
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
 %\documentclass[accumulate, fancybox]{beamer}
%\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}
}
%\pagestyle{plain}
%\markboth{left head}{right head}
%%%%%%%%%%%%%%%%%%%%%%%%%
%\ifslide
%{
%\setbeamertemplate{footline}
%{%
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%{
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%\uncover<+(1)->{
%\hskip5pt
%\fbox{
%\insertframenumber}
%{}
%\end{beamercolorbox}%
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%%%%%%%%%%%%%%%%%%%%%%
%\mode<article>{\usepackage{fullpage}}
%\usepackage{pdfpages}
%  \usetheme{}
%  \usetheme{Montpellier}
%  \usetheme{Antibes}
  % oder ...
  
%  \setbeamercovered{transparent}
  % oder auch nicht

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

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


\newcommand{\BS}{\begin{frame}}
\newcommand{\ES}{\pause\end{frame}}
\newcommand{\EB}{
\end{frame}
\begin{frame}
\frametitle{}
}
\newcommand{\FT}{\frametitle{}}
%\usepackage[russian]{babel}
%\usepackage[francais]{babel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{tpc-e.tex}
\newtheorem{theo}{\sf Theorem}%[section]



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

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







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


%\bibliographystyle{amsplain}

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

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

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

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

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


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


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





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


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

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

\newcommand{\Ar}{{\mathcal A}}
\newcommand{\Ac}{{\mathcal A}}
\newcommand{\Br}{{\mathcal B}}
\newcommand{\Cr}{{\mathcal C}}
\newcommand{\Cc}{{\mathcal C}}
\newcommand{\Ci}{{\mathcal C}^{\infty}}
\newcommand{\Dr}{{\mathcal D}}
\newcommand{\Hr}{{\mathcal H}}
\newcommand{\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{\Pc}{{\mathcal P}}
\newcommand{\Vr}{{\mathcal V}}
\newcommand{\Xr}{{\mathcal X}}
\newcommand{\X}{{\mathcal X}}





%\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}



%%%%%%%%%%%%%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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcounter{ncours}{\setcounter{ncours} {1}}
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\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}}
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\def\tcc{{\cal T}}           \def\tb{{\Bbb T}}            \def\ttf{{\bf T}}
\def\uc{{\cal U}}           \def\ub{{\Bbb U}}            \def\uuf{{\bf U}}
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\def\xc{{\cal X}}           \def\xb{{\Bbb X}}            \def\xxf{{\bf X}}
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\def\zc{{\cal Z}}           \def\zb{{\Bbb Z}}            \def\zzf{{\bf Z}}



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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ùù
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\newfam\Rus
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          depth 1.5pt\hskip 1pt}}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% TITRE
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[Analytic constructions and Eisenstein series 
]
{
Analytic constructions of
$p$-adic $L$-functions and Eisenstein series
%Doubling method and 
%Modular distributions% and their recent applications
}


\author[Alexei PANCHISHKIN]
{Alexei PANCHISHKIN\\ \small
Institut Fourier, 
  Université Grenoble-1\\
B.P.74, 38402 St.--Martin d'H\`eres, FRANCE}

\institute[Grenoble]{ 
\large 
{\magenta Conference "Zeta functions-4"}
%Explicit theory of automorphic forms, applications and computations}
%XPLICIT THEORY OF AUTOMORPHIC FORMS, APPLICATIONS AND COMPUTATIONS
% Journée « Arithmétique et ses applications »  \\ \red \large
% en l'honneur de Roland Gillard
%Séminaire de théorie des nombres de l'Institut Fourier
}



%\hskip-0.5cm
\date[
November 19 - 23, 2012] % (optional)
{
November 19 - 23, 2012}
%15 octobre 2008}
%June 18-24, 2007 
%\small\it\blue présentation est disponible à l'adresse :      
% {\tt http://www-fourier.ujf-grenoble.fr/\~{}panchish/luminySp3.pdf}




%\pgfdeclareimage[height=0.5cm]{logo-ujf}{UJF}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  INTRODUCTION
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}[shrink]
\frametitle{New ways of constructing 
complex and $p$-adic 
$L$-functions}

\

In March 2012, 
Marc Hindry,
Philippe Lebacque,
Michael Tsfasman,
and Alexey Zykin
 invited me to this fourth conference "Zeta Functions".
 
\

{\magenta Many thanks and gratitude to the organizers  for this occasion 
both to contribute again to the continuity of the interest to the theory of zeta functions, and to indicate productive directions to take research
from here.}



I would be interested in  new developpments in the theory of complex and $p$-adic zeta functions, especially related to the areas of autmorphic forms and geometry. 


\

At present stage of science and education  it is important to find possibilities to support  our exchange in the study and in further developpment  of zeta functions.


\


{\red It would be very useful to continue our efforts in this direction.
On this occasion I wish to point out that
 in July 1-5, 2013 in Grenoble,  a Conference
"Journées Arithmétiques" will be held, 
in which the theme of zeta functions will be widely presented.}

\

Welcome to Grenoble in July 2013!    

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2
 \iffalse
\begin{frame}[shrink]
\frametitle{Work of Ilya Piatetski-Shapiro and new ways \\ of constructing 
complex and $p$-adic 
$L$-functions}


In June 2011 Roger Howe invited me to 
this conference
devoted to assessing the work of Ilya Piatetski-Shapiro, especially
in the areas of autmorphic forms and geometry. 

\

{\magenta Many thanks to the organizers for this occasion 
both to review the accomplishments of Ilya Piatetski-Shapiro and his
colleagues, 
and to point to productive directions to take research
from here.}

\


I new Ilya since 1973-74 during our joint participation
 in the seminar of Manin and Kirillov on p-adic L-functions,
and attending his informal lectures on GL(3) in Moscow University in
April-May 1975.

\

After many years we met again  in Jerusalem in February 1998 during the conference "p-Adic Aspects of the Theory of Automorphic Representations". 

\

Ilya liked my construction of $p$-adic standard $L$-functions of Siegel modular forms \cite{Pa91}, and suggested to extend it to spinor $L$-functions, using the restriction of an Eisenstein series
 to the Bessel subgroup in the generalized Whittaker models (see Olga Taussky Todd memorial volume \cite{PS3}).
So we started a joint work "On $p$-adic $L$-functions for $GSp(4)$".

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3
\begin{frame}[shrink]
\frametitle{
In 1998,  a conference for Ilya Piatetski-Shapiro} was organized in the Fourier Institute
(Grenoble, France),
with participation of A.Andrianov, G.Henniart, H.Hida, J.-P.Labesse,
J.-L.Waldspurger and others.

\

In IAS, we had the most intensive period of our joint work in the Fall 
1999-2000.

My last meeting Ilya was  on January 31,
2009 
at
the  the Weizmann Institute home of 
Volodya  Berkovich and his wife Lena, who is
Grisha Freiman's daughter.
Vera brought Ilya,
Edith, and Edith's mother Ida to Berkovich's.
This
very pleasant gathering also included Grisha Freiman, his wife Nina, 
Steve and Mary Gelbart, Antoine Ducros, and my wife Marina, 
see also \cite{CGS}, p. 1268.  


\

That was the last party in Ilya's life.


\


His favorite automorphic forms were the {\magenta Eisenstein series}, 
%and there exist several ways to construct them via group theory, lattice theory,
% Galois representations, spectral theory...
and the main subject of this talk will be my new  construction 
of {\magenta $p$-adic meromorphic families of Siegel-Eisenstein series
in relation to the geometry of homogeneuos spaces}, both complex and $p$-adic, for any prime $p$.

I am glad that this construction fits into the particular subject
"Automorphic Forms and Related Geometry" of this Conference. 


\

\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4
\begin{frame}[shrink]
\frametitle{ $p$-adic Siegel-Eisenstein series and related geometry}
Let us consider the {\magenta symplectic group $\Gamma=\Sp_{m}(\Z)$} (of $(2m\times 2m)$-matrices), and prove that
the {\magenta Fourier coefficients $a_h(k)$ of the original Siegel-Eisenstein
series $E_k^m$} admit an explicit $p$-adic meromorphic interpolation on $k$
where $h$ runs through all positive definite half integral matricies
for $\det(2h)$ not divisible by $p$, where
\begin{align*}
E_k^{m}(z)&=\sum_{(c,d)/\sim}\det(cz+d)^{-k}=\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
\\ &=\sum_{h\in B_m}a_h\exp(\tr (hz))
\end{align*}
on the Siegel upper half plane  
${\mathfrak H}_m=\{z={}^t z\in M_m(\C) | \Im (z)>0\}$ of degree $m$,  $(c,d)$ runs over equivalence classes of all coprime 
symmetric couples, $\gamma= \begin{pmatrix} a&b\\c&d\end{pmatrix}$ runs over equivalence classes
of $\Gamma$ 
modulo the Siegel parabolic $P=\begin{pmatrix} *&*\\0&* \end{pmatrix}$.

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5
\begin{frame}[shrink]
\frametitle{ $p$-adic Siegel-Eisenstein series and related geometry}


{\magenta The homogeneuos  space $X=\{(c,d)/\sim\}=P\backslash \Sp_{m} $ and its $p$-adic points 
admit Siegel's  coordinates} 
$$
\nu=\det (c)\mbox{ and }
{\mathfrak R}
=c^{-1}d
$$ %as coordinates 
defined on the main affine subset  given by $\det (c)\in \mathrm{GL}_1$,
which is  used in the construction.

\

I try also to present  various {\magenta applications:  to  $p$-adic $L$-functions, 
to Siegel's Mass Formula, to $p$-adic analytic families of automorphic representations}.



\

%My idea was to prepare a general talk on 
%This talk is about on our study of  
%Complex and $p$-adic
%$L$-functions. %(Euler products and $p$-adic integration).
%, including a
%$p$-adic version of the doubling method and modular distributions


 {\magenta Eisenstein series} are basic automorphic forms,  
and there exist several ways to construct them via group theory, lattice theory, Galois representations, ...

%My geometic construction uses the symmetric spaces, both complex and $p$-adic.

\

For me, 
the Eisenstein series is the main tool of analytic constructions of
 complex and $p$-adic $L$-functions, in particular via the {\magenta doubling method}, 
see \cite{PSR}, \cite{GRPS}, \cite{Boe85}, \cite{Shi95}, \cite{Boe-Sch},\dots,   
greatly thanks to  Ilya Piatetski-Shapiro and his collaborators.
%{\large \magenta Contents}:

 
%\tableofcontents

%I Complex and $p$-adic $L$-functions \\
%II $p$-adic meromorphic continuation of Siegel-Eisenstein series\\
%III Application to Siegel's weight formula\\
%IV Link to Shahidi's method in the case of 
% GL(2)\\
%V Further applications to Ikeda-Miyawaki constructions\\

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6
\begin{frame}[shrink=1]
\frametitle{General strategy
 }
 For any Dirichlet character $\chi\bmod p^v$ 
 consider Shimura's "involuted" Siegel-Eisenstein series
 assuming their absolute convergence (i.e. $k> m+1$):
 $$
E^*_k(\chi,z)=\sum_{(c,d)/\sim}\chi(\det(c))\det(cz+d)^{-k}
%\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
=\sum_{0<h\in B_m}a_h(k, \chi)q^h.
$$
The two sides of the equality produce {\magenta dual approaches: geometric and algebraic}.
The Fourier coefficients can be computed by Siegel's method (see \cite{St81}
\cite{Shi95}, \dots )
via the singular series
\begin{align}\label{ahk}
&a_h({E}^*_k(\chi,z))\\ &\nonumber = 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
(-2\pi i)^{mk}\over 2^{m(m-1)\over 2}\Gamma_m(k)} 
%\\ &
%\times
\sum_{{\mathfrak R}\bmod 1}\chi(\nu({\mathfrak R}))\nu({\mathfrak R})^{-k}
\det h^{k-\frac{m+1} 2}e_m(h{\mathfrak R})
\end{align}
The {\magenta orthogonality relations $\bmod p^v$ produce two families of distributions}
(notice that terms in the RHS are invariant under sign changes, and (\ref{ah0}) is algebraic after multiplying by the factor in (\ref{ahk})):
\hskip-1.5cm
%\small
\begin{align}&\label{st}
{1\over \varphi(p^v)} \sum_{\chi\bmod p^v} \bar\chi (b)
 \sum_{(c,d)/\sim}{\chi(\det(c))\over \det(cz+d)^{k}}=
 \sum_{(c,d)/\sim\atop 
\det(c)\equiv b \bmod p^v}{\sgn(\det(c))^k\over \det(cz+d)^{k}} 
\end{align}
\hskip-1.3cm
\begin{align}
 \label{ah0} & 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{1\over \varphi(p^v)} \sum_{\chi\bmod p^v} \bar\chi (b)
\sum_{{\mathfrak R}\bmod 1}{\chi(\nu({\mathfrak R}))e_m(h{\mathfrak R})\over \nu({\mathfrak R})^{k}}
=
\sum_{{\mathfrak R}\bmod 1\atop 
\nu({\mathfrak R})\equiv b \bmod p^v}{e_m(h{\mathfrak R})\sgn\nu({\mathfrak R})^k\over \nu({\mathfrak R})^{k}}
\end{align}
\end{frame}
%\section{Introduction}
%I
%\section{Complex and $p$-adic $L$-functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7
\begin{frame}[shrink]
\frametitle{The use of Iwasawa theory and  pseudomeasures}

%Integrating the characters $y\mapsto y^k$ 
We express the integrals of Dirichlet characters $\theta \bmod p$
along the distributions (\ref{ah0})
through the reciprocal of a product of $L$-functions, and  elementary integral factors.
The result turns out to be an {\magenta Iwasawa function of the variable $t=(1+p)^k-1$ divided by a
distinguished polynomial} provided that $\det h$ is not divisible by $p$. 
 


Thus the second family (\ref{ah0})
comes from a  unique pseudomeasure $\mu_h^*$  which becomes a measure after multiplication by an explicit polynomial factor (in the sense of the convolution product).

\

%It follows that 
Then we deduce that
(\ref{st}) determines a unique pseudomeasure with coefficients in 
${\Q}[\![q^{B_m}]\!]$ whose moments are given by those of the coefficients (\ref{ah0})
(after removing from the Fourier expansion 
$f(z)=\sum_{h\ge 0}a_he_m(hz)$
all $h$ with $\det h$ divisible by $p$):
$$
\sum_{h>0, p\nds \det h}a_hq^h
=
p^{-m(m+1)/2}
\sum_{h_0\bmod p\atop p\nds \det h_0}\sum_{x\in S\bmod p}e_m(-h_0x/p)f(z+(x/p)).
$$
\

In this way a $p$-adic family of Siegel-Eisenstein series is geometrically produced.



\

{\blue  \large The text of this talk is closely related to that
in http://hal.archives-ouvertes.fr/hal-00688525
arXiv: 1204.3878}

\

{\large \magenta Contents}:

\
 
\tableofcontents

%I Complex and $p$-adic $L$-functions \\
%II $p$-adic meromorphic continuation of Siegel-Eisenstein series\\
%III Application to Siegel's weight formula\\
%IV Link to Shahidi's method in the case of 
% GL(2)\\
%V Further applications to Ikeda-Miyawaki constructions\\
\end{frame}
%\section{Introduction}
%I
\section{Complex and $p$-adic $L$-functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8
\begin{frame}%[shrink]
\frametitle{Generalities about $p$-adic $L$-functions}
There exist two kinds of $L$-functions
\begin{itemize}

\item Complex-analytic $L$-functions (Euler products) %$L(s, \pi, r)$
\item $p$-adic $L$-functions (Mellin transforms $L_\mu $ of $p$-adic measures) 

\end{itemize}
Both are used in order to obtain a number ($L$-value) from an automorphic form.
Usually such a number is algebraic (after normalization) via the embeddings 
$$
\overline \Q\hookrightarrow\C, \ \ \overline \Q\hookrightarrow\C_p=\widehat{\overline\Q}_p.
$$
{\magenta How to define and to compute $p$-adic $L$-functions?} 
%\pause
We use Mellin transform of a $\Z_p$-valued distribution $\mu$ on a profinite group
$$
Y=%\mathop
\lim_{\buildrel\leftarrow\over i} Y_i, \ \mu\in {\it Distr}(Y, \Z_p)=\Z_p[[Y]] =
%\mathop
\lim_{\buildrel\leftarrow\over i}\Z_p [Y_i]=:\Lambda_Y 
$$
(the {\magenta Iwasawa algebra} of  $Y$).
$$
L_\mu (x)=\int_Yx(y)d \mu, \ \ x\in X_Y=\Hom_{cont}(Y, \C_p^*) 
$$
(the {\magenta Mellin transform } of  $\mu$ on $Y$).

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9
\begin{frame}[shrink]
\frametitle{Examples of p-adic measures and $L$-functions} 
\begin{itemize}
\item $Y=\Z_p$, $X_Y=\{\chi_t: y\mapsto (1+t)^y\}$. 
The Mellin transform  $L_\mu(\chi_t)=\int_{\Z_p} (1+t)^y d\mu (y)$ of any  measure $\mu$
%on the {\magenta additive group $\Z_p$} 
on $\Z_p$ is  given by {\magenta the Amice transform},
which is the following power series
$$ 
A_\mu(t)=\sum_{n\ge 0}t^n\int_{\Z_p} {y\choose n} d\mu (y)=\int_{\Z_p} (1+t)^y d\mu (y), 
$$
e.g. $A_{\delta_m}=(1+t)^m$. Thus,
${\it Distr}(\Z_p, \Z_p)\cong\Z_p[\![T]\!]$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
%For the {\magenta multiplicative group  
$
Y=\Z_p^*=\Delta \times \Gamma= \{y=\delta (1+p)^z, \delta^{p-1}=1, z\in\Z_p\}
$ \\
$X_Y=\{\theta \chi_{(t)}\ |\ \theta \bmod p, \chi_{(t)}\}$,  
where $\chi_{(t)}((1+p)^z)=(1+t)^z$, 
$\Delta$ is the  subgroup of  roots of unity,
$\Gamma=1+p\Z_p$.\\
The {\magenta $p$-adic Mellin transform}
$L_\mu(\theta \chi_{(t)})=\int_{\Z_p^*}\theta(\delta)(1+t)^z \mu(y)$
%%%%%%%%%%%%%%%%%%%%%% 
 of a measure $\mu$ on  $\Z_p^*$ 
 is given by the collection of Iwasawa series 
$\displaystyle
G_{\theta, \mu} (t)=  
\sum_{n\ge 0} a_{n,\theta} t^n$, where $\displaystyle (1+t)^z=\sum_{n\ge 0}{z\choose n}t^n$,
\\
$\displaystyle
 a_{n,\theta}=\sum_{\delta \bmod p,\  n\ge 0}\theta (\delta)t^n\cdot
\int_{\Z_p} {z\choose n} \mu (\delta (1+p)^z) .
$



%We adopt the point of view on $p$-adic  $L$-functions as non-archimedean Mellin transforms of  %certains
%distributions attached to Euler products.


%\pause

%The {\magenta method of $p$-stabilization} of modular   distributions allows to construct in %numerous cases  
\item
A general idea is to construct
$p$-adic   $L$-functions  {\magenta directly from  Fourier coefficients} of modular forms
 (or from the Whittaker functions of automorphic forms).

\end{itemize}

\

\end{frame}





%II 
\section{$p$-adic meromorphic continuation of the Siegel-Eisenstein series}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 10
\begin{frame}[shrink]
\frametitle{Mazur's $p$-adic integral %(degree=1)
}
For any choice of a natural number $c\ge 1$ not divisible by $p$, there exists a $p$-adic measure 
$\mu_c$ on $\Z_p^*$, such that the special values
$$
\zeta(1-k)(1-p^{k-1})=%(1-\chi(c)c^{k})^{-1}
{\int_{\Z_p^*}y^{k-1}d\mu_c \over 1-c^{k}}
\in \Q, (k \ge 2\mbox{ even })
$$
produce the {\magenta Kubota-Leopoldt $p$-adic zeta-function} 
$
\zeta_p: X_p \to \cb_p
$ 
(where $X_p=X_{\Z^*_p}=Hom_{cont}(\Z_p^*, \cb_p^*)$) 
as the  {\magenta $p$-adic Mellin transform} 
%of  $\mu_c$: for any integer $c>1$ not divisible by $p$, 
$$
\zeta_p(x)=%(1-cx(c))^{-1}
{\int_{\Z_p^*}x(y)d\mu_c (y)\over 1-cx(c)}= {L_{\mu_c}(x)\over 1-cx(c)}, 
$$
with a single simple pole at 
$x=x_p^{-1}\in X_p$, 
where $\cb_p=\hat{\overline{\qb}}_p$ the Tate field, the completion of an algebraic closure of the  $p$-adic field
 ${\qb}_p$,  $x\in X_p$ (a $\cb_p$-analytic Lie group), 
$x_p(y)=y\in X_p$, and $x(y)=\chi(y)y^{k-1}$ as above.

{\large \magenta Explicitly}: Mazur's measure is given by $\mu_c(a+p^v\Z_p)$\\
$=
\frac {1}c\left[ \frac {ca}{p^v}\right] + 
\frac {1-c}{2c}=\frac{1}c B_1(\{\frac {ca}{p^v}\})-B_1(\frac {a}{p^v})$,~$B_1(x)=x-\frac 1 2$,\\
see \cite{LangMF}, Ch.XIII.
%\fi

\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 11
\begin{frame}[shrink]
\frametitle{Meromorphic $p$-adic continuation  of ${1\over \zeta(1-k) (1-p^{k-1})}$} 
%for all primes $p$}
% $p$ even $k\ge 2$.}

 For any odd prime $p$ take the Iwasawa series $G_{\theta,c}(t)$ of Mazur's measure $\mu_c$ where $\theta$ is a character $\bmod p$, 
$\displaystyle G_{\theta,c}(t):=\int_{\Z_p^*}\theta(y)\chi_{(t)}(\langle y\rangle)\mu_c
=\sum_{n=0}^\infty a_nt^n\in \Z_p[\![t]\!], 
$~and %$|t|_p<1$,
\\ $\displaystyle
\chi_{(t)}:(1+p)^z\mapsto (1+t)^z,  \  \langle y\rangle=\frac y {\omega(y)}
, \omega$  the Teichmüller character.
Mazur's integral of the character  $y^{k-1}=\omega^{k-1}\cdot \chi_{(t)}$ shows that $\theta=\omega^{k-1}, (1+t)=(1+p)^{k-1}$ 
%G_{\theta,c}(t)=
\begin{align}\label{Gthc}
\zeta(1-k)(1-p^{k-1})
={G_{\theta,c}((1+p)^{k-1}-1)  \over 1-c^k}.
\end{align}
By the {\magenta Weierstrass preparation theorem}  we have a decomposition 
$$
G_{\theta,c}(t)=U_{\theta,c}(t) P_{\theta,c}(t)
$$
with a {\magenta distinguished polynomial} $P_{\theta,c}(t)$ and {\magenta invertible power series} $U_{\theta,c}(t)$. The inversion of (\ref{Gthc}) for any even $k\ge 2$ gives :
%whose inverse denoted by $U_{\theta,c}^*$ is in the Iwasawa algebra.

\iffalse
Then $\zeta_p(1-k)=G_{\theta,c}((1+p)^{k-1}-1)(1-c^k)^{-1}$
equals $U_{\theta,c}((1+p)^{k-1}-1)(1-c^k)^{-1}$ times
the polynomial factor $P_{\theta,c}((1+p)^{k-1}-1)$, 
where $\theta=\omega^{k-1}$, and $\omega$ denotes the Teichmüller character.

The function $\zeta_p(1-k)$ has a {\magenta  simple pole  at $k=0$}  if $\omega^{k}$ is trivial.
Now it is easy to invert:
\fi
\begin{align*}
{1\over \zeta(1-k)(1-p^{k-1})}
=G_{\theta,c}((1+p)^{k-1}-1)^{-1}  (1-c^k).
\end{align*}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 12
\begin{frame}[shrink]
\frametitle{The answer: for any prime $p>2$ and  even $k\ge 2$}  
is the following Iwasawa function on $t=t_k=(1+p)^{k}-1$ divided by a distingushed polynomial:
\begin{align}\label{zki}
\frac 1 {\zeta(1-k)(1-p^{k-1})}&=\frac {U^*_{\theta,c}((1+p)^{k-1}-1) (1-c^k)}
{P_{\theta,c}((1+p)^{k-1}-1)}\\ & \nonumber
=\frac {U^*_{\theta,c}((1+t_k)(1+p)^{-1}-1) (1-c^k)}
{P_{\theta,c}((1+t_k)(1+p)^{-1}-1)}
\end{align}
which is meromorphic in the unit disc of the variable $t=(1+p)^k-1$
with a {\magenta finite number of poles} (expressed via roots of $P_{\theta,c}$) for $\theta=\omega^{k-1}$,
and
$$
U^*_{\theta,c}((1+p)^{k-1}-1):=1/U_{\theta,c}((1+p)^{k-1}-1).
$$



The above formula immediately extends to all Dirichlet $L$-functions
of characters $\chi \bmod p^v$ as the following Iwasawa function  divided by a polynomial:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\frac 1 {L(1-k, \chi)(1-\chi(p)p^{k-1})}
= \frac {U^*_{\theta,c}(\chi(1+p)(1+p)^{k-1}-1) (1-\chi(c)c^k)}
{P_{\theta,c}(\chi(1+p) (1+p)^{k-1}-1)}
$$
where 
$
U^*_{\theta,c}(\chi(1+p)(1+p)^{k-1}-1):={\displaystyle 1\over \displaystyle U_{\theta,c}(\chi(1+p)(1+p)^{k-1}-1)}
$ 

\

\end{frame}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  13
\begin{frame}[shrink]
\frametitle{%Meromorphic 
$p$-adic interpolation of $L(1-k, \chi)^{-1}(1-\chi(p) p^{k-1})^{-1}$.}
% for even $k\ge 2$.}
Thus we obtain a
meromorphic function 
$$
%\frac 1 {L(1-k, \chi)(1-\chi(p)^{k-1})}
%= 
\frac {U^*_{\theta,c}(\frac {\chi(1+p)}{1+p}(1+t)-1) (1-\chi(c)c^k)}
{P_{\theta,c}(\frac {\chi(1+p)}{1+p}(1+t)-1)}
$$
in the unit disc which is
an Iwasawa function on $t=(1+p)^{k}-1$ divided by a distingushed polynomial
with a finite number of poles (expressed via roots of $P_{\theta,c}$) for $\theta=\omega^{k-1} \chi(\omega)$.

\

Notice also the following expression for the values of the $p$-adic $L$-function $L_p(s,\chi)$ : \\ 
$
L_p(1-k, \chi)=-\frac 1 n (1-\chi\omega^{-k}(p)p^{k-1})B_{k,\chi\omega^{-k}}
$ (see \cite{Wa82}, p.58),\\
implying for all $k\ge 1$ with $\chi(-1)= (-1)^k$, 
$$
L(1-k, \chi)^{-1}(1-\chi(p)^{k-1})^{-1}=L_p(1-k, \omega^{k}\chi)^{-1}.
$$

\end{frame}
\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13
\begin{frame}[shrink=8]
\frametitle{Illustration: numerical values of $\zeta(1-2k)^{-1}(1-p^{2k-1})^{-1}$ for $p=37$}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
gp >{\tt zetap1(p,n)= -2*n/(bernfrac(2*n)*(1-{p}\^{}{(2*n-1)}+O({p}\^{}5)));}\\
gp > p=37;\\
gp > for(k=1,({p}-1)/2, print(2*k,  zetap1(p,k)))\\
\begin{tabular}{l|l}\hline
$2k$& $\zeta(1-2k)^{-1}(1-p^{2k-1})^{-1}$ \\
 \hline
2       &$25 + 24*37 + 24*37^2 + 24*37^3 + 24*37^4 + O(37^5)$\\ \hline
4       &$9 + 3*37 + 9*37^3 + 3*37^4 + O(37^5)$\\ \hline
6       &$7 + 30*37 + 36*37^2 + 36*37^3 + 36*37^4 + O(37^5)$\\ \hline
8       &$18 + 6*37 + O(37^5)$\\ \hline
10       &$16 + 33*37 + 36*37^2 + 36*37^3 + 36*37^4 + O(37^5)$\\ \hline
12       &$8 + 25*37 + 28*37^2 + 23*37^3 + O(37^5)$\\ \hline
14       &$25 + 36*37 + 36*37^2 + 36*37^3 + 36*37^4 + O(37^5)$\\ \hline
16       &$6 + 16*37 + 31*37^2 + 29*37^3 + 20*37^4 + O(37^5)$\\ \hline
18       &$3 + 4*37 + 10*37^2 + 32*37^3 + 25*37^4 + O(37^5)$\\ \hline
20      &$11 + 13*37 + 19*37^2 + 36*37^3 + 12*37^4 + O(37^5)$\\ \hline
22      &$1 + 26*37 + 15*37^2 + 35*37^3 + 9*37^4 + O(37^5)$\\ \hline
24      &$16 + 28*37 + 24*37^2 + 27*37^3 + 31*37^4 + O(37^5)$\\ \hline
26      &$4 + 17*37 + 25*37^2 + 25*37^3 + 19*37^4 + O(37^5)$\\ \hline
28      &$22 + 36*37 + 8*37^2 + 4*37^3 + 33*37^4 + O(37^5)$\\ \hline
30      &$22 + 5*37 + 35*37^2 + 9*37^3 + 5*37^4 + O(37^5)$\\ \hline
\red 32      &$36*37^{-1} + 28 + 3*37 + 19*37^2 + 18*37^3 + O(37^4)$\\ \hline
34      &$20 + 37 + 30*37^2 + 15*37^3 + 22*37^4 + O(37^5)$\\ \hline
\green 36      &$36*37 + 29*37^2 + 35*37^3 + 5*37^4 + 37^5 + O(37^6)$\\ 
\hline
\end{tabular}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\

\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{frame}
%\end{document}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 15
\begin{frame}[shrink]
\frametitle{%Meromorphic 
$p$-adic interpolation of the Eisenstein series 
(elliptic case)}

The above formula %obviously 
extends to all Dirichlet $L$-functions
of character $\chi \bmod p^v$.

It follows directly that all $p$-regular Fourier coefficients of the Eisenstein series with
such Dirichlet character $\chi$ that $\chi (-1)=(-1)^k, k>2$,
$$
%E_{k,\chi}
E_k(\chi,z)
=1+{2\over L(1-k, \chi)}\sum_{n=1}^\infty \sum_{d|n}\chi(d)d^{k-1}q^n
$$
admit a $p$-adic  meromorphic continuation
as follows:
$$
1+{2 U_{\theta_k, c}^*(\chi(1+p)(1+p)^{k-1}-1)(1-\chi(c)c^k) \over P_{\theta_k, c}(\chi(1+p)(1+p)^{k-1})((1-\chi(p)p^{k-1})}
\sum_{n=1}^\infty \sum_{d|n}\chi(d)d^{k-1}q^n
$$
(for all $n$ not divisible by p) where $\theta_k=\omega^{k-1} \chi(\omega)$, $\chi(\omega)$ is the 
%tame part of $\chi$, i.e. its 
restriction on $\Delta\subset \Z_p^*$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Notice that for any $d\in \Z_p^*$, the function of $t=(1+p)^k-1$
\begin{align}\label{ak}
&k\mapsto d^k  =  \omega(d)^k\langle d\rangle^k=
%\omega(a)^k(((\langle a\rangle-1)+1))^k
%\\ \nonumber
% &=\omega(a)^k\sum_{n=0}^\infty{k\choose n}(\langle a\rangle-1)^n=
 \omega(d)^k 
 (1+p)^{k{\log \langle d\rangle\over \log(1+p)}}
% \\ \nonumber
% &
% =\omega(d)^k 
% (((1+p)^k-1)+1)^{\log \langle d\rangle\over \log(1+p)}
% \\& \nonumber
% =\omega(a)^k\sum_{n=0}^\infty{\log \langle a\rangle \over \log(1+p)}\choose n}((1+p)^k-1)^n
 \\& \nonumber
 =\omega(d)^k 
 (1+t)^{{\log \langle d\rangle\over \log(1+p)}}
 =\omega(d)^k\sum_{n=0}^\infty{{\log \langle d\rangle \over \log(1+p)}\choose n}t^n  
%\ (\mbox{ where }(1+p)^k= 1+t)
\end{align} 
is a $p$-adic analytic Iwasawa function  denoted by $\tilde d(t)\in \Z_p[\![t]\!]$, of $t=(1+p)^k-1$ with $\omega^k$ fixed, where
${x\choose n}$=${x(x-1)\cdots (x-n+1)\over n!}$. 

\

\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 16
\begin{frame}[shrink]
\frametitle{%Meromorphic 
$p$-adic interpolation of the Eisenstein series at the cusp 0}
%(the elliptic case)}
%The above formula obviously extends to all Dirichlet $L$-funcions
%of character $\chi \bmod p^v$.
In order to make these formulae amenable for geometric constructions
let us
 consider the "involuted" Eisenstein series
 assuming their absolute convergence (i.e. $k> 2$):
 $$
E^*_k(\chi,z)=\sum_{(c,d)\atop \gcd (c,d)=1}\chi(c)(cz+d)^{-k}
%\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
=\sum_{h>0} a_h({E}^*_k(\chi,z)) q^h
$$
 then their Fourier coefficients can be computed by Hecke's method \cite{He27}, \cite{Miy89}, 
 Theorem 7.1.3, but also using 
 directly Siegel's method (see \cite{Si39}, \cite{St81}
\cite{Shi95}, \dots )
via the singular series
\begin{align}\label{ahk2}
a_h({E}^*_k(\chi,z))%\\ &\nonumber = 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
={
(-2\pi i)^{k}\over  2\Gamma(k)} 
%\\ &
%\times
\sum_{{\mathfrak R}\bmod 1}\chi(\nu({\mathfrak R}))\nu({\mathfrak R})^{-k}
h^{k-1}e(h{\mathfrak R})
\end{align}
The {\magenta orthogonality relations $\bmod p^v$ produce two families of distributions on
$\Z_p^*$}
(notice that terms in the RHS are invariant under sign changes, and (\ref{ah0}) is algebraic after multiplying by the factor in (\ref{ahk2})):
\small
\begin{align}&\label{st1}
\hskip-1.2cm
{1\over \varphi(p^v)} \sum_{\chi\bmod p^v} \bar\chi (b)
 \sum_{\large (c,d)\atop \gcd(c,d)=1} %{{ \hskip-1pt \vskip 1pt} *}
 {\chi(c)\over {}(cz+d)^{k}}=
 \sum_{ \gcd(c,d)=1\atop 
c\equiv b \bmod p^v}%{}^{*}
{\sgn({}c)^k\over (cz+d)^{k}} \\ \label{ah1} & 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hskip-1.3cm
{
 1\over \varphi(p^v)} \sum_{\chi\bmod p^v} \bar\chi (b)
\sum_{{\mathfrak R}\bmod 1}{\chi(\nu({\mathfrak R}))e(h{\mathfrak R})\over \nu({\mathfrak R})^{k}}
=
\sum_{{\mathfrak R}\bmod 1\atop 
\nu({\mathfrak R})\equiv b \bmod p^v}{e(h{\mathfrak R})\sgn\nu({\mathfrak R})^k\over \nu({\mathfrak R})^{k}}
\end{align}
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 17
\begin{frame}[shrink]
\frametitle{
$p$-adic interpolation of the Eisenstein series\\ at the cusp 0 (continued)}
Let us use
formulas for the Fourier coefficients, see \cite{Miy89}, Theorem 7.1.3,
  %\cite{Ka76},   
\cite{PaIAS}, p. 2367
%for $\det(2h)\not = 0$}  
%\end{document}
\begin{align*}
a_h({E}_k^*(\chi,z))&= 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
(-2\pi i)^{k}
\over 
\Gamma(k)
L(k, \bar \chi)%\prod_{i=1}^{[m/2]}\zeta(2k-2i)
}
\sum_{d|h} \chi(d) 
(h/d)^{k-1}=
\end{align*}
Suppose that $p\nd h$ and $\chi=\chi_0$
is the trivial character $\bmod p$, then
\begin{align*}&
a_h({E}_k^*(\chi_0,z))= 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
(-2\pi i)^{k}
\over 
\Gamma(k)L(k, \chi_0)
}
\sum_{p\nd d|h} %\chi(d) 
(h/d)^{k-1}
\\ &\nonumber
={
(-2\pi i)^{k}
\over 
\Gamma(k)
\zeta (k)(1-p^{-k})
}
\sum_{d|h} %\chi(d) 
(h/d)^{k-1}
\\ &\nonumber
={(1-p^{k-1})
\over 
(1-p^{-k})
}\cdot
{1
\over 
\zeta (1-k)(1-p^{k-1})
}
\sum_{d|h} %\chi(d) 
(h/d)^{k-1}
\end{align*}
Therefore, for all $h\ge 1$ not divisible by $p$, the normalised Fourier coefficients
\begin{align}&
{(1-p^{-k})
\over 
(1-p^{k-1})
}\cdot a_h({E}_k^*(\chi_0,z))= 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\\ &\nonumber
{1
\over 
\zeta (1-k)(1-p^{k-1})
}
\sum_{d|h} %\chi(d) 
d^{k-1}
\end{align}
are given by the value at $t=(1+p)^k-1$ of a $p$-adic meromorphic function (\ref{zki})
times an elementary Iwasawa function (the sum over divisors of $h$).

\

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 18
\begin{frame}%[shrink]
\frametitle{The Siegel-Eisenstein series}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us consider the symplectic group $\Sp_{m}(\Z)$, and prove that
all the Fourier coefficients $a_h(k)$ of the original Siegel-Eisenstein
series $E_k^m$ admit an explicit $p$-adic meromorphic interpolation on $k$
where $h$ runs through all positive definite half integral matricies
with ${}(2h)$ not divisible by $p$.

More precisely, consider for any $k > m+1$ the absolutely convergent holomorphic 
Siegel-Eisenstein series defined by 
$$
E_k^{m}(z)=\sum_{(c,d)/\sim}\det(cz+d)^{-k}=\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
%=\sum_{T\in B_m}a_Tq^T
$$
on the Siegel upper half plane of degree $m$ 
$\Hb_m=\{z\in M_m(\C) | \Im (z)>0\}$, where $(c,d)$ runs over equivalence classes of all coprime 
symmetric couples, $\gamma= \begin{pmatrix} a&b\\c&d\end{pmatrix}$ runs over equivalence classes
of $\Gamma={\mathrm Sp}_m$ modulo the Siegel parabolic $P=\begin{pmatrix} *&*\\0&* \end{pmatrix}$.

\end{frame}
\fi
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14
\begin{frame}[shrink]
\frametitle{Fourier expansion of the Siegel-Eisenstein series}
has the form
$$
E_k^{m}(z)=\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
=\sum_{h\in B_m}a_hq^h, %\mbox{ with }
% a_T=a_T(k)=a_T(E_k^{n})
$$
where $a_h=a_h(k)=a_h(E_k^{m})$, $q^h=e^{2\pi i\tr (hz)}$, $h$ runs over semi-definite half integral $m\times m$ matrices.
%$a_T=a_T(k)=a_T(E_k^{n})$.

\

The {\magenta rationality of the coefficients $a_h$} 
was established in Siegel's pioneer work \cite{Si35}  in connection with a study of local densities for quadratic forms. 
Siegel expressed
$a_h(k)$ as a product of local factors over all primes and $\infty$. 

In a difficult later work \cite{Si64b} Siegel proved the {\magenta boundedness} of their denominators, 
and S.Boecherer \cite{Boe84} gave a simplified proof of a more precise result in 1984.
M.Harris extended the rationality to  
wide classes of Eisenstein series on Shimura varieties
\cite{Ha81}, \cite{Ha84}. Their relation to the Iwasawa Main Conjecture
and $p$-adic $L$-functions on the unitary groups
was established in   \cite{HLiSk}.

\

\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  20
\begin{frame}[shrink=6]
\frametitle{Explicit $p$-adic continuation of $a_h(k)$}
as  {\magenta Iwasawa functions on $t=(1+p)^{k}-1$ divided by distinguished polynomials}.
Let $a_h^{(p)}(k)$ denote the {\magenta $p$-regular part of the coefficient $a_h(k)$} 
(i.e. with the Euler $p$-factor removed from the product). 
%of local densities). 
Namely, for any even $k$, $a_h^{(p)}(k)=a_h(E_k^{m})$ times 
$$
\begin{cases} 1/((1-p^{k-1})(1+\psi_h (p)p^{k-\frac m 2-1})\prod_{i=1}^{(m/2)-1}(1-p^{2k-2i-1})) & \\
= (1-\psi_h (p)p^{k-\frac m 2-1})/((1-p^{k-1})\prod_{i=1}^{m/2}(1-p^{2k-2i-1})),& m \mbox{ even}
\\ 
1/((1-p^{k-1})\prod_{i=1}^{(m-1)/2}(1-p^{2k-2i-1})),& m \mbox{ odd},  
\end{cases}
$$ 
where the {\magenta $p$-correcting factor is a $p$-adic unit}, 
and $\displaystyle \psi_h (n):=\left({\det(2h)(-1)^{m/2} \over n}\right)$.

\begin{theorem}[\hskip-1pt\inserttheoremnumber) (A.P., 2012]\label{pMSE}
Let $h$ be any positive definite half integral matrix
with $\det(2h)$ not divisible by $p$.
Then there exist explicitly given distinguished polynomials $P^E_{\theta,h}(T)\in \Z_p[T]$
and Iwasawa series $S^E_{\theta,h}(T) \in \Z_p[\![T]\!]$
such that the $p$-regular part $a_h^{(p)}(k)$ of the Fourier coefficient $a_h(k)$ 
%of the original Siegel-Eisenstein
%series $E_k^n$ 
admit the following $p$-adic meromorphic interpolation on all even $k$ with $\theta=\omega^k$ fixed 
$$
a_h^{(p)}(k)=
{S^E_{\theta,h}((1+p)^{k-1}-1)\over P^E_{\theta,h}((1+p)^{k-1}-1)}
$$
with a finite number of poles expressed via the roots of  $P^E_{\theta,h}(T)$
where the denominator depends only on 
$\det(2h) \bmod 4p$ and $k   \bmod p-1$.
\end{theorem}



%$p$-adic version of  Shahidi's integral
%("The p-adic  Eisenstein measure and Shahidi-type $p$-adic integral for
%SL(2)", in preparation jointly with  S. Gelbart, \dots
%S. Miller,  F. Shahidi

%\item %{1)} 
%$p$-adic interpolation of the special  $L$-values  of
%for Rankin and Garrett (jointly with S.Böcherer, ...)
%\fi

%\item{} Proof of a $p$-adic Miyawaki Modularity Lifting 
%Conjecture by pullback of families  of modular  distributions with values in Siegel modular forms (jointly with
%Hisa-Aki Kawamura, \dots)
%%%%%%%%%%%%%%%%%%%%%%%%

%-Ikeda's constructions (2001, 2006) extend the doubling method to pullbacks of cusp forms instead of pullbacks of Eisenstein series. 
%Both p-adic and complex analytic L-functions are produced in this way.

%-Thus obtained cuspidal p-adic measures generalize the Eisenstein measure, and produce families of cusp forms

%-A version of this construction Klingen-Eisestein series  and Langlands Eisenstein series.



%\item %{1)} 
%$p$-adic interpolation of the special  $L$-values  
%and $p$-adic $L$-functions of two variables
%for
%${\rm GSp}(4)\times \GL(2)$ 
%(a joint project with S.Böcherer, B.Heim, ...)
%Liens avec les unités elliptiques et les points de Stark-Heegner (un
%nouveau groupe de travail avec G.Robert) 
%\fi

\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 21
\begin{frame}[shrink]
\frametitle{Computation of the Fourier coefficients}
% of the Siegel-Eisenstein series}
Recall that Siegel's computation of the coefficients $a_h=a_h(E_k^{m})$ : 
$$
E_k^{m}(z)=\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
=\sum_{h\in B_m}a_hq^h
$$
is based on the Poisson summation formula giving the equality
(see \cite{Maa71}, p.304): 
%2^{-(m(m-1)/2}, 
% [Sie39, p.652, [St, p.338]­
$$
\sum_{a\in S_m}\det(z+a)^{-k}={(-2\pi i)^{mk}\over 2^{m(m-1)\over 2}\Gamma_m(k)} \sum_{h\in C_m}\det(h)^{k-\frac {m+1} 2}
e^{2\pi i\tr (hz)},
$$
where $\Gamma_m(k)=\pi^{m(m-1)/4}\prod_{j=0}^{m-1}\Gamma(s-\frac j 2)$,
 $q^h=e^{2\pi i\tr (hz)}$, 
  $h$ runs over the set $C_m$ of positive definite half integral $m\times m$ symmetric matrices, and 
  $a$ runs over the set $S_m$ of integral $m\times m$ symmetric matrices,  see \cite{Si39}, p.652, \cite{St81}, p.338.
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 22
\begin{frame}[shrink]
\frametitle{Formulas for the Fourier coefficients for $\det(2h)\not = 0$}  
%\end{document}
\begin{align*}
a_h({E}_k^{m})&= 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
(-2\pi i)^{mk}\Gamma_m^{-1}(k)
\over 
\zeta(k)\prod_{i=1}^{[m/2]}\zeta(2k-2i)
}
%\end{align*}
%\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%
\\ & %2^{-\frac{m(m+1)}2}
\times \det(2h)^{k-\frac {m+1} 2}M_h(k)
\begin{cases} \displaystyle 
%2\Gamma(k-\frac m 2+\mu)
L(k-\frac m 2,\psi_h ),
%\over (-2\pi i)^{k-\frac m 2}}
 & m \mbox{ even}, \\ 
1,& m \mbox{ odd}.  
\end{cases}
\end{align*}
%\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{frame}
%\end{document} 
The {\magenta integral factor}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    (2.132)
 %   \begin{align*}  %\label{EQfactMhchi}
  $ \displaystyle   M_h(k) = \prod_{{\ell} \in P(h)} M_{\ell}(h,{\ell}^{-k})$ 
 %   \end{align*}
  is a {\magenta  finite Euler product}, extended over primes ${\ell}$ in the set $P(h)$ of
  prime divisors %of the number $N$ and 
  of all elementary divisors of the
  matrix $h$. The important property of the product is that for each ${\ell}$ we
  have that $M_{\ell}(h,t) \in \Z[t]$ is a polynomial with integral coefficients.
  
  Notice the {\magenta  $L$-factor $L(k-\frac m 2,\psi_h )$} depends on the  index $h$
of the Fourier coefficient;
this makes a difference to the case of odd $m$; the case of $GL(2)$ corresponds to $m=1$.
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 18
\begin{frame}[shrink]
\frametitle{Proof: \small the use of the normalized Siegel-Eisenstein series}
defined as in \cite{Ike01}, \cite{PaSE} and \cite{Pa91} by \\ %the equality
$$
{\mathcal E}_k^{m} =
{
{E}_k^{m}(z)
%(2\pi i)^{-mk}\Gamma_m^{}(k)
2^{m/2}
\zeta(1-k)\prod_{i=1}^{[m/2]}\zeta(1-2k+2i), 
}
$$
I show that it produces a nice $p$-adic family, namely:
%\end{frame}
%\end{document}
%\begin{cases} {\Gamma^{-1}(k-\frac m 2+\mu)\over(1-p^{k-1})\prod_{i=1}^{m/2}(1-p^{2k-2i-1})(1+\psi_T (p)p^{k-1})},& %m \mbox{ even}, \\ &\\ 
%{1\over(1-p^{k-1})\prod_{i=1}^{m/2}(1-p^{2k-2i-1})},& m \mbox{ odd},  
%\end{cases}
%$M_T(k)=\prod_{\ell | \det (2T)} M_{\ell}(\ell^k)$
\begin{prop}[\hskip -1pt\!\!\inserttheoremnumber]\label{NSES}
\begin{itemize}
%\item[(1)] 
%The following equality holds
%E_k^{n}(z)=
%{\mathcal E}_k^{n}(z){2^{m/2}\over 
%\zeta(1-k)\prod_{i=1}^{[m/2]}\zeta(1-2k+2i)}
%$

%\end{itemize}
%\end{prop}
%\end{frame}
%\end{document}
\item[(1)] 
For any non-degenerate matrice $h\in C_m$ the following equality holds
\begin{align}\label{ah}&
a_h({\mathcal E}_k^{m})=
2^{-\frac{m}2}\det h^{k-\frac {m+1} 2} M_h(k)
\\ & \nonumber
%2^{-\frac{m}2}\det T^{k-\frac {m+1} 2} M_T(k)
\times
\begin{cases} L(1-k + {\frac m 2}, \psi_h )C_h^{\frac m 2 - k + (1/2)},& m \mbox{ even}, \\ 
1,& m \mbox{ odd}, 
\end{cases} 
\end{align}
where $C_h$ is the conductor of $\psi_h$.
%\end{itemize}
%\end{prop}
%\end{frame}
%\end{document}
\item[(2)]  
for any prime $p>2$, and $\det(2h)$ not divisible by $p$, 
define the {\magenta $p$-regular part} $a_h({\mathcal E}_k^{m})^{(p)}$  of the coefficient 
$a_h({\mathcal E}_k^{{m}})$ of  ${\mathcal E}_k^{{m}}$  by introducing the factor
%2^{-\frac{m(m+1)}2}\det T^{k-\frac {m+1} {2}} M_T(k)
%\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{prop}
%\end{frame}
%\end{document}
%$2^{-\frac{m}2}\det h^{k-\frac {m+1} 2} M_h(k)$\\
$\begin{cases} 
%L(1-k+\frac m 2,\psi_h)
(1-\psi_h(p)p^{k-\frac m 2-1}),%{C_h^{\frac m 2 - k + (1/2)}}
& m \mbox{ even}, \\ 
1,& m \mbox{ odd.}  
\end{cases}
$

Then $a_h({\mathcal E}_k^{m})^{(p)}$ 
is a $p$-adic analytic Iwasawa function of $t=(1+p)^k-1$ for all $k$ with $\omega^k$ fixed,
 and divided by the elementary factor $1-\psi_h(c_h)c_h^{k-\frac m 2}$. 
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{prop}
%\end{frame}
%\end{document}
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19
\begin{frame}[shrink]
\frametitle{Proof of (1) of Proposition \ref{NSES}}
Proof of (1) is deduced like at p.653 of \cite{Ike01}
from the Gauss duplication formula 
$$\Gamma (\frac s 2 )\Ga ( \frac {s+1} 2)=2^{1-s}\sqrt{\pi}\Gamma (s),$$
the definition 
$$
\Gamma_m(k)=\pi^{m(m-1)/4}\prod_{j=0}^{m-1}\Gamma(s-\frac j 2)
$$
and the functional equations
\begin{align*}&
\zeta(1-k)={2(k-1)!\over (-2\pi i)^k} \zeta(k), \\ &
\zeta(1-2k+2i)={2 (2k-2i-1)!\over (-2\pi i)^{2k-2i}} \zeta(2k-2i),
%\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{frame}
%\end{document}
\\ & 
L(1-k+\frac m 2, \psi_h)
 ={2 (k-\frac m 2-1)!\over (-2\pi i)^{k-\frac m 2}} 
 L(k-\frac m 2, \psi_h) C_h^{k-\frac m 2-\frac 1 2} 
\end{align*}



\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20
\begin{frame}[shrink]
\frametitle{Proof of (2) of Proposition 2} is then deduced easily : 
%for all non-degenerate matrices $T\in C_m$
% the Fourier coefficients of the difference
%between the left and the right hand sides in (1) vanish. 
%Hence this difference vanish identically because it is
%a singular modular form of a big weight $k>m+1$.

Notice that for any $a\in \Z_p^*$, the function of $t=(1+p)^k-1$
\begin{align}\label{ak}
k\mapsto a^k  &=  \omega(a)^k\langle a\rangle^k=
%\omega(a)^k(((\langle a\rangle-1)+1))^k
%\\ \nonumber
% &=\omega(a)^k\sum_{n=0}^\infty{k\choose n}(\langle a\rangle-1)^n=
 \omega(a)^k 
 (1+p)^{k{\log \langle a\rangle\over \log(1+p)}}
 \\ \nonumber
 &
 =\omega(a)^k 
 (((1+p)^k-1)+1)^{\log \langle a\rangle\over \log(1+p)}
 \\& \nonumber
% =\omega(a)^k\sum_{n=0}^\infty{\log \langle a\rangle \over \log(1+p)}\choose n}((1+p)^k-1)^n
% \\& \nonumber
 =\omega(a)^k\sum_{n=0}^\infty{{\log \langle a\rangle \over \log(1+p)}\choose n}t^n  
\end{align} 
is a $p$-adic analytic Iwasawa function  denoted by $\tilde a(t)\in \Z_p[\![t]\!]$, of $t=(1+p)^k-1$ with $\omega^k$ fixed, where
${x\choose n}$=${x(x-1)\cdots (x-n+1)\over n!}$. 

Then
Mazur's formula applied to $L(1-k+\frac m 2,\psi_h)(1-\psi_h(p)p^{k-\frac m 2-1})$ shows that this function  is a $p$-adic analytic Iwasawa function of $t=(1+p)^k-1$ with $\omega^k$ fixed
(a single simple pole may occur at $k=\frac m 2$ only if $\omega^{k-\frac m 2}$ is trivial).   


\


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 21
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\frametitle{Proof of Main Theorem}
Let us use the equality
$$
{E}_k^{m} =
{
\mathcal E}_k^{m}(z)\cdot
%(2\pi i)^{-mk}\Gamma_m^{}(k)
{2^{-m/2}\over
\zeta(1-k)\prod_{i=1}^{[m/2]}\zeta(1-2k+2i) 
}
$$
and the properties of the normalized series
${\mathcal E}_k^{n}(z)$ in Proposition \ref{NSES}.

First let us compute the reciprocal 
of the product of $L$-functions
$$
\zeta(1-k)\prod_{i=1}^{[m/2]}\zeta(1-2k+2i)
$$ 
using the above:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for even $k\ge 2$,  
\begin{align}\label{zk}&\zeta(1-k)^{-1}(1-p^{k-1})^{-1}  
={
U^*_{\theta_{k},c}((1+p)^{k-1}-1) (1-c^k)
\over P_{\theta_k,c}((1+p)^{k-1}-1)}\\ & \nonumber
\zeta(1-2k+2i)^{-1}(1-p^{2k-2i-1})^{-1}\\ &  \label{z2k}
={
U^*_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1) (1-c^{2k-2i})
\over P_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1)}
\end{align} 
which is meromorphic in the unit disc
with a finite number of poles (expressed via roots of $P_\theta$) for $\theta_k=\omega^{k-1}$.


\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 22
\begin{frame}[shrink]
\frametitle{Proof of Main Theorem (continued)}
Let us use again the notation $1+t=(1+p)^k$ with $k\in \Z_p$.
\begin{align}\label{pNF}& \nonumber
{2^{-m/2}\over
{\zeta(1-k)(1-p^{k-1})\prod_{i=1}^{[m/2]}\zeta(1-2k+2i)(1-p^{2k-2i-1})} 
}
%\\ &\nonumber
%\cdot
%{1 \over {(1-p^{k-1})\prod_{i=1}^{[m/2]}(1-p^{2k-2i1})}}
\\ &
={{
U^E_{\omega^k}(t)
%{U^*_{\theta_{k},c}(\frac{1+t}{1+p}-1) (1-c^k)
%\prod_{i=1}^{[m/2]}
%U^*_{\theta_{2k-2i},c}({\frac{(1+t)^2}{(1+p)^{2i+1}}-1}) (1-c^{2k-2i})
}
\over {
P^E_{\omega^k}(t)}}
%P_{\theta,c}({\frac{1+t}{1+p}}-1)
%\prod_{i=1}^{[m/2]} P_{\theta_{2k-2i},c}({\frac{(1+t)^2}{(1+p)^{2i+1}}-1})}
%}
%\\ &\nonumber{
%U^*_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1) (1-c^{2k-2i})
%\over P_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1)}
\end{align}
where  the {\magenta numerator} is (an Iwasawa function)
%\begin{align*}&
$U^E_{\omega^k}(t)=$
$$U^*_{\theta_{k},c}(\frac{1+t}{1+p}-1) (1-c^k)
\prod_{i=1}^{[m/2]}
U^*_{\theta_{2k-2i},c}({\frac{(1+t)^2}{(1+p)^{2i+1}}-1}) (1-c^{2k-2i}), 
$$
%\end{align^*}
and
$$
P^E_{\omega^k}(t)=
P_{\theta,c}\left({\frac{1+t}{1+p}}-1\right)
\prod_{i=1}^{[m/2]} P_{\theta_{2k-2i},c}\left({\frac{(1+t)^2}{(1+p)^{2i+1}}-1}\right)
$$
is the {\magenta polynomial denominator}  which depends only on $k \bmod p-1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 23
\end{frame}
\begin{frame}[shrink]
\frametitle{Proof of Main Theorem: control over the conductor of $\psi_h$}
%%%%%%%%%%%%%%%%%%%%%%%
%Moreover, 
Mazur's formula applied to $L(1-k+\frac m 2,\psi_h)(1-\psi_h(p)p^{k-\frac m 2-1})$ ({\magenta in the numerator}) shows that
for all $h$ with $\det(2h)$ not divisible by $p$, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}\label{Lpsi}
L(1-k +\frac m 2, \psi_h)&(1-\psi_h(p)^{k-\frac m 2-1})\\ &\nonumber
={ 
G_{\theta, h}((1+p)^{k-\frac m 2-1}-1) 
\over
1-\psi_h(c_h)c_h^{k-\frac m 2}} 
\end{align} which is meromorphic in the unit disc 
with a possible single simple pole at $k=\frac m 2$  for all $k$ with $\theta=\omega^{k-1}$.
It comes from Mazur's measure on the finite  product $\displaystyle\prod_{{\ell}\in P_h}\Z_{\ell}^*$ extended over primes ${\ell}$ in the set $P_h=P(h)\cup \{p\}$; recall that $P(h)$ is the set of
  prime divisors %of the number $N$ and 
  of all elementary divisors of the
  matrix $h$ as above. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Indeed, for any choice of a natural number $c_h > 1$ coprime to $\prod_{{\ell}\in P_h}{\ell}$, there exists a $p$-adic measure 
$\mu_{c_h,h}$ on $\Z_p^*$, such that the special values
\begin{align*}&
L(1-k+\frac m 2, \psi_h)(1-\psi_h(p)p^{k-1-\frac m 2})=
{\int_{\Z_p^*}y^{k-\frac m 2-1}d\mu_{c_h, h}\over 
1-\psi_h(c_h)c_h^{k-\frac m 2}}
\\ & \nonumber
: =
(1-\psi_h(c_h)c_h^{k-\frac m 2})^{-1}
\int_{
\prod_{{\ell}\in P_h}\Z_{\ell}^*}\psi_h(y) y_p^{k-\frac m 2-1}d\mu_{c_h},  
\end{align*}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24
\begin{frame}[shrink]
\frametitle{Proof of Main Theorem  (continued)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where Mazur's measure $\mu_{c_h}$ extends on the product 
$\displaystyle \prod_{{\ell}\in P_h}\Z_{\ell}^* {\buildrel y_p   \over \longrightarrow} \Z_p^*$ 
%comes from Bernoulli distribution and given by the same formula as above 
(see \S 3,Ch.XIII of \cite{LangMF}): 
$$
\mu_{c_h}(a+(N))
=
\frac {1}{c_h}\left[ \frac {c_ha}{N}\right] + 
\frac {1-c_h}{2c_h}=\frac{1}{c_h} B_1(\{\frac {{c_h}a} {N}\})-B_1(\frac {a}{N})
%,~$B_1(x)=x-\frac 1 2
$$
for any natural number $N$ with all prime divisors in $P_h$.

 The regularizing factor is the following Iwasawa function which depends on $c_h \bmod 4$ and $k\bmod p-1$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}\label{ck1}
&1-\psi_h(c_h) c_h^{k-\frac m 2} =1-(\psi_h\omega^{k-\frac m 2})((\frac{(1+p)^{k}}
{(1+p)^{\frac m 2}}
-1)+1)^{\log \langle c_h\rangle\over \log(1+p)}
 \\& \nonumber
 =1-(\psi_h\omega^{k-\frac m 2})(c_h)\sum_{n=0}^\infty{{\log \langle c_h\rangle \over \log(1+p)}\choose n}(
 %(1+p)^k-1
 (\frac{(1+p)^{k}}
{(1+p)^{\frac m 2}}
-1
 )^n
 \\& \nonumber
 =1-(\psi_h\omega^{k-\frac m 2})(c_h)\sum_{n=0}^\infty{{\log \langle c_h\rangle \over \log(1+p)}\choose n} (\frac{(1+t)}
{(1+p)^{\frac m 2}}
-1)^n
% \\ & \nonumber
% = 1-\psi_h (c_h) \tilde c_h(t)
 \in \Z_p[\![t]\!],
\end{align}
%\fi
where we write $c_h$ in place of $i_p(c_h)$ and use the notation $1+t=(1+p)^{k}$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{frame}
%\end{document}
The function (\ref{ck1}) is divisible by $t$ or invertible in $\Z_p[\![t]\!]$
according as $\omega^{k-\frac m 2}\psi_{c_h}$  is trivial or not
%(explain)
because $t=0 \Eq k=0$ and $1+t=(1+p)^k$.
\\

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\iffalse
\begin{frame}[shrink]
\frametitle{Proof of Theorem \ref{pMSE} (continued)}
A formula for the reciprocal 
of the product of $L$-functions follows directly from (\ref{zk}), (\ref{z2k}): 
%\zeta(1-k)\prod_{i=1}^{[m/2]}\zeta(1-2k+2i)=
\begin{align*}&
{
\zeta(1-k)^{-1}\prod_{i=1}^{[m/2]}\zeta(1-2k+2i)^{-1}
\over
(1-p^{k-1})\prod_{i=1}^{[m/2]}(1-p^{2k-2i-1})
}
=
\\ & 
{
U^*_{\theta_{k},c}((1+p)^{k-1}-1) (1-c^k)
\over P_{\theta_k,c}((1+p)^{k-1}-1)}\times %\\ & \nonumber
\\ & 
\prod_{i=1}^{[m/2]}
{
U^*_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1) (1-c^{2k-2i})
\over P_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1)}
\\ &
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{U^*_{\theta_{k},c}((1+t)(1+p)^{-1}-1) (1-c^k)
%\over P_{\theta_k,c}(1+t)((1+p)^{-1}-1)}\times
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
={
U^*_{\theta_{k},c}((1+t)(1+p)^{-1}-1) (1-\tilde c(t))
\over P_{\theta_k,c}((1+t)(1+p)^{-1}-1)} %\\ & \nonumber
\\ & 
\prod_{i=1}^{[m/2]}
{
U^*_{\theta_{2k-2i},c}((1+t)^2(1+p)^{-2i-1}-1) (1-\tilde c(t)^2 c^{-2i})
\over P_{\theta_{2k-2i},c}((1+t)^2(1+p)^{-2i-1}-1)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{align*}
and the remaining factors  from  (\ref{Lpsi}) and (\ref{ck1}) %, (\ref{ch}) 
allow to collect everything together in the product.
Let us give the answer for the coefficients for the $p$-regular parts  of the Fourier coefficients
\begin{align*}&
a_h^{(p)}(k)=a_h(E_k^{n}) \times  %\\ &\times
\begin{cases} \displaystyle
%1/((1-p^{k-1})(1+\psi_h (p)p^{k-\frac m 2-1})\prod_{i=1}^{(m/2)-1}(1-p^{2k-2i-1})) & \\
{(1-\psi_h (p)p^{k-\frac m 2-1})\over(1-p^{k-1})\prod_{i=1}^{m/2}(1-p^{2k-2i-1})},& m \mbox{ even}
\\ 
\displaystyle
{1\over(1-p^{k-1})\prod_{i=1}^{(m-1)/2}(1-p^{2k-2i-1})},& m \mbox{ odd},  
\end{cases}
\end{align*}
where $\displaystyle \psi_h (n)=\left({\det(2h)(-1)^{m/2} \over n}\right)$. 
\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 25
\begin{frame}[shrink]
\frametitle{Elementary factors}
%Namely,  (\ref{ak}) gives
%\iffalse 
%Notation:  
%$1-c_h^k=t u_{c_h}^{-1} (t)$$, if p-1 divides k, 
%invertible otherwise :

Notation: 
\begin{align*}
u_{c_h}(t)=
\begin{cases}
(1-\psi_h(c_h)\tilde c_h(t))/t, &\mbox{ if } \omega^{k-\frac m 2}\psi_h  \mbox{ is trivial}, \\
1- \psi_h(c_h) \tilde c_h(t),&  \mbox{ otherwise}.
\end{cases}
\end{align*}
By (\ref{ck1}) we have that  $u_{c_h}(t)\in \Z_p[\![t]\!]^*$, and we denote by $u^*_{c_h}(t)$ its inverse. 
%\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Moreover,  (\ref{ah}) gives
the elementary factor 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
{\mathcal M}_h((1+p)^k-1)= 2^{-\frac{m}2}\det h^{k-\frac {m+1} 2} \prod_{{\ell}|P(h)} M(h, {\ell}^{-k})C_h^{k-\frac {m+1} 2}
$$
which is also an {\magenta Iwasawa function} as above:
 $$
 {\mathcal M}_h((1+p)^k-1)={\mathcal M}_h(t)\in \Z_p[\![t]\!].$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{frame}
%\end{document}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 26
\begin{frame}%[shrink]
\frametitle{Proof of Main Theorem: the numerator}
It follows that
$$
a_h^{(p)}(k)=
{S^E_{\theta,h}((1+p)^{k}-1)\over P^E_{\theta,h}((1+p)^{k}-1)}={S^E_{\theta,h}(t)\over P^E_{\theta,h}(t)}, 
$$
where 
%\end{frame}
%\end{document}
\begin{align*}&
S^E_{\theta,h}= u^*_{c_h}((1+p)^{k}-1)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\mathcal M}((1+p)^{k}-1)
\\ & \times
 U_{\theta, h}((1+p)^{k-\frac m 2-1}-1) (1-c_h^k)
 U^*_{\theta_{k},c}((1+p)^{k-1}-1) 
 \\ & \times
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\\ & \nonumber
\prod_{i=1}^{[m/2]}
U^*_{\theta_{2k-2i},c}((1+p)^{2k-2i-1}-1) (1-c_h^{2k-2i})
 \\ &
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
= u^*_{c_h}(t)
{\mathcal M}(t) U_{\theta, h}((1+t)(1+p)^{-\frac m 2-1}-1) 
\\ & \times
(1-\tilde c_h (t))%\\ & \times
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
U^*_{\theta_{k},c_h}((1+t)(1+p)^{-1}-1) %\times
\\ & \nonumber 
\times \prod_{i=1}^{[m/2]}
U^*_{\theta_{2k-2i},c_h}((1+t)^2(1+p)^{-2i-1}-1) 
(1-\tilde c_h^{2}(t)c_h^{-2i}), 
\end{align*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 27
\begin{frame}[shrink]
\frametitle{Proof of Main Theorem  (end)}
{\large \blue The denominator is the following distinguished polynomial}

\begin{align*}&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\ & 
P^E_{\theta, h} ((1+p)^{k-1}-1)= 
(1+((1+p)^{k-1}-2)\delta(\omega^{k-\frac m 2}\psi_{c_h}))
%(1-\psi_h(c)c_h^{k-\frac m 2})
\\ & \times
P_{\theta_k,c_h}((1+p)^{k-1}-1) %\\ & \nonumber
\prod_{i=1}^{[m/2]}
 P_{\theta_{2k-2i},c_h}((1+p)^{2k-2i-1}-1)
 \\ &
 = %u_{c_h}(t) (1-\psi_h(c_h) \tilde c_h (t)  c_h^{-\frac m 2})
 %\\ & \times
 (1+(t-1)\delta(\omega^{k-\frac m 2}
\psi_{c_h}))
P_{\theta_k,c_h}((1+t)(1+p)^{-1}-1) \\ & \nonumber
\times \prod_{i=1}^{[m/2]}
 P_{\theta_{2k-2i},c}((1+t)^2(1+p)^{-2i-1}-1), \mbox{ where}
\end{align*} 
%\end{frame}
%\end{document}
$%\begin{align}
\delta(\omega^{k-\frac m 2}\psi_{c_h})=
\begin{cases}
1, &\mbox{ if } \omega^{k-\frac m 2}\psi_{c_h}  \mbox{ is trivial}, \\
0,&   \mbox{ otherwise},  
\end{cases}
$ 
so that\\ 
$%\begin{align}
1+(t-1)\delta(\omega^{k-\frac m 2}
\psi_{c_h})=
\begin{cases}
t, &\mbox{ if } \omega^{k-\frac m 2}\psi_{c_h}  \mbox{ is trivial}, \\
1,&   \mbox{ otherwise}.
\end{cases}
$ .\\
{\magenta
It remains to notice that different choices of $c_h$ coprime to $p \det (2h)$  give the same polynomial factors  $P^E_{\theta, h}$
 (up to invertible Iwasawa function). Indeed they all give the same single simple zero.
 }
\qed  

\

\end{frame}
\section{Pseudomeasures and their Mellin transform }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 28
\begin{frame}[shrink]
\frametitle{Interpretation: Mellin transform of a pseudomeasure}
Pseudomeasures were introduced by J.Coates \cite{Co} as elements of the {\magenta fraction field ${\mathcal L}$}  of the 
Iwasawa algebra. 
Such a pseudomeasure is defined  by its Mellin transform 
which is a ring homomorphism and we can extend it by universality
(the extension of the integral along measures in $\Lambda=\Z_p[[T]]$ to the whole fraction field ${\mathcal L}$). 

The $p$-adic meromorphic function 
$$
a_h^{(p)}(k)=
{S^E_{\theta,h}((1+p)^{k}-1)\over P^E_{\theta,h}((1+p)^{k}-1)}={S^E_{\theta,h}(t)\over P^E_{\theta,h}(t)}, 
$$
%\end{frame}
%\end{document}
is attached to an explicit {\magenta pseudomeasure}: 
$$
\rho_h^E={\mu_h^E \over \nu_h^E},\ \ 
 {S^E_{\theta,h}(t)\over P^E_{\theta,h}(t)}={\int_{\Z_p^*}\theta \chi_{(t)} \mu_h \over 
 \int_{\Z_p^*}\theta \chi_{(t)} \nu_h }
$$
%\end{frame}
%\end{document}
\begin{itemize}
\item ${\mathcal S}(x)= \int_{\Z_p^*} x \mu_h^E$ is given by the collection of Iwasawa 
 functions $S_{\theta,h}(t)=\int_{\Z_p^*}\theta\chi_{(t)} \mu_h^E$ (the {\magenta numerator}),
\item  ${\mathcal P}(x)= \int_{\Z_p^*} x \nu_h^E$ is given by the collection of polynomials 
$P_{\theta,h}(t)=\int_{\Z_p^*}\theta\chi_{(t)} \nu_h^E$ (the {\magenta denominator}). 
\end{itemize}
   
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 29
\begin{frame}[shrink]
\frametitle{Pseudomeasure $\rho$ as a family of distributions}
A pseudomeasure $\rho$ can be described as a certain {\magenta family of distributions, 
parametrized by the   set $X_p$ of $p$-adic characters}.
%\end{frame}
%\end{document}


%Let $X^+$ denote the union of those components $X_\theta$ in $X_p$ where $P_\theta$ vanishes identically.
For any $x\in X_p$ 
%outside the  set   
%$$ 
%X^+\cup(
%\{ \mbox{ zeros of  }\mathcal P\}\times  X^{ \mathrm{tors}}_p
%$$                 
we have a distribution given by the formula
$$
\rho_{h,x}^E (a+(p^v))=
\frac 1 {\varphi(p^v)} 
\sum^{}_{\hskip-.1cm\chi\bmod p^v}{\hskip-.3cm{}'}
\chi(a)^{-1}\frac {{\mathcal S}^E(\chi x)}
{{\mathcal P}^E(\chi x)}
$$
where ${}'$ means that the terms with ${\mathcal P}(\chi x)=0$ are omited.
It follows that
$$
\int_{\Z_p^*} \chi  \rho_{h,x}^E  
%{\int_{\Z_p^*} \chi x \mu_{h,x}^E \over \int_{\Z_p^*} \chi x \nu_{h,x}^E}
%$$
%equals
%$$
= \begin{cases}
\displaystyle 
{{\mathcal S}^E(\chi x) \over {\mathcal P}^E(\chi x)}
, &\mbox{ if } {\mathcal P}^E(\chi x) \not =0 \\
0, &\mbox{otherwise}, 
\end{cases}
$$
%\end{frame}
%\end{document}
where 
\begin{align*}&
{\mathcal S}^E_h(\chi x)=S^E_{(\chi x)_\Delta, h}(\chi x(1+p)-1)
=S^E_{\theta,h}((1+p)^t-1), \\ &\theta=(\chi x)_\Delta, (\chi x)(1+p)=1+t, 
%\zeta_{p^v}=\chi (1+p),
%\end{frame}
%\end{document}&
\\ &
{\mathcal P}^E_h(\chi x)
=P^E_{(\chi x)_\Delta, h}(\chi x(1+p)-1)
=P^E_{\theta,h}((1+p)^t-1). 
\end{align*}
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\iffalse
\begin{frame}%[shrink]
\frametitle{\large Complex valued distributions attached to arbitrary Euler products}
were discovered already in 1980, are certain  Fourier transforms of natural zeta-distributions coming from partial zeta values, see
(\cite{Pa81}, \cite{Pa82}; \cite{Co-PeRi}). 
They were used by	 Mazur-Tate-Teitelbaum, J.Coates, B.Perrin-Riou, R.Greenberg, \dots.

In the cases we are interested in these values are in  $\Z$,  
%$\Q\cap  \Z_p$,  
or in a  $\Z$-module of finite rank in $\cb$ (corresponding to "critical values" of $s$).

We consider a  Dirichlet seres
\begin{equation}  \label{EQserDir}
    \dc(s) = \sum_{n=1}^{\infty} a_n n^{-s} ~~~~ (s, a_n \in \cb)
  \end{equation}
with the following  multiplicativity property of the 
coefficients $a_n$:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.69)
 \begin{equation}  \label{EQDirDrs}
    \dc(s) = %\prod_{q \in S} 
    F_p(p^{-s})^{-1} \sum_{n=1\atop(n,p)=1}^{\infty} a_n n^{-s},
  \end{equation}
where  $F_p(X)$ is a  polynomial of degree  %quelconque 
$d$ and of the constant term %égale à 
$1$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.70)
\small  \begin{equation}
    F_p(X) = 1 + \sum_{i=1}^{{d}} a_{p,i} X^i, \mbox{ in particular, } a_{p,i}=-a_p
  \end{equation}\normalsize
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink] 
\frametitle{Higher Modular Symboles and Complex valued distributions} 
Let us define (possibly via a complex-analytic continuation over $s$)
%pour $s \in \cb$ tel que la  série (\ref{EQserDir}) converge absoluement
\begin{equation}
    \dc(s,\chi) = \sum_{n=1}^{\infty} \chi(n) a_n n^{-s}.
  \end{equation}
and define the {\it higher modular symbols} (difficult to compute!) as the following function $P_s : \qb \rightarrow \cb$
 \begin{equation}  \label{EQserPs}
    P_s(x) = \sum_{n=1}^{\infty} e(nx) a_n n^{-s} ~~~~ (e(x) = \exp(2i\pi x)).
  \end{equation}

Starting from the functions (\ref{EQserPs}) one constructs distributions $\mu_s$ with values in $\cb$
 on the compact group  $\zb_p^{\times}$ such that for any primitive   Dirichlet  character $\chi \bmod p^m$
viewed as a homomorphism $\chi : \zb_p^{\times} \rightarrow \cb^{\times}$, the values of the seies  $\dc(s,\chib)$ at $s$ %avec $\Re(s) > 1+c$ 
are expressed in the form of  the integral 
$$ 
\int_{\zb_p^{\times}} \chi \dd \mu_s = \sum_{%\stackrel
     {a \mod p^m}\atop {(a,p)=1}} \chi(a) \mu_s(a+(p^m)).  
$$
\end{frame}

\begin{frame}[shrink]  
\frametitle{Zeta distributions and their Fourier transform } 
%\end{frame}
%\iffalse 
Let $\al=\al(p)\ne 0$ be an inverse root of the  polynomial 
$F_p(X) = 1 + \sum_{i=1}^{{d}} a_{p,i} X^{i}=F_p(X)=(1-\al_1X)\cdots  (1-\al_{{d}}X)$
 with $\al=\al_1$, \\
and let
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.73)
$   
H_p(X) := 1 + \sum_{i=1}^{{d}-1} b_{p,i} X^i=F_p(X)/(1-\al_1X)\in \cb[X].
$
\begin{theo}[A.P., 1980, \cite{Pa81}, \cite{Pa82},  \cite{Co-PeRi}]  \label{THEOdistr} 
  (a) For any choice of an inverse root $\al \neq 0$, and for all  $s\in \cb$ in the domain of definition of the series définition des
   $\dc(s,\chi)$,  
there exists a distribution $\mu_s = \mu_{s,\al}$ on $\zb_p^{\times}$ such that
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.78)
  \begin{equation}  \label{EQdistmu}       
\mu_s(a+(p^m)) = \frac{p^{m(s-1)}}{\al(p)^m} \sum_{i=0}^{{d}-1} b_{p,i}
      P_s\left( \frac{ap^i}{p^m} \right) p^{-is}.
    \end{equation}

  (b) For any primitive Dirichlet character $\chi\bmod p^{m_\chi}$ 
  one has
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.79)
 $
      \int_{\zb_p^{\times}} \chi d\mu_s =     
      %\prod_{q \in S \setminus S(\chi)}
      \left( 1-\chi(p)\al(p)^{-1}p^{s-1} \right)    
      H_p \left( \bar\chi(p) p^{-s}\right) 
      \frac{C_{\chi}^{s-1}}{\al^{m_\chi}} G(\chi) \dc(s,\chib),
$
where
    $ G(\chi) = %\displaystyle 
    \sum_{a \bmod C_{\chi}} \chi(a) \exp \left(2\pi i %\frac
    ({a}/
    {C_{\chi}})
       \right)$
the Gauss sum, $C_{\chi}=p^{m_\chi}$.
the conductor of $\chi$.
\end{theo}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{A useful special case: $E$ an  elliptic curve of multiplicative reduction at $p$,  voir %\cite{BerDa09}, 
\cite{PollR}}
In this case $E$ has the conductor
$N$, $p\| N$, $k=2$, $s=1$, $\al=a(p)$, $d=1$,$H_p(X) = 1$,  
$$
\mu_1(a+(p^m)) = \dfrac{1}{a(p)^m} 
      P_1\left( \dfrac{a}{p^m} \right).
$$  
Let $K/\Q$ be a real quadratic field in which 
$p$ is inert: $[K_p:\qb_p]=2$.    

{\it \red The Stark-Heegner points} is a $p$-adic  version of Heegner points, 
and they are  (conjecturelly) defined over Hilbert class field of $K$.
These points are defined using the {\it \red  $p$-adic upper half plane}, 
$\hc_p = \cb_p \setminus \qb_p$
(notice that  $\cb \setminus \rb$ is equal to the union of two  copies of  $\hc$).

The distribution property makes it possible  
to compute efficiently certain rational points (see \cite{PollR} for details)
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\fi

\begin{frame}[shrink=1]
\frametitle{Modular distributions (A.P., 2001): examples}
(these are distributions $\Phi$ taking values in spaces of modular forms). 

(a) {\magenta Eisenstein distributions :}
$$
E_k(a+(p^m)):= c_0 +  \sum_{\scriptstyle n\ge 1}
\sum_{ \scriptstyle d\equiv a (\bmod p^m)}d^{k-1}q^n\in \mc_k(p^{m}), \ \ c_0 =c_0 (a+(p^m)).
$$
(b) {\magenta Partial modular forms.} For any  $f=\sum_{n\ge 0}a_n(f)q^n$ $\in \mc_k(\Gamma_1(N))$ let us define
$$
\Phi_f(a+(p^m)):= \sum_{\scriptstyle n\ge 0\atop \scriptstyle n\equiv a (\bmod p^m)}a_n(f)q^n \in
\mc_k(Np^{2m}),
$$
then for any Dirichlet character
$\chi\bmod p^m$ viewed as a function on $\zb_p^*$ with values in
$\cb$,  the integral  
$$
\int_{\zb_p^*}\chi (y)\, d\Phi_f = \Phi_f(\chi)=\sum_{n\ge 0}\chi(n) a_n(f)q^n \in
\mc_k(Np^{2m})\subset\mc_k(\Ga_1(N p^{2m}))
$$
coincides with the twisted modular form  $f_\chi$.

(c) {\magenta Partial theta series} 
%(aussi avec un polynôme sphérique), 
%see~\cite{PaIAS}, [Hi85], \S 1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}  
\frametitle{%Passage 
From modular forms to algebraic  numbers
("$p$-stabilization")}
Consider   a  $p$-adic distribution $\Phi$ on $\zb_p^*$ with values in an infinite dimensional 
 $\Qb$-vector space  
$\mc=\mc (\Qb)\subset \mc(\cb_p)$
  (of {\it   modular forms}), 
$$
\mc (\Qb):=\bigcup_{{m} \ge 0} \mc_k
(\Gamma _0(N p^{m} ), \psi, \Qb )  \mbox{ ("the Igusa tower") }
$$
Starting from the distribution $\Phi$, we construct a distribution $\pi(\Phi)$ with values in spaces in a finite dimensional space  $\mc^\alpha\subset \mc(Np)$.
We use the caracteristic projection $\pi$ on the characteristic
subspace $\mc^\alpha$ (of generalized eigenvectors) associated to a non-zero eigenvalue $\alpha $ Atkin's $U$-operator.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In particular,  for any  $\al\in\Qb^*$, 
the known $p$-adic distributions 
(of Manin, Amice-Vélu,  Mazur-Tate-Teitelbaum, Vishik, $\dots$), can be expressed in the form of the composition:
$$
\mu_{\Phi,\al}
=\ell (\pi_{\al}(\Phi)),\  \ 
%(j=0, 1, \cdots, k-2)
$$
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}  
\frametitle{Canonical projector} 
\begin{itemize}
\item
$\pi_{\al}$ is a canonical $\Qb$-projector onto the subspace $\mc^\al= \mc^\al(\Qb)$
  {\it of finite dimension} (the characteristic subspace) 
of Atkin's operator: 
$$
 U\left(\sum_{n\ge 0} b_nq^n
\right )= \sum_{n\ge 0} b_{pn} q^n,
$$
\item
 $\ell \in (\mc^\al)^*$ is a certain $\Qb$- linear form. 
 \end{itemize}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
We use the following  commutative diagram:
$$
\begin{matrix}
&\mc_k(Np^{{m} +1}, \psi) &\displaystyle\mathop{\longrightarrow}_ {
{\pi _{\alpha ,{{m} }}}}
&\mc_k^\alpha (Np^{{m} +1}, \psi)&\cr
&U^{{m} } \Big\downarrow&&\Big\downarrow\!\!\wr  U^{{m} }&\cr
&\mc_k(Np, \psi) &
\displaystyle\mathop{\longrightarrow}_ {
{\pi _{\alpha ,{0}}}}
&\mc_k^\alpha (Np, \psi)
\end{matrix}
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 $
\pi_\alpha(g){\buildrel {def}\over =}(U^ \alpha)^{-{m} }\pi _{\alpha ,0}(U^{m} (g))\in
 \mc^\alpha(\Gamma_0(Np),\psi, \cb)
$ \\
({\it \magenta the canonical $\alpha$-caracteristic projection} of a modular form
 $g\in \mc_k (\Gamma_0(Np^{{m} +1}, \psi)) $ with a Dirichlet character $, \psi\bmod Np$).
 \end{frame}
\begin{frame}  
\frametitle{Passage %de valeurs dans les formes modulaires 
to numerical values}
Now the construction of $p$-adic distributions reduces to properties of the Fourier coefficients of modular forms, after application of an {\it \magenta algebraic linear form} $\ell$. 


%In many cases the integrals of these $p$-adic distributions coincide  
%with thoses of zeta  distributions;  
%it gives a  {\it conceptual explanation} of the above formulas.

In fact, for a certain linear eigenform  $\ell \in U^*\in (\mc(Np)){}^*$, 
the formula much simplifies to 
\begin{align*}&
\mu_{\Phi,\al}(a+(p^m))=\ell(\pi_{\al}(\Phi)(a+(p^m))
\\ &=\ell((U^ \alpha)^{-{m'}}\pi_{\alpha ,0}(U^{m'} \Phi(a+(p^m))))=\al^{-m'}\ell((U^{m'}\Phi(a+(p^m))).
\end{align*} 
for the modular forms $\Phi(a+(p^m))\in \mc$ (without $\pi_{\alpha ,0})$. %, explaining the factor $\al^{-m'}$.
In fact, the last line follow from the fact that for $m'\gg 0$ the difference 
\begin{align*}&
h=U^{m'}\Phi(a+(p^m))-\pi_{\alpha ,0}((U^{m'}\Phi(a+(p^m))))\\ &\in 
{\rm Im}(I-\pi_{\alpha ,0})\subset \Ker(\pi_{\alpha ,0})\subset \Ker\ell.
\end{align*}
\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 30
\begin{frame}[shrink=1]
\frametitle{A geometric construction: Siegel's method and duality %of the Siegel-Eisenstein series
 }
 For any Dirichlet character $\chi\bmod p^v$ 
 consider Shimura's "involuted" Siegel-Eisenstein series
 assuming their absolute convergence (i.e. $k> m+1$):
 $$
E^*_k(\chi,z)=\sum_{(c,d)/\sim}\chi(\det(c))\det(cz+d)^{-k}
%\sum_{\gamma\in P\backslash \Gamma}\det(cz+d)^{-k}
=\sum_{h\in B_m}a_h(E^*_k(\chi,z))q^h
$$
The series on the left is {\magenta geometrically defined}, and the Fourier coefficients on the right can be computed by Siegel's method (see \cite{St81}
\cite{Shi95}, \dots )
via the singular series
\begin{align}\label{ahk3}
&a_h({E}^*_k(\chi,z))\\ &\nonumber = 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
(-2\pi i)^{mk}\over 2^{m(m-1)\over 2}\Gamma_m(k)} 
%\\ &
%\times
\sum_{{\mathfrak R}\bmod 1}\chi(\nu({\mathfrak R}))\nu({\mathfrak R})^{-k}
\det h^{k-\frac{m+1} 2}e_m(h{\mathfrak R})
\end{align}



%\iffalse
%The {\magenta orthogonality relations $\bmod p^v$ produce two families of distributions}
%(notice that terms in the RHS are invariant under sign changes, and (\ref{ah0}) 
%is algebraic after multiplying by the factor in (\ref{ahk})):
%\small
%\begin{align}&\label{st}
%\hskip-1.2cm{1\over \varphi(p^v)} \sum_{\chi\bmod p^v} \bar\chi (b)
% \sum_{(c,d)/\sim}{\chi(\det(c))\over \det(cz+d)^{k}}=
% \sum_{(c,d)/\sim\atop 
%\det(c)\equiv b \bmod p^v}{\sgn(\det(c))^k\over \det(cz+d)^{k}} \\ \label{ah0} & 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\hskip-1.3cm{
% 1\over \varphi(p^v)} \sum_{\chi\bmod p^v} \bar\chi (b)
%\sum_{{\mathfrak R}\bmod 1}{\chi(\nu({\mathfrak R}))e_m(h{\mathfrak R})\over \nu({\mathfrak R})^{k}}
%=
%\sum_{{\mathfrak R}\bmod 1\atop 
%\nu({\mathfrak R})\equiv b \bmod p^v}{e_m(h{\mathfrak R})
%\sgn\nu({\mathfrak R})^k\over \nu({\mathfrak R})^{k}}
%\end{align}
%\fi

If $\chi=\chi_0\bmod p$ is trivial and $p\nd \det h$ then
% not divisible by $p$ give
%\magenta
\begin{align}\label{ahk4}
&a_h({E}^*_k(\chi_0,z))\\ &\nonumber = 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{
(-2\pi i)^{mk}\over 2^{m(m-1)\over 2}\Gamma_m(k)} 
%\\ &
%\times
\sum_{{\mathfrak R}\bmod 1}\chi_0(\nu({\mathfrak R}))\nu({\mathfrak R})^{-k}
\det h^{k-\frac{m+1} 2}e_m(h{\mathfrak R})=
\\ & \nonumber
a_h(E_k^m)\times
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cases} (1-p^{-k})(1+\psi_h (p)p^{-k+\frac m 2})\prod_{i=1}^{(m/2)-1}(1-p^{-2k+2i}),
%& \\= (1-p^{-k})\prod_{i=1}^{m/2}(1-p^{-2k+2i})/(1-\psi_h (p)p^{-k+\frac m 2}),
& m \mbox{ even}
\\ \\
(1-p^{-k})\prod_{i=1}^{(m-1)/2}(1-p^{-2k+2i}),& m \mbox{ odd}.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{cases}
\end{align}  

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 31
\begin{frame}[shrink=1]
\frametitle{A geometric construction (continued)
%of the Siegel-Eisenstein series
 }
The formula (\ref{ahk4}) means that the series
${E}^*_k(\chi_0,z)$  
coincides with $E_k^m$ after
removing $h$ with $\det h$ divisible by $p$ and normalizing by the factor in
(\ref{ahk4}).
Moreover, the {\magenta Gauss reciprocity law} shows that
the normalizing factor depends only on $\det h\bmod 4p= \det h_0\bmod 4p$, where 
$h_0\equiv h \bmod 4p$ runs through a representative system.
Let us denote this factor by $C^+(h_0,k,4p)$: 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for the trivial character $\chi=\chi_0\bmod p$ and $\det h$ not divisible by $p$
\begin{align}\label{ahk5}
&a_h({E}^*_k(\chi_0,z))= 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a_h(E_k^m)C^+(h_0,k,4p),  \mbox{ where }\\ & \nonumber
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
C^+(h_0,k,4p)=
\begin{cases} (1-p^{-k})(1+\psi_h (p)p^{-k+\frac m 2})\prod_{i=1}^{(m/2)-1}(1-p^{-2k+2i}),
%& \\= (1-p^{-k})\prod_{i=1}^{m/2}(1-p^{-2k+2i})/(1-\psi_h (p)p^{-k+\frac m 2}),
& m \mbox{ even}
\\ \\
(1-p^{-k})\prod_{i=1}^{(m-1)/2}(1-p^{-2k+2i}),& m \mbox{ odd}.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{cases}
\end{align}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\large \blue From the Fourier coefficients to modular forms: }

If we remove in the Fourier expansion 
$\magenta
E_k^m(z)=\sum_{h\ge 0}a_he_m(hz)
$ 
all terms with $\det h$ divisible by $p$, the equality of Fourier coefficients (\ref{ahk5}) transforms to the equality of the series
\begin{align}\label{chi0}&
{E}^*_k(\chi_0,z)=
%\sum_{h>0, p\nds \det h}a_hq^h
(4p)^{-m(m+1)/2}\sum_{h_0\bmod 4p\atop p\nds \det h_0}C^+(h_0,k,4p)\times
\\ & \nonumber
\sum_{x\in S\bmod 4p}
e_m(-h_0x/4p)E_k^m(z+(x/4p)).
\end{align}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 32
\begin{frame}[shrink=1]
\frametitle{A geometric construction 
%of the Siegel-Eisenstein series
 }
Let us apply the interpolation theorem  (Theorem \ref{pMSE})
to all the coefficients 
\begin{align}\label{chi02}
& a_h^{(p)}(k)=a_h(E_k^{m})C^-(h_0,k,4p), \mbox{ where }
\\ & \nonumber C^-(h_0,k,4p)=
\begin{cases} %{1\over(1-p^{k-1})(1+\psi_h (p)p^{k-\frac m 2-1})\prod_{i=1}^{(m/2)-1}(1-p^{2k-2i-1})} %& \\
{1-\psi_h (p)p^{k-\frac m 2-1}\over(1-p^{k-1})\prod_{i=1}^{m/2}(1-p^{2k-2i-1})},& m \mbox{ even}
\\  \\
{1\over(1-p^{k-1})\prod_{i=1}^{(m-1)/2}(1-p^{2k-2i-1})},& m \mbox{ odd},  
\end{cases}
\end{align}
and (\ref{chi0}) becomes a "geometric-algebraic equality"
of two families of modular forms 
\begin{align}\label{chi01}&
{E}^*_k(\chi_0,z)=
%\sum_{h>0, p\nds \det h}a_hq^h
(4p)^{-m(m+1)/2}\sum_{h_0\bmod 4p\atop p \nds \det h_0}
C^+(h_0,k,4p)
\times
\\ & \nonumber
C^-(h_0,k,4p)\sum_{x\in S\bmod 4p}
e_m(-h_0x/4p)E_k^m(z+(x/4p)).
\end{align}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 33
\begin{frame}[shrink=1]
\frametitle{A geometric construction (end)}
We deduce by the orthogonality that 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}\label{chi03}&
\sum_{x'\in S\bmod 4p}
e_m(-h_0x'/4p)E^*_k(\chi_0, z+(x'/4p))=
\\ \nonumber &
C^+(h_0,k,4p)
\sum_{x\in S\bmod 4p}
e_m(-h_0x/4p)
E_k^m(z+(x/4p)).
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Each series
$
C^-(h_0,k,4p)\sum_{x\in S\bmod 4p}
e_m(-h_0x/4p)E_k^m(z+(x/4p))
$
in (\ref{chi01})
 determines a {\magenta unique pseudomeasure with coefficients in 
${\bar\Q}[\![q^{B_m}]\!]$} whose moments are given by those of the coefficients (\ref{chi02})
(the unicity means that a pseudomeasure is determined by its Mellin transform).
It is also a family of distributions geometrically defined by the series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
{C^-(h_0,k,4p)\over C^+(h_0,k,4p)}
\sum_{x\in S\bmod 4p}
e_m(-h_0x/4p)E^*_k(\chi_0, z+(x/4p)).
$$ 
%accomplishing the geometric construction.

\

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%III 
\section{Application to Minkowski-Siegel Mass constants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 34
\begin{frame}[shrink=1]
\frametitle{$p$-adic version of Minkowski-Siegel Mass constants.
%for unimodular latticies \cite{ConSl98}, p.409
}
%\begin{itemize}
%\item
An application of the construction is the  $p$-adic version of Siegel's Mass formula.
It expresses the Mass constant through the  above product of $L$-values.
This    %extends to the the symplectic case 
product can be viewed as the proportionality coefficient between two kinds of Eisenstein series in the symplectic case extending Hecke's
result (1927) 
of the two kinds of Eisenstein series and 
the relation between them. 
However, there is no direct analogue of Hecke's computation in the symplectic case.

Thus this mass constant admits an 
explicit product expression through the values of the functions (\ref{zki}) at $t_j=(1+p)^j-1$, 
for $j=k$, and $j=2,4,\dots, 2k-2$.

%a meromorphic $p$-adic interpolation (as a certain pseudomeasure) of $k$.
%\item Measures coming from the theta-series 
%are related to the measures coming from the Siegel-Eisenstein series ; 
%the proportionality coefficient between the two measures coinsides with a product of $L$-values. 
%\end{itemize}

\


Recall that (\cite{ConSl98}, p.409)
\begin{quote}
unimodular latticies have the property that there are explicit formulae, the mass formulae, which give appropriately weighted sums of the theta-series of all the inequivalent latticies of a given dimension.
In particular,  the numbers of inequivalent latticies is given by Minkowski-Siegel Mass constants for unimodular latticies.
\end{quote} 

\

%Siegel's mass formula is a
%formula for the sum of the weights of the lattices (quadratic forms) in a genus, 
%weighted by the reciprocals of the orders of their automorphism groups
%(\cite{ConSl98}, Ch.16).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 35
\begin{frame}[shrink]
\frametitle{Application to Minkowski-Siegel Mass constant
(continued)}

In the particular case of even unimodular
quadratic forms of rank $m = 2k \equiv 0(\bmod 8)$, this formula means that
there are only finitely many such forms up to equivalence for each $k$ and that,
if we number them $Q_1, \dots ,Q_{h_k}$ , then we have the relation
$$
\sum_{i=1}^{h_k}\frac 1 {w_i}\Theta_{Q_i}(z)=m_kE_k
$$
where $w_i$ is the number of automorphisms of the form $\Theta_{Q_i}$ is the theta
series of $Q_i$, $E_k$ the normalized Eisenstein series of weight $k = m/2$ (with
the constant term equal to 1),

The dimension of lattices is $2k$ and the Mass formula 
express  an identity of a sum of {\magenta weighted theta functions}
 and a {Siegel-Eisenstein series} of weight $k$, % and genus $m$, 
 multiplied by the Mass constant 
 $$
 m_k=2^{-k}
\zeta(1-k)\prod_{i=1}^{k-1}\zeta(1-2k+2i)= 
(-1)^{k}\frac {B_k} {2k} 
\times \prod_{j=1}^{k-1}\frac {B_{2j}} {4j}
 $$
which is related %exactly 
the above normalising coefficient.\\
{\tt 
gp > mass(4)\\
\% = 1/696729600\\
gp > mass(8)\\
\% = 691/277667181515243520000
}

\



\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 36
\begin{frame}[shrink]
\frametitle{Application to Minkowski-Siegel Mass constant
(continued)}

The present result says that the $p$-regular part of $1/m_k$ is a product of values of the 
$p$-adic meromorphic functions (\ref{zki}) at $t_j=(1+p)^j-1$, $j=k$ and $j=2, 4, \dots, 2k-2$.  
%(with $\omega^k$ fixed).

It is known that the rational number $m_k$ becomes very large rapidly, 
when $k$ grows (using the functional equation). 
It means that the denominator of $1/m_k$ becomes {\magenta enormous}.
The explicit formula (\ref{pNF}) 
applied to  the reciprocal of the product of $L$-functions as above 
shows that these are {\magenta only irregular primes} which contribute to the denominator, 
and this contribution can be evaluated for {\magenta all primes} knowing the {\magenta Newton polygons} of the polynomial part $P_\theta$, which can be found directly from 
the Eisenstein measure.
Precisely, for the distingushed polynomial $P(t)=P_{\theta}(t)=a_dt^d+\dots+a_0$, 
$\ord_p a_d=0$, and $\ord_p a_i>0$ for $0\le i\le d-1$, and $\ord_p(t_j)=\ord_p j+1$, where $t_j=(1+p)^j-1$  for $j=k$ and $j=2, 4,\dots,2k-2$.
Then 
$$
\ord_p P(t_j)= \mathop{\min}_{i=0, \dots, d} \left(\ord_p a_{i,k}+i(\ord_p j+1)\right).
$$
the values $\ord_p a_{i,k}$ for $0\le i\le d$ come from the Iwasawa series in the denominator in the left hand side of (\ref{pNF}).
Also, it gives an important information about the location
 of zeroes  of the polynomial part as in (\ref{pNF})). 
However $P(t_j)\ne 0$ in our case because all the $L$-values in question do not vanish. 

 \

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 37
\begin{frame}[shrink]
\frametitle{Application to Minkowski-Siegel Mass constant \\ (numerical illustration)}
{\tt
for(k=1,10,print(2*k,~factor(denominator(1/mass(2*k)))))\\
{2}       1\\
4       1\\
6       1\\
8       [691, 1]\\
10       [691, 1; 3617, 1; 43867, 1]\\
12       [131, 1; 283, 1; 593, 1; 617, 1; 691, 2; 3617, 1; 43867, 1]\\
14       [103, 1; 131, 1; 283, 1; 593, 1; 617, 1; 691, 1; 3617, 1; 43867, 1; 6579
31, 1; 2294797, 1]\\
16       [103, 1; 131, 1; 283, 1; 593, 1; 617, 1; 691, 1; 1721, 1; 3617, 2; 9349,
 1; 43867, 1; 362903, 1; 657931, 1; 2294797, 1; 1001259881, 1]\\
18       [37, 1; 103, 1; 131, 1; 283, 1; 593, 1; 617, 1; 683, 1; 691, 1; 1721, 1;
 3617, 1; 9349, 1; 43867, 2; 362903, 1; 657931, 1; 2294797, 1; 305065927, 1; 
 1001259881, 1; 151628697551, 1]\\
20      [103, 1; 131, 1; 283, 2; 593, 1; 617, 2; 683, 1; 691, 1; 1721, 1; 3617,
1; 9349, 1; 43867, 1; 362903, 1; 657931, 1; 2294797, 1; 305065927, 1; 
1001259881, 1; 151628697551, 1; 154210205991661, 1; 26315271553053477373, 1]
}

\

\iffalse
Siegel concluded his 1935 paper by the phrase:

 "Identität zwischen Functionen verschiedener Herkunft zu besitzen, nämlich zwischen gewissen Thetareihen und anderen durch Eisensteinsche Reihen definierten Modulformen; und jede einfache nicht-triviale Identität bedeutet ja für eine complicierte Theorie bereits eine Entschuldigung".
\\  
  "\dots to obtain identities between functions of different nature such as theta series and Eisenstein seris; each such simple identity means an excuse even for a complicated theory".  
\fi 
 


%\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 38
\end{frame}
%IV 
\section{Link to Shahidi's method in the case of 
 SL(2)  and regular prime $p$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 43
\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), and

(5) Shahidi's method.

There exist already advances for (1) to (4), and we are also trying to develop  (5).
%,  see \cite{GPS}.

We use the Eisenstein series on classical groups and  $p$-adic integral of Shahidi's type
 for the reciprocal of a product of certain $L$-functions.
 
 \
 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  40
\begin{frame}[shrink]
\frametitle{Link to Shahidi's method in the case of 
 SL(2)  and regular prime $p$}
 
The starting point here is the Eisenstein series
$$
E(s,P,f,g)=\sum_{\gamma\in P\smallsetminus G}
f_s(\gamma g),
$$
on a reductive group $G$  and a {\magenta maximal  parabolic subgroup} $P=MU^P$
 (decomposition of Levi). 

This series generalizes
$$
E(z,s)=\frac{1}{2}\sum\frac{y^s}{|cz+d|^{2s}},\quad (c,d) =1.
$$

Here $f_s$ is an appropriate function in the induced representation space 
 $I(s, \pi)=Ind_{P_\A}^{G_\A}(\pi\otimes |det_{M}(\cdot)|^s_{\A}) )$, see
 (I.2.5.1) at p. 34 of \cite{GeSha}.
 
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 40
\begin{frame}[shrink]
\frametitle{
Computing a non-constant term  (a Fourier coefficient)}
 of this Eisenstein series provides an analytic continuation and the functional equation  for many Langlands
 $L$ functions $L(s,\pi,r_j)$. 
 
 \
 
In this way the  $\psi$-th Fourier coefficient  (with $\psi$ of type $\psi(x)=\exp(2\pi i n x), n\in \N, n\ne 0$) of the series $E(s,P,f,e)$ is determined by the  Whittaker functions $W_v$  in the form (see \cite{GeSha}, (II.2.3.1), p.78):
$$
E^\psi(e,f,s)= \prod_{v\in S}W_v(e_v)\prod^m_{j=1}\frac{1}
{L^S(1+js,\pi, {r}_j)},
$$
where ${r}_j$ are certains fundamental representations of the dual group ${}^LM$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 41
\begin{frame}[shrink]
\frametitle{}
\begin{theo}[a complex version] 
With the data $G=SL(2)$, $M =\{\mat a, 0,  0, a^{-1}, \}\cong \GL_1 $, $\pi=I$, and $\psi$ a  non-trivial character of the group 
$U(\ab)/U(\qb)$, $U=\{\mat 1, *,  0, 1, \}\cong \G_a$, 
let
$
E^{\psi}(s,f,e)= \int E(s,f,n){\psi(n)} dn,
$
the integration on the  quotient  space of $U(\ab)$ by $U(\qb)$.
Then the first Fourier coefficient has the form
$$
E^{\psi}(s,f,e)={W_{\infty}}(s)\frac{1}{\zeta(1+s)}, 
$$
for a certain Whittaker function ${W_{\infty}}(s)$ (see \cite{Kub}, p.46).
\end{theo}
%Montrons, comment transférer ce result pour le groupe p-adique $SL(2)$.
\begin{theo}[a $p$-adic  version, work in progress with S.Gelbart, S.Miller, F.Shahidi] %, \cite{GPS}] 
Let $p$ be a regular prime. 
Then there exists an explicitly given distribution   $\mu^*$ on  $\zb^{*}_p$ 
such that for all   $k \geq 3$ and for all primitive Dirichlet characters $\chi\bmod p^v$ with $\chi(-1)=(-1)^k$ one has 
$$
\int_{\zb^{*}_p} \chi y^{k}_{p} \mu^*=\frac{1}{(1-\chi(p) p^{k-1})L(1-k, \chi)}, 
$$
where  $L(s, \chi)$ 
is the Dirichlet $L$-function.
 More precisely, the distribution $\mu^{*}$ can be expressed through the non-constant Fourier coefficients of a certain   Eisenstein series $\Phi^{*}$.
 %, la distribution d'une variable attachée à la distribution $E^{*}$ de deux variables
 %(définie par une série  d'Eisenstein usuelle).
%To prove the Theorem we follow closely the "method of modular distributions" introduced in the 2003 {\it %Inventiones} paper of Panchishkin.
\end{theo}
\hskip-3pt{\large \blue Remark.}   Using Siegel's method for the symplectic groups
$GSp_m$, and for all primes $p$, this result also follows from Main Theorem (\ref{pMSE})
by specializing to the case of regular $p$ and $m=1$.

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


\

\end{frame}
\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{How to define the   distribution $\mu^*$?}
It is given by
%%%%%%%%%%%%%%%%%%%%%%%%%%
all its  moments for $k\ge 3$
$$
\int_{a+(p^m)}y^k\mu^*(y)=
\ell (\pi_1(\Phi^*_k(a+(p^m)))
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   $$\mu^{*}(a+(p^m))=l(\pi_{1}(E^*(a+(p^{m}))),$$
where $\Phi^*_k(a+(p^m))$
is a convolution of certain Eisenstein distributions $E^*_k(a+(p^m))$ defined using a natural summation procedure
\begin{align*}&
E^*_k(a+(p^m))=
\\ &
 \Gamma(k) p^{m(k-1)}\sum_{x,b \bmod p^m} e(-\frac {ax}{p^m})E^{*}_{k,p^m}(p^m z,x,b) \in \mc,
\end{align*}
obtained from the absolutely convergent Eisenstein series, 
$$
E^{*}_{k,p^m}(z,x,b)=\sum_{pgcd(c,d)=1 \atop (c,d)\equiv (x,b)\bmod p^m} (cz+d)^{-k},
$$
$\mc$ is a certain  {\it modular Igusa tower}.

A natural explanation of this distribution comes from the division by the constant term, and can be expressed 
in group-theoretical terms.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Computation of the   distribution $\mu^*$}
\begin{itemize} 

\item $\pi_{1}:\mc\to \mc^{1}$ is the (canonical) projection onto the characteristic subspace  
 $\mc^{1}$ defined by all the generalized eigenvectors of eigenvalue 1 
of Atkin's operator $U$, and
 
 
\item  $\ell:\mc^{1}\to{\qb}_p$ is a linear  form given by a certain linear combination 
of Fourier coefficients $\ell_m(\sum a_{n}q^n)=a_m$ for $m\ge 1$.
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}

\item In order to prove that  
$$
\int_{{\zb}^{*}_p} y^{k}_{p} \mu^*
$$ 
defines a bounded  distribution, one needs to check Kummer-type congruences for certain   non-constant Fourier coefficients 
 $a_{n}(\Phi^*(a+(p^m)))$ of modular forms $\Phi^*(a+(p^m))$, and varying integers $n$ and $m$.
\end{itemize}  
 
\end{frame}

\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}
\frametitle{Further applications: we only mention 
proof of a $p$-adic Myawaki Modularity Lifting 
Conjecture for Families  of Siegel Modular Forms %(a joint work with H.Kawamura)
} 

%%%%%%%%%%%%%%%%%%%%%%%%%%
\item{} Proof of a $p$-adic Miyawaki Modularity Lifting 
Conjecture by pullback of families  of modular  distributions with values in Siegel modular forms (jointly with
Hisa-Aki Kawamura, \dots)
%%%%%%%%%%%%%%%%%%%%%%%%

-Ikeda's constructions (2001, 2006) extend the {\magenta doubling method} to pullbacks of cusp forms instead of pullbacks of Eisenstein series. 
Both p-adic and complex analytic L-functions are produced in this way.

-Thus obtained cuspidal p-adic measures generalize the Eisenstein measure, and produce families of cusp forms

-A version of this construction Klingen-Eisestein series  and Langlands Eisenstein series.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For a prime $p$ and a positive integer ${n}$, we study some constructions of $p$-adic families of Siegel modular forms of genus ${n}$.
In a previous paper, 
by making use of certain lifting procedures, two conjectures on the existence of the modularity liftings from  
${\rm GSp}_{\,r} \times {\rm GSp}_{\,2m}$ to ${\rm GSp}_{\,r+2m}$ were formulated 
for some positive integers $r$ and $m$.% by the second author.
We describe a proof of the first of these $p$-adic Myawaki modularity lifting 
Conjectures, using the recent construction of the $p$-adic Ikeda modularity lifting by H.Kawamura.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\begin{frame}
\frametitle{Ikeda Lifts of Siegel Modular Forms}

Let us recall the Ikeda constructions (\cite{Ike01}, \cite{Ike06}):
let $k$ be an even positive integer,
${{\ph}}\in S_{2k}(\Gamma_1)$ a normalized cusp Hecke eigenform of weight $2k$,
%$F_2\in S_{k+1}(\Gamma_2) =Maass({{\ph}})$ the Maass lift of ${\ph}$,  and consider the cusp form\\
$g:=F_{2{m}}({\ph})\in S_{k+{m}}(\Gamma_{2{m}})$ the {\it Duke-Imamoglu--Ikeda lift} of ${{\ph}}$ of genus $2{m}$
(under the assumption $k\equiv {m}\bmod 2$, ${m}\in \mathbb{N}$).
It turns out that  this lift can be functorially characterized by the equality of $L$-functions
(see \cite{Ike01}):
$$
L(g, St, s)=L(F_{2{m}}(\ph), St, s)=\zeta (s) \prod_{i=1}^{2m}L(\ph, s+k+m-i).
$$ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\begin{frame}
\frametitle{$p$-adic versions of Ikeda constructions}
Consider a $p$-adic family 
$$
{\bold \varphi}=\{\ph_{2k}\} : 2k{}\mapsto {\ph}_{2k}=\sum_{n=1}^\infty a_{n}({2k})q^n \in {\overline{\mathbb{Q}}}[\![q]\!] \subset \mathbb{C}_p[\![q]\!], 
$$ 
where the Fourier coefficients $a_n(2k)$ of the normalized cusp Hecke eigenform ${\ph}_{2k}$ and one of the Satake $p$-parameters  $\alpha({2k}):=\alpha_p({2k})$ are given by  
certain $p$-adic analytic functions $k{}\mapsto a_n({2k})$ for $(n,p)=1$.  

Then we know that the Fourier expansions of the modular forms 
$
F=F_{2m}(\ph_{2k})
$
%\mbox{ and } 
%{\cal F}_{{\ph},f}={\cal F}_{{\ph}_{2k},f_{k+{m}+r}^{r}},
%$$
can be explicitly evaluated. It turns out that this sequence provide an example of a $p$-adic family of Siegel modular forms.
%(under certain congruence conditions on $k$, ${m}$ and $r$). % of positive slope.

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

\

%In order to state the first result, 
%consider  a $p$-adic family  $\{\ph_{2k}\}$ of elliptic modular forms of
% even weight $2k$ and of level 1.
%Here we make no assumption that the family $\{\ph_{2k}\}$ is ordinary or  non-ordinary. 
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\begin{theo}[\cite{Kawa}]
\label{pIk} After a suitable regularization  
the family 
\begin{align*}%\label{pIk}
F(\bold \varphi)=\{  g_{k+{m}}^{2{m}}\}, g_{k+{m}}^{2{m}} :=F_{2{m}}({\ph_{2k}})\in S_{k+{m}}(\Gamma_{2{m}})
\end{align*}
becomes a $p$-adic family of Siegel modular forms of weight $k+{m}$ and genus ${2{m}}$.
%(under certain congruence assumptions on $k$ and $m$).
\end{theo} 

Nice numerical examples for this theorem were given %recently 
by Hisa-Aki Kawamura
%(Hokkaido University (Sapporo, Japan)/ Institute Fourier (Grenoble, France)), see
in \cite{Kawa8}
%{\tiny \tt
%http://www-fourier.ujf-grenoble.fr/\~{}panchish/KawamuraCong11-19-23.pdf}\\
for certain $p$-adic families containing the Ramanujan cusp form $\Delta(z)=\ph_{12}(z)$, and the Hecke cusp eigenforms $f_{18}$ and $f_{20}$ with respect to ordinary primes $p=11$, $p=19$, $p=23$. %, and for a non-ordinary prime $p=5$. 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\fi

%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Ikeda-Miyawaki constructions (\cite{Ike01}, \cite{Ike06})}


Let $k$ be an even positive integer,
${{\ph}}\in S_{2k}(\Gamma_1)$ a normalized Hecke eigenform of weight $2k$,
%$F_2\in S_{k+1}(\Gamma_2) =Maass({{\ph}})$ the Maass lift of ${\ph}$,  and consider the cusp form\\
$g:=F_{2{m}}({\ph})\in S_{k+{m}}(\Gamma_{2{m}})$ the {\it Duke-Imamoglu--Ikeda lift} of ${{\ph}}$ of genus $2{m}$
(under the  assumption $k\equiv {m}\bmod 2$, ${m}\in \mathbb{N}$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next let
$f\in S_{k+{m}+r}(\Gamma_{r})$  be an arbitrary   Siegel cusp eigenform of genus $r$ and weight $k+{m}+r$,  with ${m},r\ge 1$. 

Let us consider the lift $F_{2{m}+2r}({\ph})\in S_{k+{m}+r}(\Gamma_{2{m}+2r})$, and the integral  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\small
\begin{align*}&
{\cal F}_{{{\ph}},f}=\\ &  \int_{\Gamma_r\backslash {\mathbb H}_r} F_{2{m}+2r}({\ph})\left(
\begin{pmatrix}z&\\ & z'
 \end{pmatrix}\right)
 f(z')(\det {\rm Im }(z'))^{k+{m}+r-1} dz'  \in S_{k+{m}+r}(\Gamma_{2{m}+r}),   
\end{align*}
which defines the {\it Ikeda--Miyawaki lift},
where $z\in {\mathbb H}_{2{m}+r}, z'\in {\mathbb H}_r$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
Under the non-vanishing condition on ${\cal F}_{{{\ph}},f}$, one has the equality
$$
L({\cal F}_{{{\ph}},f}, St, s)= L({f}, St, s)\prod_ {i=1}^{2{m}}L({\ph}, s+k+{m}+r-i).
$$

%\subsection{Examples of Non-vanishing of Ikeda-Miyawaki Lifting}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\frametitle{}
Let $k$ be an even positive integer,
${{\ph}}\in S_{2k}(\Gamma_1)$ a normalized Hecke eigenform of weight $2k$,
%$F_2\in S_{k+1}(\Gamma_2) =Maass({{\ph}})$ the Maass lift of ${{\ph}}$,  and
$F_{2m}({\ph})\in S_{k+m}(\Gamma_{2m})$ the Ikeda lift of ${{\ph}}$ of genus $2m$
(we assume $k\equiv m\bmod 2$, $m\in \mathbb{N}$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\

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

\begin{theo}[in progress]
\label{pMiyIk} Let $m=r=1, n=m+2r=3$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Consider two $p$-adic families 
\begin{align*}&
{\bold \varphi}=\{\ph_{2k}\}: 
k{}\mapsto \ph_{2k} =\sum_{n=1}^\infty a_{n}({2k})q^n \in {\overline{\mathbb{Q}}}[\![q]\!] \subset \mathbb{C}_p[\![q]\!], \\ &
{\bold f}=\{f_{k}\}:
k{}\mapsto {f}_{k+m+r}=\sum_{n=1}^\infty b_{n}({k+n+r})q^n \in {\overline{\mathbb{Q}}}[\![q]\!] \subset \mathbb{C}_p[\![q]\!]
\end{align*}

where the Fourier coefficients $a_n(2k)$ and $b_{n}({k+n+r})$
%of the normalized cusp Hecke eigenform ${\ph}_{2k}$
% and one of the Satake $p$-parameters  $\alpha({2k}):=\alpha_p({2k})$ 
are given by  
certain $p$-adic analytic functions. % $k{}\mapsto a_n({2k})$ for $(n,p)=1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
After a suitable regularization  
the family 
\begin{align*}&
{\cal F}_k:=G_{k+r+{m}}^{r+2{m}}
:=
{\cal F}_{{\ph}_{2k},f_{k+m+r}}\in S_{k+r+m}(\Gamma_{r+2m})
%F_{2{m}}({\ph_{2k}})\in S_{k+{m}}(\Gamma_{2{m}})
%\\ &
%(f, g)=(f, F_{2m}({\ph}))\mapsto {\cal F}_{{{\ph}_{2k}},f}\in S_{k+r+m}(\Gamma_{r+2m}), \\ &
%(f, g)=(f, F_{2m}({\ph}_{2k}))\in   S_{k+r+m}(\Gamma_{r})\times S_{k+m}(\Gamma_{2m}).
%(g, F_{2m})\in S_{k+3m}(\Gamma_{2})\times S_{k+m}(\Gamma_{2}).
\end{align*}
becomes a $p$-adic family of Siegel modular forms of weight $k+r+{m}$ and genus $r+{2{m}}$.
%(under certain congruence assumptions on $k$ and $m$).
%%%%%%%%%%%%%%%%%%% ***28/4
\end{theo} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=6]
\frametitle{Proof is very similar to our construction of $\Lambda$-adic Klingen-Eisenstein series (Section 5 of \cite{PaSE})}
with Koji Kitagawa (Hokkaido University).
We  constructed a  
% $p$-adic measure 
modular distribution 
 coming from 
 the Klingen--Eisenstein series on the symplectic group
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$
G=\GSp_{(2\cdot){n}}
$
% =\left\lbrace
%\al\in \GL _{2{n}}\left\vert\right. {}^t \al J_{n}\al =\nu(\al)J_{n},
%\nu (\al )
%\in \GL_1
%\right\rbrace,
%$$  
 %over $\Q$ where 
%$
%J_{n}=\begin{pmatrix}0 & I\\ -I & 0
%\end{pmatrix}.
%$
This distribution takes values in a space of $\Lambda$-adic Siegel modular forms and
 defines a
 ${\cal L}$-adic Siegel modular form where 
${\cal L}$ denotes
 the field of fractions of the Iwasawa algebra $\Lambda = {\Z}_{p}[[T]]$.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ùù

For $k>m+r+1$ and $m\ge r$ then the
 Klingen -- Eisenstein series is defined as the following absolutely convergent
 series  
$$
E_k^{m,r}(z,f, \chi)=\sum_{\ga \in \De_{m,r}\cap \Ga_n \backslash \Ga_n}
 \chi (\det (d_{\ga})) f(\w^{(r)} (\ga z))j(\ga ,z)^{-k},\eqno(5.1)
$$ 
 with $z\in  {\mathbb H}^n$, $\w (z)^{(r)}$ being 
 the upper left
 corner of $z$ of size $ r\times r$, $n=m+r$, $\ga =\begin{pmatrix}a_{\ga}&b_{\ga}\\ c_{\ga}&d_{\ga}
\end{pmatrix}$ and
$\De_{m,r}$ denotes the set of
 elements in $\Ga_n$ having 
 the form 
$\begin{pmatrix}*&*\\ 0_{{n}-r,{n}+r}&*
\end{pmatrix}$
 [Kl]. 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\frametitle{}
  Let $D(s, f, \chi)$ be the standard zeta function of $f\in \Sc_k^r(\Ga)$ as above (with local
factors of degree $2r+1$) and $\chi$ be a Dirichlet character. 
Then the essential fact for our construction is the Boecherer--Garrett--Shimura
 %following 
identity:
$$
\La (k, \chi)D(k-r, f, \eta)E_k^{m,r}(z,f, \chi)=
\big< f^\prime (w), 
E_k^{m+r, 0}(\diag [z, w])\rr. %\eqno (5.3)
$$
Here $\La  (k, \chi)$ is a product of special values of Dirichlet
 $L$-functions and
 $\Ga$-functions, $\eta$ is a certain Dirichlet character,
 $E_k^{m+r, 0}$ a series of type (5.2) transformed by a suitable element of
 $G^{m+r}$,
 $f^\prime$ an easy transform of $f$, $(z, w)\in \Hb_m\times \Hb_r$
 (see \cite{Boe85}, [Shi95, (7.4), p.572]). 
 
The construction is based on the fact the  Fourier coefficients of the series $E_k^{m+r, 0}(\diag [z, w])$ %(5.2)  
produce   $p$-adic
 measures (the Siegel--Eisenstein measure).
 %, see [Pa7].
  \end{frame}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Modular Forms %, Involution 
and $q$--expansions}
We put 
 $G^{n}=Sp_{n}({\Q})$ and 
 $G^{m,r}=Sp_{m}({\Q})\times Sp_{r}({\Q})$.
We define the congruence subgroup $\Gamma^{n}_{0}(p^{\alpha})$ in a usual
 way and put
$\Gamma^{m,r}_{0}(p^{\alpha})
   =\Gamma^{m}_{0}(p^{\alpha})\times \Gamma^{r}_{0}(p^{\alpha})$.
Let $B_{n}$ be the semigroup of symmetric, semi--definite, half--integral
 matrices of size $n$. 
We put $B_{m,r}=B_{m}\times B_{r}$.

Let $G$ be one of the groups $G^{n}\;(n=1,2,3, 4)$ or $G^{3,1}$. 
Let $\Gamma_{0}(p^\al)$ be the one of $\Gamma^{n}_{0}(p^{\alpha})\;(n=1,2,3, 4)$
 or 
$\Gamma^{3,1}_{0}(p^{\alpha})$ according to $G$. 
Let $B$ be the one of $B_{n}\;(n=1,2,3, 4)$ or $B_{3,1}$ according to $G$. 
Then we may consider classical and $\Lambda$-adic modular forms
\begin{align*}&
 M_{k}(\Gamma_{0}(p^{\alpha}),\psi)\subset \overline \Q[[q^B]]\ \ 
 \mbox{and}  \\ &  M(G, \Lambda)\subset \Lambda[[q^B]] 
 \\ & \mbox{where}\ \ 
 M^{3,1}_{k}(\Gamma_{0}^{3,1}(p^{\alpha}),\psi)
=\{f\in M^{3,1}_{k}(\Gamma_{1}^{3,1}(p^{\alpha}))\ \ 
\\ &
\big| \ \ 
          f|_{k}(g_{3},g_{1})=\psi (\det(d_{3})\cdot d_{1})f
            \;\;{\rm for\  any\  }\;(g_{3},g_{1})\in \Gamma_{0}^{3,1}
(p^{\alpha})\} 
 \end{align*}
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{$\Lambda$-adic modular forms.}
Let $\chi=\omega^{a}$ with some $a$ $(0\leq a <p-1)$,
 $\omega: \Z_p^\times \to \mu_{p-1}$ the
 Teichm\"uller character.
Let $G$ be one of the groups $G^{n}\;(n=1,2,3,4)$ or $G^{3,1}$. 
Let $\Gamma_{0}(p^\al)$ be the one of the groups
 $\Gamma^{n}_{0}(p^{\alpha})\;(n=1,2,3, 4)$ or 
 $\Gamma^{3,1}_{0}(p^{\alpha})$ according to $G$. 
Let $B$ be the one of $B_{n}\;(n=1,2,3, 4)$ or $B_{3,1}$ accoding to $G$. 
{\magenta Let $\Pc$ be a Zariski dense subset of $\Spec\,\Lambda (\bar{\Q}_{p})$.
We call an element $F$ of $\Lambda [[q^{B}]]$ a $\Lambda$-adic modular
 forms on $G$ 
 with character $\chi$ with respect to $\Pc$
 if $F((1+p)^{k}-1)$ gives a $q$-expansion of modular forms in 
 $M_{k}(\Gamma_{0}(p^{\alpha}),\epsilon\chi\omega^{-k})$ for all $(k,\epsilon)$
 such that
 $P_{k,\epsilon}\in {\Pc}$.  
We shall take ${\Pc}={\Pc}(5)=\{P_{k,\epsilon};k\geq 5\}$.}
We denote by $M(G,\chi;\Lambda)$ the $\Lambda$--submodule of 
 $\Lambda [[q^{B}]]$ generated by $\Lambda$-adic modular forms on $G$.

We use the symbol $M^{n}(\chi;\Lambda)\;(n=1,2,3, 4)$ or $M^{3,1}(\chi;\Lambda)$
 for $M(G,\chi;\Lambda)$ according to the group $G$. 
Then formally one has
$$
A[[q^{B_{3,1}}]]=A[[q^{B_{3}}]]\hat\otimes_{A}A[[q^{B_{1}}]] .
$$
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{frame}
\frametitle{ The pullback of $\Lambda$-adic modular forms}
Let $\pi:B_{4}\rightarrow B_{3}\times B_{1}$ be a map given by
$$
\begin{pmatrix} a_{11} & a_{12} & a_{13}& a_{14} \\ 
                                         a_{21} & a_{22} & a_{23} & a_{24}  \\
                                         a_{31} & a_{32} & a_{33} & a_{34} \\
                                         a_{41} & a_{42} & a_{43} & a_{44}   
\end{pmatrix}                                         
  \mapsto 
\left(\begin{pmatrix}   a_{11} & a_{12} & a_{13} \\
                                         a_{21} & a_{22} & a_{23 }\\
                                         a_{31} & a_{32} & a_{33}          
\end{pmatrix}, a_{44}\right)
$$
We note that for each $(\xi_{3},\xi_{1})\in B_{3,1}$ the fiber 
$\pi^{-1}(\xi_{3},\xi_{1})$ is a {\it finite} set. 
For $F=\sum a(\xi_{4},F)q^{\xi_{4}}\in \Lambda [[q^{B_{4}}]]$ we define
$\Phi(F)\in \Lambda [[q^{B_{3,1}}]]$ by 
$$
a(\xi_{3},\xi_{1},\Phi(F))
 =\sum_{\xi_{4}\in \pi^{-1}(\xi_{3},\xi_{1})}a(\xi_{4},F), 
 $$
where $\Phi$ is a $\Lambda$--module homomorphism. 
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{
We let the Hecke operator $T_{1}(p)$ act on 
$M^{3,1}(\Gamma_{0}^{3,1}(p^{\alpha}),\psi)$.} 
We put ${\displaystyle e_{1}=\lim_{n\rightarrow\infty} T_{1}(p)^{n!}}$, and
$$
 M^{3,1-ord}(\Gamma_{0}^{3,1}(p^{\alpha}),\psi)
  =e_{1}M^{3,1}(\Gamma_{0}^{3,1}(p^{\alpha}),\psi) 
  =\pi_{ord}M^{3,1}(\Gamma_{0}^{3,1}(p^{\alpha}),\psi).
$$
We call $f\in M^{3,1}(\Lambda,\chi)$ $1$-ordinary if its specialization at 
$P_{k,\epsilon}$ is in
 $M^{3,1-ord}(\Gamma_{0}^{2,1}(p^{\alpha}),\psi;{\cal O})$ for any
 $P_{k,\epsilon}\in {\Pc}$.
We denote by $M^{3,1-ord}(\Lambda,\chi)$ the space of $1$-ordinary
 $\Lambda$-adic modular forms,
 and $M^{ord}(\Lambda,\chi)$ the space of
 ordinary
 $\Lambda$-adic modular forms in the sense of Hida.
 \begin{prop}
$$
M^{3,1-ord}(\chi; \Lambda)
  = M^{3}(\chi; \La)\otimes_{\Lambda} M^{ord}(\chi; \Lambda)
$$
 \end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{
Proof.}

Let $f^i\in M^{ord}(\chi;\La)$ ($i=1, 2, \dots, n$) be a basis.
Define $\La$-adic linear forms 
$$
l^i: M^{ord}(\chi;\La)\to \La \ \ \ (i=1, 2, \dots, n)
\ \ \ {\rm  such\ \  that}\ \  l^i(f^j)=\de_{i, j}.
$$
Let $P=P_{k, \ep}\in Pc$, $G\in M^{3,1-ord}(\chi;\La)$,
$G_P\in M_k^{3,1-ord}(\Ga _0(p^\al), \chi \ep \w^{-k};\Oc)$.
We have that $M_k^{3,1-ord}(\Ga _0(p^\al), \chi \ep \w^{-k};\Oc)$
$$
\cong M_k^{3}(\Ga _0(p^\al), \chi \ep \w^{-k};\Oc)\otimes_{\Oc}
M_k^{ord}(\Ga _0(p^\al), \chi \ep \w^{-k};\Oc),
$$
and there exist $g^i_P$ such that  $G_P=\sum_{i=1}^nf_P^i\otimes g^i_P$.
Put 
$$
R_P^i=l_P^i\otimes id\ :\ G_P\mapsto (l_P^i\otimes id)(\sum_{j=1}^n 
f_P^j\otimes g^j_P) =
1\otimes g_P^i = g_P^i.
$$
This gives an $\Oc$--module homomorphism $R_P^i=l_P^i\otimes id$ :
$$
M_k^{3,1-ord}(\Ga _0(p^\al), \chi \ep \w^{-k}; \Oc)
\to 
M_k^{3}(\Ga _0(p^\al), \chi \ep \w^{-k};\Oc).
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{A $\La$-adic lift of $R_P^i$ or of $g_P^i$.}
We apply the following patching lemma to $q$-expansions of $\La$-adic forms,
 coefficient by coefficient:


%Lemma. 
\begin{lemma}
 Suppose that for each $P\in \Pc$ we are given a $g_P\in \La/P\La$ such that
they all are compatible (i.e. $g_P$ and $g_Q$ map to the same $g_{P+Q}\in \La/(P+Q)\La$).
Then there exist $g\in \La$ such that 
$$
\forall P\in \Pc \ \ \ g_P\equiv g\bmod P\ \in \ \La/P\La.
$$
\end{lemma}
(see [Wi87], [Hi93], p. 232).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink] 
\frametitle{$\Lambda$-adic Peterson inner product}

We follow Hida's idea of algebraic Peterson inner product ([Hi93,
 p.222]). 
Let $S^{ord}(\chi; \Lambda)$ denote the space of $\Lambda$-adic cusp forms with character
$\chi$.  
Let $h^{ord}(\chi; \Lambda)$ be the $p$-adic Hecke algebra with character $\chi$.
We note that $S^{ord}(\chi; \Lambda)$ and $h^{ord}(\chi; \Lambda)$ are the 
$\Lambda$--dual of each other by the pairing 
$$
\langle f, h\rangle = a(1,f|h).
$$
We put 
$$
D=h^{ord}(\chi; {\cal L})=h^{ord}(\chi; \Lambda)\otimes_{\Lambda}{\cal L}.
$$
Since $D$ is semi--simple there is a non  degenerate pairing $(\cdot,\cdot)$ on $D$ given by
$$
(h,g)=Tr_{D/{\cal L}}(hg).
$$
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{By duality we have the dual pairing $(\cdot,\cdot)$ on $S^{ord}(\chi;{\cal L})$.}
If $F\in S^{ord}(\chi;\Lambda)$ is the normalized eigenform then $(F,G)$
%$$
%c(F,G)={(F,G)\over (F,F)} \in {\cal L}
%$$
 is well defined.
We write $M(\chi;  \Lambda)=M^{1}(\chi;  \Lambda)$. 
Let $M^{ord}(\chi;  \Lambda)$ denote the ordinary part and let $\pi_{ord}$
 denote the
 projection to the ordinary part.
The Hecke algebra $H^{ord}(\chi;  \Lambda)$ acts on $M^{ord}(\chi;  \Lambda)$.
 $H^{ord}(\chi;  {\cal L})=H^{ord}(\chi;  \Lambda)\otimes_{\Lambda}
 {\cal L}$ is a direct sum of
 $h^{ord}(\chi;  {\cal L})$ and the Hecke algebra over $\cal L$ corresponding
 to Eisenstein series. 
Let $1_{cusp}$ denote the idempotent corresponding to
 $h^{ord}(\chi;  {\cal L})$. 
We define a
 $\Lambda$--bilinear map $\langle \cdot, \cdot \rangle:
    S^{ord}(\chi;  \Lambda)\times M(\chi;  \Lambda)\rightarrow {\cal L}$ by
$$
\langle F,G\rangle=(F,1_{cusp}\cdot \pi_{1-ord}(G)).
$$
 \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink] 
\frametitle{$\Lambda$-adic kernel function}
Let $G\in M^{3,1}(\chi;  \Lambda)$. We write 
$$
G=\sum_{j\in J} g_{j}\otimes h_{j}
$$
with $g_{j}\in M^{3}(\chi;  \Lambda)$ and $h_{j}\in M^{1}(\chi;  \Lambda)$. 
We define a $\Lambda$--linear map
$R(G):S^{ord}(\chi;  \Lambda)\rightarrow
 M^{3}(\chi;  \Lambda)\otimes {\cal L}$ by
$$
R(G)=\sum_{j\in J} g_{j}\langle \cdot,h_{j}\rangle.\eqno(**)
$$
In the definition (**) we used the $\Lambda$--bilinear map
 $\langle \cdot, \cdot \rangle:
    S^{ord}(\chi;  \Lambda)\times M(\chi;  \Lambda)\rightarrow {\cal L}$
 of the previous section.  
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[shrink]
\frametitle{A $\Lambda$-adic Ikeda-type Integral}
\begin{defi}%[$\Lambda$-adic Klingen--Eisenstein series]
We may call
$$
{\cal F}_\La ({\bf f}, \chi) = %\frac 1 {\big< {\bf f}, {\bf f} \big>} 
 R(\Phi(F_{\Lambda}(\bold \varphi))({\bf f})\in M^{3}(\chi;  \Lambda)\otimes_{\Lambda}{\cal L}
$$
 the $\Lambda$-adic Ikeda-type Integral for a
 $\Lambda$-adic cusp form 
${\bf f}\in S^{ord}(\chi;  \Lambda)$. 
\end{defi} 
It follows from our construction that there is a precise relation of 
$$
%\frac 1 {\big< {\bf f}, {\bf f} \big>} 
 R(\Phi(F_{\Lambda}(\bold \varphi))({\bf f}) \eqno (*)
$$
with the complex-analytic Ikeda lift:
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
{ Let $m=1$, $r=1$.} For all algebraic points $P=P_{k, \ep}\in {\cal P}$ we have that  
the specialization of (*) at $P$ is given by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*}
{\cal F}&_{{{\ph_{2k}}},f_{k+n+r}}= \\ & \int_{\Gamma_r\backslash {\mathbb H}_r} F_{2{m}+2r}({\ph_{2k}})\left(
\begin{pmatrix}z&\\ & z'
 \end{pmatrix}\right)
 f_{k+n+r}(z')(\det {\rm Im }(z'))^{k+{m}+r-1} dz' \\ & \in S_{k+{m}+r}(\Gamma_{2{m}+r})
 = S_{k+2}(\Gamma_{3}),   
\end{align*}

{\sl Proof } %of (5.15)
 (in case $m=1$, $r=1$) is based on this identity, the definition of
 $E_{\Lambda}(\chi)$,
 of the kernel function 
$$
 R(G) : S^{ord}(\chi;  \Lambda)\rightarrow 
 M^{3}(\chi;  \Lambda)\otimes {\cal L}.
$$ 
%of 5.6.
\begin{coro} The non-vanishing condition is satisfied for infinitely many cases
$F^3_{k+2}, k\equiv 12 \mod (p-1)p^{m}$
\end{coro}
{\it Questions}~1)~Is $L(s,F^3_{k+2})
=L(s-k+1, f_{k+2})L(s-k, f_{k+2})
\times L(s, f_{k+2}\otimes \ph_{2k})$ ?\\
2) Compute the Fourier coefficients of 
$F^3_{k+2}$. %Is there theta function attached?
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%







%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{III. $p$-adic interpolation of the special  $L$-values  
and $p$-adic $L$-functions of two variables
for ${\rm GSp}(4)\times \GL(2)$ 
(a joint project with S.Böcherer, B.Heim, ...)} 

%A similar subject (for fixed weights in the critical strip) 
%was treated by Boecherer and Heim using formulas for the special values of such $L$-function.


Let $Z_{G\otimes h}(s)$  denote the tensor product $L$-function attached to a Siegel
modular form $G$ of weight $k$ and an elliptic cusp form $h$ of weight $l$. 
Assume that the first Fourier-Jacobi coefficient of $G$ is not identically zero. 
Then Deligne's
conjecture is fully proven in \cite{BH06} for $Z_{G\otimes h}(s)$when $l \le 2k-2$. 
%and partly for the remaining case.
%\iffalse

A conjecture is discussed which  is similar in nature
to the "unbalanced case" for triple-$L$-functions, 
%\iffalse
According to M.Harris and S.Kudla, 
The central critical value of a triple product $L$--function, 
Annals of Math. 133 (1991), 605-672,  under
the  restriction $l + m< k$
the corresponding normalized critical value is a square in the field ${\mathbb Q}(f, g, h)$ generated by the
Fourier coefficients of $f$, $g$, $h$.
%\fi
\end{frame}
\begin{frame}
\frametitle{Main ingredients of the construction}
\begin{itemize}
 \item  Critical values of L-functions on $GSp_2\times Gl_2$
Math. Z. (2006) 254: 485-503 Math. Z. by S.Böcherer, B.Heim

This paper deals with Deligne's conjecture on the critical values of
$L$-functions. 

\item ``$p$-adic $L$-function for ${\rm GSp}(4)\times{\rm GL}(2)$''
by Mahesh Agarwal, 
PhD Thesis,  The University of Michigan
2007, Advisor: Christopher M. Skinner.

Here a $p$-adic interpolation of the spacial values in the "weight direction" was constructed using $\Lambda$-adic Siegel-Eisenstein series
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Siegel modular forms of genus 3 and $p$-adic $L$-functions
}

Let  $\Gamma={\rm Sp}_{n}(\Z)\subset {\rm SL}_{2{n}}(\Z)$ be the   Siegel modular group of genus  ${n}$, and for a prime $p$, let  ${\mathcal L}_{n,p}$ %$\mathbf{T}(p)=\mathbf{T}(\underbrace{1, \cdots, 1}_{{n}}, \underbrace{p, \cdots, p}_{{n}})$
denote the  $p$-local Hecke algebra  of $\Gamma$.
Let  $\displaystyle f=\sum_{\mathcal{T}\in B_n} a(\mathcal{T})q^\mathcal{T}\in \mathcal{S}_k^n$ be a Siegel cusp modular form of weight $k$ and of genus $n$
on the Siegel upper-half plane $\mathbb{H}_n=\{z={}^tz\in \mathrm{M}_n(\mathbb{C})  | \mathrm{Im}(z) >0 \}$, where 
$
q^{\mathcal T}= \exp(2\pi i \, {\rm tr}({\mathcal T}z))$,  
and
$\mathcal{T}$ is in the semi-group
 $
B_n =\{{\mathcal T}={}^t{\mathcal T}\ge 0\vert {\mathcal T}\hbox{ half-integral}\}.
$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose that   $f$ is an eigenfunction of all  {\it Hecke operators}, and
$L(Sp(f), s)$ is the spinor  $L$-function of $f$ (for $s\in \mathbb{C}$, and for
all Dirichlet characters) $\chi$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
According to  \cite{Pa94}, \cite{Y}, and to general Deligne's conjecture \cite{De},
the $L$-function satisfy a functional equation  %conjecturale
determined by the following $\Gamma$-factors  for $n=3$ and $k\ge 5$, $\Lambda(Sp(f), s)=\Lambda(Sp(f), 3k-5-s)$, 
$
\Lambda(Sp(f), s)=\Gamma_\C(s)\Gamma_\C(s-k+3)\Gamma_\C(s-k+2)\Gamma_\C(s-k+1)L(Sp(f), s).
$
For $k\ge 5$ the critical values in the sense of Deligne \cite{De} are
$
s=k, \cdots, 2k-5.
$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsubsection*{Critical values, periods and $p$-adic %$L$-functions for ${\rm Sp}_3$}
%, see \cite{Boe-Pa6}, \cite{Ha-Li-Sk}, \cite{Pa94}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{We compute the critical values  for  three following $L$-functions}
\begin{align*}&
L(Sp(F_{12}), \chi, s)= L(s-9, \chi,\Delta)L(s-10,  \chi,  \Delta)L(s, \chi,\Delta\otimes g_{20}),\ \\ &(\mbox{the critical strip }s=12, \dots, 19), \\ &
 L(Sp(F_{14}), \chi, s)= L(s-12, \chi,\Delta)L(s-13,\chi, \Delta)L(s, \chi,\Delta\otimes g_{26}),\\ & (\mbox{ the critical strip }s=14, \dots, 23)\\ & \mbox{ and }
 L(f\boxtimes g,\chi;s), 
 (\mbox{the critical strip }s=k, \dots, 2k-5)
\end{align*}
(the Miyawaki $L$-functions,  and the  $L$-function
 of Boecherer-Heim $L(f\boxtimes g,\chi;s)$, where $f$ is an elliptic cusp eigen form of weight $k-2$, and $g$ is a Siegel cusp eigen form of degree 2 and of weight $k\ge 5$, see \cite{Mi92},\cite{BH06}).
The $L$-function 
$L(f_{k-2}\boxtimes g_{k}^{(2)},\chi;s)$ presumably coincides with 
$L(Sp(F), \chi, s)$
 for a Siegel modular form $F$ of genus 3 and of weight $k$.
 Using the techniques of \cite{Boe-Pa9}, we construct the attached $p$-adic  $L$-function by interpolating 
 the special values, computed in \cite{BH06}.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
\begin{frame}
\frametitle{ $L$-functions for $GSp(4)\times GL_2$ in the case of high $GL_2$ conductor}
%\iffalse
A. Pitale and R Schmidt \cite{PiSch9}
used an integral representation for the degree 8 $L$-function
of   $GSp(4)\times GL_2$  given by Furusawa \cite{Fu93} who carried out the unramified calculation. 
%\fi
The local $p$-adic zeta integrals were calculated  under the assumption that the
%\fi
$GSp(4)$ representation $\pi$ is unramified 
%\fi
and the $GL_2$ representation $\tau$ has conductor $p$;
they generalized to the case where the $GL_2$ representation has arbitrarily
high conductor. Their result is that the zeta integral represents the local Euler factor
$L(s, \pi \times \tau)$ in all cases. 
%\fi
As a global application they obtained a special value result for a $GSp(4)\times GL_2$ global $L$-function coming from classical holomorphic cusp forms with arbitrarily high level for the elliptic modular form.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
\fi
\fi


%V 
\section{Further applications: doubling method and Ikeda-Miyawaki constructions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 42
\begin{frame}[shrink]
\frametitle{Further applications: we only mention the proof of the $p$-adic Miyawaki Modularity Lifting Conjecture}
 by pullback of 
families  %of modular  distributions with values in 
Siegel modular forms (jointly with
Hisa-Aki Kawamura), see \cite{Kawa}, \cite{PaIsr11}.
%%%%%%%%%%%%%%%%%%%%%%%%

\

Ikeda's constructions (\cite{Ike01}, \cite{Ike06}) extend the {\magenta doubling method} to pullbacks of cusp forms instead of pullbacks of Eisenstein series. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
In the Fall 1999 in IAS, Ilya was much inspired by the preprint of the first Ikeda's  lifting,  and tried to interpret it representation-theoretically.

Indeed, it extends his own work \cite{PS1} on Saito-Kurokawa lifting from genus 2 to arbitrary genus $2m$.

In fact, there is a relation of Ikeda's work to Arthur's conjecture \cite{Ar89}.  

\

In the same period, Ilya studied  the preprint of 
{\magenta \cite{KMS2000} on $p$-adic Rankin-Selberg $L$-functions in an informal  seminar in his office in IAS
together with me and other participants:  Jim Cogdell, Siegfried Böcherer, Reiner Schulze-Pillot, \dots} 
\fi

\

%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 43
%\begin{frame}[shrink]

{\magenta The use of the 
The Eisenstein family 
${\mathcal E}_k^{(n)}$} as above plays a crucial role in Ikeda's work:
the idea was to substitute the Satake parameter $\alpha_p(k)$ of a cusp form in place of the 
parameter $k$ in the Siegel-Eisenstein family.  

Both $p$-adic and complex analytic $L$-functions are produced in this way.

Thus obtained cuspidal $p$-adic measures generalize the Eisenstein measure, and produce families of cusp forms.

A version of this construction produces Klingen-Eisenstein series 
 and Langlands Eisenstein series, see \cite{PaSE} ($p$-adic Peterson product of a cusp form with a pullback of the constructed family), more recently used by Skinner-Urban \cite{MC}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\

For genus two, my student P.Guerzhoy found  %in 1998 
a $p$-adic version of the {\magenta holomorphic Maass-Saito-Kurokawa lifting} \cite{Gue}, answering a question of E.Freitag.  
%P.Guerzhoy visited Ilya here in Yale in 1999. 

\

\end{frame}

\section*{ Références}

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

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\

\ifslide
{\large\blue References}
\fi


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






%\newpage
\bibliographystyle{plain}
\begin{thebibliography}{10xxxx}



\iffalse
\bibitem[Ag07]{Ag07}
{\sc Mahesh Agarwal},
$p$-adic $L$-function for ${\rm GSp}(4)\times{\rm GL}(2)$,
PhD Thesis,  The University of Michigan
2007, Advisor: Christopher M. Skinner.
\fi



\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, 119 - 131


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%J. ARTHUR, 
\bibitem[Ar89]{Ar89}
{\sc Arthur}, J., 
{\em Unipotent automorphic representations: Conjectures.} in:
 Orbites unipotentes et représentations, II, Astérisque 171-172, Soc. Math. France,
Montrouge, 1989, 13-71. %MR 1021499

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

%\bibitem[An74]{An74}
%{\sc Andrianov, A.N.}, 
%{\em Euler products corresponding to Siegel modular forms of genus 2},
%Russian Math. Surveys, 29:3 (1974), pp. 45--116, (Uspekhi Mat. Nauk 29:3 (1974) pp. 43--110).

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

\iffalse

\bibitem[BerDa98]{BerDa98}
{\sc Bertolini},  M., {\sc Darmon}, H.: 
Heegner points, $p$-adic $L$-functions and the
\v Cerednik-Drinfeld uniformization. Invent. Math. 131(3), 453-491 (1998)



\bibitem[BerDa09]{BerDa09}
{\sc Bertolini},  M., {\sc Darmon}, H.: 
The rationality of Stark-Heegner points over genus fields of real quadratic fields.  Ann. of Math. (2)  170  (2009),  no. 1, 343--370.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Boe-Pa6]{Boe-Pa6}
{\sc B\"ocherer}, S.,  {\sc Panchishkin},  A.A., 
{\em Admissible $p$-adic measures attached to triple products of elliptic cusp forms}.
Documenta Math. Extra volume~: John H.Coates' Sixtieth Birthday (2006), 77-132.
%\ifslide
%\newpage
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\end{document}
