshop entitled "Arithmetic of automorphic forms and special L-values" at Durham University, on Monday 26th  and Tuesday 27th of March 2018, 
		%the Department of Mathematics Colloquium 
		%of Purdue University,
		%on the occasion of The 2017 Paul J. Sally Jr. Midwest Representation Theory Conference Honoring the 70th Birthday of Freydoon Shahidi,
		to Siegfried Boecherer (Mannheim), Sho Takemori (MPIM) and Emmanuel Royer (University Clermont Auvergne) 
		%for the invitation,
		%to  the University Clermont Auvergne, 
		%to  the Mathematics Laboratory  Blaise Pascal (laboratoire de mathématiques Blaise Pascal, Unité mixte de recherche 6620 UCA et CNRS)
		%Maxim Aleksandrovich Korolev for the invitation,
		%to Steklov Mathematical Institute of Russian Academy of Sciences, to the	 
		%Faculty of Mechanics and Mathematics of
		%	Moscow State University Lomonosov,
		for valuable discussions and observations.}	
\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[ArChu]{ArChu}
%{\sc Archipov}, G. I. , {\sc Chubarikov} V. N. 
%{\it On the mathematical works of professor A. A. Karatsuba.}
%Proc. Steklov Inst. Math., vol. 218, 1997. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%\bibitem[AChK]{AChK}
%{\sc Arkhipov} G. I., {\sc Chubarikov} V. N., {\sc Karatsuba} A. A., 
%Trigonometric sums in number theory and analysis (English). 
%Berlin-New-York: Walter de Gruyter, 2004, 554 p.


\bibitem[Boe85]{Boe85}
{\sc B\"ocherer}, S., \ {\em \"Uber die Funk\-tio\-nal\-glei\-chung 
	auto\-mor\-pher $L$--Funk\-tio\-nen zur Sie\-gel\-scher Modul\-gruppe.} 
J. reine angew. Math. 362 (1985) 146-168

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 21
\bibitem[BoeNa13]{BoeNa13}
{\sc Boecherer,} S.,  {\sc  Nagaoka},  S. , 
{\em On $p$-adic properties of Siegel modular forms}, in: Automorphic Forms. Research in Number
Theory from Oman. Springer Proceedings in Mathematics and
Statistics 115. Springer 2014. %, see also arXiv:1305.0604 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem[Boe-Pa9]{Boe-Pa9}
%{\sc B\"ocherer}, S.,  {\sc Panchishkin},  A.A.,
%{\it $p$-adic Interpolation of Triple $L$-functions: Analytic Aspects}. %In: Automorphic Forms and $L$-functions II: Local Aspects -- David Ginzburg, 
%Tel-Aviv University, 
%Erez Lapid, %The Hebrew University of Jerusalem, 
%and David Soudry, %Tel-Aviv University, Editors, AMS, 2009, 313 pp.; pp.1-41
%\fi

\bibitem[Boe-Pa11]{Boe-Pa11}
{\sc B\"ocherer}, S.,  {\sc Panchishkin},  A.A.,
{\it Higher Twists and Higher Gauss Sums}
Vietnam Journal of Mathematics 39:3 (2011) 309-326

%\bibitem[BH06]{BH06}
%{\sc B\"ocherer}, S.,  {\sc Heim},  B., Critical values of L-functions on $GSp_2\times Gl_2$
%Math. Z. (2006) 254: 485-503 

\bibitem[BS00]{BS00}
{\sc B\"ocherer, S.},  and {\sc Schmidt, C.-G.}, 
{\em $p$-adic measures attached to Siegel modular forms}, 
Ann. Inst. Fourier 50, \Numero 5, 1375-1443 (2000).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Bou14]{Bou14}
{\sc Bouganis T.} {\em Non-abelian p-adic L-functions and Eisenstein series of unitary groups; the CM
	method}, Ann. Inst. Fourier (Grenoble), 64 no. 2 (2014), p. 793-891.


\bibitem[Bou16]{Bou16}
{\sc Bouganis T.} {\em 
p-adic Measures for Hermitian Modular Forms and the Rankin-Selberg Method.}
in Elliptic Curves, Modular Forms and Iwasawa Theory - Conference in honour of the 70th birthday of John Coates,  pp 33-86
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 22
 \bibitem[Hur899]{Hur899}
 {\sc Carlitz,  } L.  , {\em  The coefficients of the lemniscate function}, Math. Comp., 16 (1962), 475-478.
\bibitem[CourPa]{CourPa}
{\sc Courtieu,}M.,  {\sc Panchishkin },A.A.,
{\em Non-Archimedean $L$-Functions and Arithmetical Siegel Modular Forms},
Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

\bibitem[CoWi77]{CoWi77}
{\sc  Coates }, J.  and {\sc Wiles}, A., 
{\em  On the conjecture of Birch and Swinnerton-Dyer}, Inventiones math. {\bf 39}, 223-251

\bibitem[Coh17]{Coh15}
{\sc Cohen, H.} {\em
	Computing L -Functions: A Survey.} Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 3 , p. 699-726



%%%%%%%%%%%%%%%%%%%%%
%\bibitem[De84]{De84}
%{\sc Deligne, P.},  {\em  Les corps locaux de caractéristique p, limites %de corps locaux de caractéristique 0}. Representations of reductive %groups over a local field, 119-157, Travaux en Cours, Hermann, Paris, %1984.

	\bibitem[De79]{De79}
	{\sc Deligne} P., 
	{\em Valeurs de fonctions $L$ et p\'{e}riodes
		d'int\'egrales,} Proc.Sympos.Pure Math. vol. 55. Amer. Math. Soc.,
	Providence, RI, 1979 , 313-346.


\bibitem[EE]{EE}
{\sc Eischen}, Ellen E., {\em $p$-adic Differential Operators on
	Automorphic Forms on Unitary Groups.} Annales de l'Institut
Fourier 62, No.1 (2012) 177-243.

	\bibitem[EHLS]{EHLS}
	{\sc Eischen} Ellen E., {\sc  Harris}, Michael,  {\sc Li}, Jian-Shu,
	{\sc Skinner}, Christopher M., {\em 
		p-adic L-functions for unitary groups},
	{\tt	arXiv:1602.01776v3 [math.NT]}

\bibitem[EZ85]{EZ85}  
{\sc Eichler,} M., {\sc Zagier,} D.,
{\em The theory of Jacobi forms}, Progress in Mathematics, vol. 55 (Birkhäuser,
Boston, MA, 1985).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 23
\bibitem[Ike01]{Ike01}
{\sc Ikeda, T.},
{\em On the lifting of elliptic cusp forms to Siegel cusp forms of degree $2n$}, Ann. of
Math. (2) \textbf{154} (2001), 641-681.


\bibitem[Ike08]{Ike08}
{\sc Ikeda, T.},
{\em  On the lifting of Hermitian modular forms}, 
Compositio Math. \textbf{144}, 1107-1154,  (2008)

%\bibitem[Fu93]{Fu93}
%{\sc Furusawa M.},  {\em On L-functions for $GSp(4)\times GL(2)$ 
%and their special values}, 
%J. Reine Angew. Math. 438 (1993) 187-218.



%\bibitem[GPS]{GPS}
% {\sc Gelbart,} S.,  {\sc Panchishkin,} A., and  {\sc Shahidi,} S., %{\it The $p$-Adic Eisenstein Measure and Fourier coefficients for %SL(2)}
%   Arxiv NT, 1001.1913 (2010) 
\bibitem[Iw]{Iwa72} K. {\sc Iwasawa},
{\it Lectures on $p$-Adic $L$-Functions},
Ann. of Math. Studies, N{${}^\circ$} 74. Princeton Univ. Press (1972).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Pa14]{Pa14}
 {\sc Panchishkin,} S.,
{\em Analytic constructions of
$p$-adic $L$-functions and Eisenstein series.} Travaux du
Colloque "Automorphic Forms and Related Geometry,   Assessing the Legacy of I.I.Piatetski-Shapiro
(23-27 April, 2012, Yale University in New Haven, CT)",  345-374, 2014

\bibitem[GMPS14]{GMPS14}
{\sc Gelbart,} S., {\sc Miller,} S.D, {\sc Panchishkin,} S., and  {\sc Shahidi}, F.,
{\em 
A $p$-adic integral for the reciprocal of $L$-functions.} Travaux du Colloque "Automorphic Forms and Related Geometry, Assessing the Legacy of I.I. Piatetski-Shapiro" (23 - 27 April, 2012, Yale University in New Haven, CT), Contemporary Mathematics, 345-374 (avec Stephen Gelbart, Stephen D. Miller, and Freydoon Shahidi), 53-68, 2014.


\bibitem[GeSha]{GeSha}
{\sc Gelbart,} S.,  and  {\sc Shahidi}, F.,
{\em Analytic Properties of Automorphic $L$-functions}, Academic Press, New York, 1988.

\bibitem[GPSR]{GPSR}  {\sc Gelbart S.,Piatetski-Shapiro I.I., Rallis S.}
{\it Explicit
	constructions of automorphic $L$-functions.}
Springer-Verlag, Lect. Notes in Math.
N 1254 (1987) 152p.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Ge16]{Ge16}  {\sc Guerberoff, L.}, 
{\it Period relations for automorphic forms on unitary groups and critical values of $L$-functons}, Preprint, 2016.

\bibitem[GH16]{GH16}  {\sc Grobner, H. and Harris, M.}
{\it  Whittaker periods, motivic periods, and special values of tensor product $l$-functions}, Journal of the Institute of Mathematics of Jussieu  Volume 15, Issue 4, October 2016, pp. 711-769


\bibitem[Ha81]{Ha81}
{\sc Harris, } M., {\em
	Special values of zeta functions attached to Siegel modular
	forms. }Ann. Sci. Ecole Norm Sup. 14 (1981), 77-120. 
 \bibitem[Ha97]{Ha97}
 {\sc Harris, } M., {\em L-functions and periods of polarized regular motives.} J. Reine Angew. Math, (483):75-161, 1997.
 
 \bibitem[Ha14]{Ha14}
 {\sc Harris, } M., {\em
 	Automorphic Galois representations and the
 	cohomology of Shimura varieties.}
 Proceedings of the International Congress of Mathematicians, Seoul, 2014
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
 \bibitem[Hur1899]{Hur1899}
 {\sc Hurwitz,  } A., {\em Über die Entwicklungskoeffizienten der lemniskatischen Funktionen}, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhaeuser, Basel, 1962-1963, see Vol. 2, No. LXVII. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibitem[Kawa8]{Kawa8} {\sc  Kawamura,} Hisa-Aki, Private communication.
%(Hokkaido University (Sapporo, Japan)/ Institute Fourier (Grenoble, France)), 
%July 2008, 
%{ \tt
%http://www-fourier.ujf-grenoble.fr/\~{}panchish\\
%/KawamuraCong11-19-23.pdf}


\bibitem[Ich12]{Ich12}
{\sc Ichikawa, T.},
{\em Vector-valued $p$-adic Siegel modular forms}, J.
reine angew. Math., DOI 10.1515/ crelle-2012-0066.

%\bibitem[Ich13]{Ich13}
%{\sc Ichikawa, T.},
%{\em  Arithmeticity of vector-valued Siegel modular
%	forms in analytic and p-adic cases}. Arxiv: 1508.03138v2  [MathNT].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24
%\bibitem[Kar]{Kar} {\sc
%	Karatsuba}, A.A. (1975) "Introduction to the analytical number theory"
%(in Russian). Moscow, Nauka (1975). English transl. of
%the 2nd edition: Basic Analytic Number Theory. Berlin-Heidelberg-New York: Springer-Verlag, 1993] was inspiring for the generations of mathematicians studing  number theory.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Ka76]{Ka76}  
{\sc  Katz, N.M.}, {\em $p$-adic interpolation of real analytic Eisenstein
	series.} Ann. of Math. 104 (1976) 459--571

%\bibitem[Ka77]{Ka77}  
%{\sc  Katz, N.M.},
%{\em The Eisenstein measure and $p$-adic interpolation.}
% Amer. J. Math. 99 (1977) 238--311

\bibitem[Ka78]{Ka78}  
{\sc  Katz, N.M.}, {\em $p$- adic $L$-functions for CM-fields.} Invent.
Math. 48 (1978) 199-297

\bibitem[KiNa16]{KiNa16}  
{\sc Kikuta}, Toshiyuki,  {\sc Nagaoka}, Shoyu,  {\em 
	Note on mod $p$ property of Hermitian modular forms} arXiv:1601.03506 [math.NT]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù%%
\bibitem[Kl]{Kl}
{\sc Klosin },K., 
{\em Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive
	of a modular form}, 
Journal of the Mathematical Society of Japan, Vol. 67, No. 2 (2015) pp. 797-860.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem[Ko80]{Ko80}
{\sc Koblitz}, Neal, 
{\em p-adic Analysis. A Short Course on
	Recent Work}, Cambridge Univ. Press, 1980
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[KuLe64]{KuLe64} 
% [204]. 
Kubota, T., Leopoldt, H.--W. (1964): Eine $p-$adische Theorie
der Zetawerte. I. J. reine u. angew. Math., \textbf{214/215}, 328-339 (1964).
%Zbl.186,91.

\iffalse
\bibitem[Kr44]{Kr44}
{\sc Krasner}, M., {\em Une généralisation de la notion de corps-corpoïde. Un corpoïde remarquable de la
théorie des corps valués}, C. R. Acad. Sci. Paris Sér. IMath. 219 (1944), 345-347.

\bibitem[Kr50]{Kr50}
{\sc Krasner}, M., {\em Quelques méthodes nouvelles dans la théorie des corps valués complets.}
Algèbre et Théorie des Nombres, 29-39. Colloques Internationaux du Centre National
	de la Recherche Scientifique, no. 24, Centre National de la Recherche Scientifique,
	Paris, 1950.

\bibitem[Kr66]{Kr66}
{\sc Krasner}, M., Prolongement analytique uniforme et multiforme,
Collogue C.N.R.,S. n° 143, Clermont-Ferrand, 1963, Paris, Ed. C.N.R.S.,
1966, p. 97-141


\bibitem[Kr74]{Kr74}
{\sc Krasner}, M., 
{\em Rapport sur le prologement analytique dans les corps
	values complete par la methode des e1ements analytiques quasiconnexes},
Bull. Soc. Math. France, Mem. 39-40 (1974) 131-254.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bibitem[Kr80]{Kr80}
{\sc Krasner}, M., 
{\em  Anneaux gradués généraux}, Colloque d'Algèbre Rennes (1980), 209-308.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù%%

\bibitem[KrKa51]{KrKa51}
{\sc Krasner}, M., {\sc Kaloujnine}, L.  
{\em  Produit complet des groupes de permutations et problème d'extension de groupes II }, Acta Sci. Math. Szeged , 14 (1951) p. 39-66 et 69-82.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem[KrVu87]{KrVu87}
{\sc Krasner}, M., {\sc Vukovi\'c}, M. 
{\em Structures paragraduées (groupes, anneaux, modules)}, Queen's Papers in Pure and Applied Mathematics, 77, Queen's University, Kingston, Ontario, Canada, 1987.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fi


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 25
\bibitem[LangMF]{LangMF}
{\sc
	Lang}, Serge. {\em Introduction to modular forms. With appendixes by D. Zagier and Walter Feit.} %Corrected reprint of the 1976 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 222. 
Springer-Verlag, Berlin, 1995

\bibitem[Ma73]{Ma73} {\sc  Manin, Yu. I.},
{\em  Periods of cusp forms and p-adic Hecke series}, Mat.
Sbornik, 92 , 1973, pp. 378-401

\bibitem[Ma76]{Ma76} {\sc  Manin, Yu. I.},
{\em  Non-Archimedean integration and Jacquet-Langlands p-adic L-functions}, Uspekhi Mat. Nauk, 1976,
Volume 31, Issue 1(187), 5-54


\bibitem[MaPa]{MaPa}{\sc  Manin, Yu. I.},{\sc Panchishkin, A.A.},
{\em Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories} (Encyclopaedia of Mathematical Sciences), Second Edition, 504 p., Springer (2005) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\bibitem[Ma-Vi]{Ma-Vi}
{\sc Manin},  Yu.I., {\sc Vishik}, M. M., 
{\em  $p$-adic Hecke series of imaginary quadratic fields}, 
(Russian) Mat. Sb. (N.S.) 95(137) (1974), 357-383. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[MTT]{MTT} 
{\sc Mazur}, B.,   {\sc Tate} J.,   {\sc Teitelbaum}, J.,  
On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer. 
Invent. Math. 84, 1-48 (1986).

\bibitem[Mi-St]{Mil-Sta74} J. {\sc Milnor}, J. {\sc Stasheff},
{\it Characteristic Classes},
Ann. of Math. Studies N{${}^\circ$} 76, Princeton Univ. Press. (1974), p 231-264.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    26

\bibitem[Pa88]{Pa88}{\sc Panchishkin, A.A.},
{\em Non-Archimedean automorphic zeta functions},
Moscow University Press (1988).


\bibitem[Pa91]{Pa91}{\sc Panchishkin, A.A.},
{\em Non-Archimedean $L$-Functions of Siegel and Hilbert Modular Forms.}
 Volume 1471 (1991)
 
\bibitem[Pa94]{Pa94}
 {\sc Panchishkin,  A.}, 
 {\em Motives over totally real fields and $p$--adic
 $L$--functions}.  Annales de
 l'Institut Fourier, Grenoble, {\bf 44}, 4 (1994), 989--1023 

%\bibitem[PaSE]{PaSE}  {\sc Panchishkin, A.A.},
%{\it On the Siegel-Eisenstein measure and
%	its applications},
%Israel Journal of Mathemetics, 120, Part B (2000) 467-509.

\bibitem[PaMMJ]{PaMMJ} {\sc Panchishkin,   A.A.},
{\em A new method of constructing $p$-adic $L$-functions associated 
	with modular forms}, 
Moscow Mathematical Journal, 2 (2002), Number 2, 1-16


\bibitem[PaTV]{PaTV} {\sc Panchishkin,   A. A.},
{\em Two variable $p$-adic $L$ functions attached to eigenfamilies of positive slope},
Invent. Math. v. 154, N3 (2003), pp. 551 - 615



%\bibitem[PaB1]{PaB1}
%{\sc Panchishkin,  A.A.}, 
%{\em The Maass--Shimura differential operators and 
%congruences between 
%arithmetical   Siegel modular forms},  
%Moscow Mathematical Journal,  
% v. 5,  N 4,  883-918 (2005).

	\bibitem[RG78]{RG78} {\sc Robert, Gilles},  
	{\em Nombres de Hurwitz et unités elliptiques. Un critère de régularité pour les extensions abéliennes d'un corps quadratique imaginaire.}
 Annales scientifiques de l'École Normale Supérieure, Sér. 4, 11 no. 3, 1978
 p. 297-389 


\bibitem[Sha69]{Sha69}  {\sc Shafarevich}, I.R.
{\em Zeta Function},
Moscow University Press (1969).


	\bibitem[Sl]{Sl} {\sc Sloane N.J.A,},  
	{\em A047817. Denominators of Hurwitz numbers $H_n$} The On-Line Encyclopedia of Integer Sequences {\tt https://oeis.org/A047817}. 

\bibitem[Se73]{Se73}
{\sc Serre,  J.--P.}, 
{\em Cours d'arithmétique}. Paris, 1970.


	\bibitem[Se70]{Se70} {\sc Serre,  J.--P.},  
	{\em Facteurs locaux des fonctions z\^eta
	des vari\'et\'es alg\'eb\-riques (d\'efinitions et conjectures)}.
	Sém. Delange - Pisot - Poitou, exp. 19, 1969/70.

\bibitem[Se73]{Se73}
{\sc Serre,  J.--P.}, 
{\em Formes modulaires et fonctions z\^eta $p$-adiques}, 
Lect Notes in Math. 350 (1973) 191--268 (Springer Verlag)





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 27
[\bibitem[Shi97a]{Shi97a}
{\sc Shimura} G.,
{\em  Euler Products and Eisenstein series}, CBMS Regional Conference
Series in Mathematics, No.93, Amer. Math. Soc, 1997.

\bibitem[Shi97b]{Shi97b}
{\sc Shimura} G.,
{\em Colloquium Paper: Zeta functions and Eisenstein series on classical groups},
Proc Nat. Acad Sci U S A. 1997 Oct 14; 94(21): 11133-11137




\bibitem[Shi00]{Shi00}
{\sc Shimura} G.,
{\em Arithmeticity in the theory of automorphic forms},
Mathematical Surveys and Monographs, vol. 82 (Amer. Math.
Soc., Providence, 2000).


%\bibitem[SkU6]{SkU6}
%{\sc Skinner},  C.  and {\sc Urban}, E.  {\em Sur les déformations $p$-adiques de certaines %représentations automorphes},
%Journal of the Inst. of Math. Jussieu 5 (2006), no. 4, 629-698.

\bibitem[MC]{MC}
{\sc Skinner},  C.  and {\sc Urban}, E.  {\em
	The Iwasawa Main Cconjecture for  GL(2)}.
%{\tt http://www.math.jussieu.fr/\~{}urban/eurp/MC.pdf}
Invent. Math. 195 (2014), no. 1, 1-277.
MR 3148103
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[U14]{U14}
{\sc  Urban,  E.},
{\em Nearly Overconvergent Modular Forms},
in: Iwasawa Theory 2012.
State of the Art and Recent Advances,
%Thanasis Bouganis, Otmar Venjakob Eds.,
 Contributions in Mathematical and Computational Sciences book series (CMCS, Vol. 7), pp. 401-441 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[VoKar]{VoKar}
{\sc Voronin},  S.M., {\sc Karatsuba},  A.A., {\it The Riemann zeta-function}, Moscow, Fizmatlit,
1994.

%\bibitem[Vo]{Vo}
%{\sc Voronin},  S.M.,  {\it Izbrannye Trudy (Selected Works). Mathematics.} / Ed. by A.A.Karatsuba, Moscow, Edition MGTU N.E.Bauman,(2006).

\bibitem[Wa]{Was82} Washington,L.,
{\it Introduction to Cyclotomic Fields},
Springer (1982).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[YS]{YS}
{\sc Yoshida, H.}, {\em Review on Goro Shimura, Arithmeticity in the
	theory of automorphic forms \cite{Shi00}}, 
Bulletin (New Series) of
the AMS, vol. 39, N3 (2002), 441-448.


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\end{thebibliography}

\end{document}
