

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
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%\documentclass[a4paper, 14pt, noxcolor]{article}
%\documentclass[handout, accumulate, noxcolor]{beamer} 
%\documentclass[accumulate, xcolor, 12pt]{beamer} 
\documentclass[handout]{beamer}

%\documentclass{article}
\usepackage{beamerarticle}
\usepackage{beamerseminar}
\usepackage{beamertexpower}
%\usepackage{noxcolor}
%\usepackage{beamerfoils}
%\mode<presentation>
\mode<presentation>
{
  \usetheme{Boadilla}% Madrid Singapore Montpellier
  \setbeamercovered{transparent}
}
%\pagestyle{plain}
%\markboth{left head}{right head}
%%%%%%%%%%%%%%%%%%%%%%%%%
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%{
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%{%
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%\begin {center}
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%{
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%\uncover<+(1)->{
%\hskip5pt
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%\insertframenumber}
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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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{graphics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\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}[theo]{\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\Fr{{\rm Fr}}
%\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{\Cp}{{\mathbb C}_p}
\newcommand{\Qp}{{\mathbb Q}_p}
\newcommand{\Zp}{{\mathbb Z}_p}
\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{\Sr}{{\mathcal S}}
\newcommand{\Sc}{{\mathcal S}}
\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 \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 \exa  {\medskip\noindent {\sc\blue Example.}\ }
\def \sol  {\medskip\noindent {\it\blue Solution.}\ }
\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 \Det{{\rm Det}}
\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{\PP}{\mathbb P}
\newcommand{\QQ}{\mathbb Q}
\newcommand{\RR}{\mathbb R}
\newcommand{\ZZ}{\mathbb Z}


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

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


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

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



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

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

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

\font\tenirus=wncyi10
\font\eightirus=wncyi8
\newfam\Irus
\textfont\Irus=\tenirus
\scriptfont\Irus=\eightirus
\def\iru{\fam\Irus\tenirus}
%\def\sir{\fam\Irus\eightirus}
\font\tenrus=wncyr10
\font\sevenrus=wncyr7
\newfam\Rus
\textfont\Rus=\tenrus
\scriptfont\Rus=\sevenrus
\def\ru{\fam\Rus\tenrus}
\def\scr{\fam\Rus\sevenrus} 
\input cyracc.def
%%%%%%%%%%%%%%%%%%%%%%
\def\i1{\accent'044i}
\def\I1{\accent'044I}
\def\e1{\accent'040e}
%\newcommand{\eo}{e1}
%\newcommand{\EO}{E1}
\def\l1{l{}p1}
\def\fraco #1,#2,{{{#1}\over {#2}}}%\fi







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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[p-adic L-functions and modular forms 
]
{
%Effective Arithmetic,  Motives and $L$-Functions
p-adic L-functions and modular forms 
}


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

\institute[Grenoble]{ \large 
Lectures for the International 
 Summer School and Conference \\ \red
 p-adic Analysis and its Applications
% Mathematics, Algorithms and Proofs
\\
August 31th
to September 18th, 2009
(Trieste - ITALY)
}


%\hskip-0.5cm
\date[ICTP, September,2009] % (optional)
{September, 2009}
%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}
%\logo{\pgfuseimage{logo-ujf}}



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}
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\def\ZZ{\mathbb Z}

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\addtocounter{ncours}{1}}
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  \frametitle{Contents
%Table des matières
}}
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  % Die Option [p% %%ausesections] könnte nützlich sein.
\end{frame}%\end{document}
\fi
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  INTRODUCTION
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Introduction}

%\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=20]
\frametitle{Abstract}

\

 {\large\blue 1) Congruences and  $p$-adic numbers. Hensel's lemma. The Tate field} 

\
 
 {\large\blue 2) Continuous  and  $p$-adic analytic functions. Mahler's criterion. Newton polygons} 
 
 Zeroes of analytic functions. The Weierstrass preparation theorem and its generalizations. 

\
 
 {\large\blue 3) Distributions, measures, and the abstract Kummer congruences.}
 
 The Kubota and Leopoldt $p$-adic $L$-functions  and the Iwasawa algebra 

\
 
{\large\blue 4) Modular forms and  $L$-functions. %, modular symbols. 
} 

Congruences  %Galois representations 
and families of modular forms. 
%Applications to  the proof of Serre's Modularity Conjectures

\

{\large\blue 5) Method of canonical projection of modular distributions.}

 Examples of construction of $p$-adic $L$-functions 

\end{frame}
\begin{frame}[shrink]\frametitle{Related topics (for discussions, not included in the text of these materials)}

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


\

 {\large\blue 6) Other approaches of constructing p-adic L-functions} by Mazur-Tate-Teitelbaum, J.Coates, P.Colmez, H.Hida ... (using modular symbols by Manin-Mazur and their generalizations; Euler systems, work of  D.Delbourgo, T.Ochiai, L.Berger, ..., overconvergent modular symbols by R.Pollack, G.Stevens,  H.Darmon, ...)

\

 {\large\blue 7) Relations to the Iwasawa Theory} 

\

 {\large\blue 8) Applications of $p$-adic $L$-functions to Diophantine geometry} 

\

 {\large\blue 9) Open questions and problems in the theory of $p$-adic $L$-functions} 
(Basic sources:  Coates 60th Birthday Volume, Bourbaki talks by P.Colmez, J.Coates ...)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
 {\large\blue 
 6)  Local and global methods in arithmetic. The Birch and Swinnerton-Dyer Conjecture. \\ Constructing rational points using complex and $p$-adic $L$-functions. 
Fermat's Last Theorem and Modularity of Elliptic Curves
}


see
'Efficient computations of
$p$-adic $L$-functions via
overconvergent modular symbols
(and applications to Stark-Heegner points)' by 
Robert Pollack, Boston University, 
http://math.bu.edu/\~{}rpollack/

 William Stein, `An Explicit Approach to 
Number Theory'; 

Henri Cohen,
Number Theory Vol I: Tools and Diophantine Equations Vol II: Analytic and Modern Tools,  Springer-Verlag - Graduate Texts in Mathematics 239 and 240, May 2007)





 {\large\blue 7)  Open questions and problems in the theory of $p$-adic $L$-functions}
\fi

\end{frame}

\iffalse
\begin{frame}{Abstract (continued)}



{\large\blue 3) Motivic $L$ functions and Galois representations.} 

Weil conjectures. Complex and 
$p$-adic $L$-functions attached to motives and their use. Automorphic $L$-functions and Artin's Conjecture.

(see \cite{Ma-Pa05}, Ch. 6)

\

 {\large\blue 4) Modular forms and congruences. Deformation theory of Galois representations and families of modular forms. Applications to  the proof of Serre's Modularity Conjectures} 
 
 (Khare-Wintenberger-Kisin et al., 2007,  
%(http://en.wikipedia.org/wiki/Serre\_conjecture%28number_theory%29
http://www.math.utah.edu/\~{}shekhar/papers.html)

The relation to the proof of the Sato-Tate Conjecture (2006,  L.Clozel, M. Harris, N.Shepherd-Barron and R.Taylor).
%(see \cite{Ma-Pa05}).


\end{frame}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  lecture1  6.2.3.




\section{Congruences and  $p$-adic numbers. Hensel's lemma. The Tate field}
%$L$-values attached to modular forms}

\begin{frame}[shrink=10]
\frametitle{\ncours {$p$-adic numbers and congruences} }
Originally $p$-adic numbers were invented by Hensel as a tool of solving congruences 
modulo powers of a prime number $p$. 

\exa $p=7$.\ Solve the congruence $x^2\equiv 2\bmod 7^n$.



\sol If $n=1$, put $x_0=\pm 3$ then $x_0^2\equiv 2\mod 7$.


If $n=2$, put $x_1=x_0+7t_1, x_0=3$ then $(x_0+7t_1)^2\equiv 2\mod 7^2$ gives:
\\
$
9+6\cdot 7t_1+7^2t_1^2 \equiv 2\mod 7^2 \Rightarrow 9+6t_1\equiv 0\mod 7\Rightarrow t_1=1
$
\\
$
\Rightarrow x_1=3+7\cdot 1=10.
$

If $n=3$, put $x_2=x_1+7^2t_2, x_1=10$ then $(10+7^2t_2)^2\equiv 2\mod 7^3$ gives:
\\
$
100+20\cdot 7^2t_2+7^4t_2^2 \equiv 2\mod 7^3 \Rightarrow  t_2\equiv -2/20\mod 7 \Rightarrow t_2\equiv 2\mod 7
$
\\ 
$
\Rightarrow x_2=3+7\cdot 1+2\cdot 7^2=108.
$

In this way we obtain a sequence $x_0, x_1, x_2, \dots$, so that $x_n\equiv x_{n+1} \bmod p^n$.

This is in strong analogy with approximation of a real number by rationls, for example:\\
$
 \sqrt 2=1.414213562373095048801688724\cdots
 $ \\
 $=1+4\cdot 10^{-1}+1\cdot 10^{-2}+4\cdot 10^{-3}+2\cdot 10^{-4}+\cdots
$

\end{frame}
\begin{frame}%[shrink=3]
\frametitle{$p$-adic numbers as a completion of rationals}

The idea of extending the field $\Q$
 appears in algebraic number theory
 in various different guises.
For example,
 the embedding $\Q \subset \R$
 often gives useful necessary conditions
 for the existence of solutions
 to Diophantine equations over $\Q$ or $\Z$.
The important feature of $\R$ is its
 completeness:\index{Field!Complete --}
 every Cauchy sequence $\bra {\al_n},_{n=1}^\infty$ in $\R$
 has a limit $\al$
 (a sequence is called
 Cauchy\index{Cauchy sequence}
 if for any $\ep > 0$ we have
  $|\al_n - \al_m|<\ep$ whenever $n$ and $m$
  are greater than some large $N = N(\ep)$).
Also, every element of $\R$ is the limit of some Cauchy sequence
 $\bra {\al_n},_{n=1}^\infty$ with $\al_n \in \Q$. 


An analogous construction exists using the
 $p$--adic absolute value
 $|\cdot|_p$ of $\Q$: % (see \S 2):
\begin{align*}&
|\cdot|_p:\Q \to \R_{\ge  0}=\bra {x\in \R\ |x\ \ge 0}, \cr
&|a/b|_p=p^{\ord_pb-\ord_pa},\ \ |0|_p=0,
\end{align*}
 where $\ord_pa$ is the highest power of $p$ dividing the integer $a$.
 \end{frame}
 \begin{frame}
This general construction of
 ``adjoining the limits of Cauchy sequences''
 to a field $k$ with an absolute value $|\cdot|$
 leads to a
 completion\index{Completion of a field}\index{Field!Completion of a --}
 of $k$.
This completion, often denoted $\hat k$,
 is complete, and contains $k$ as a dense subfield
 with respect to the extended absolute value
 $|\cdot|$, 
\cite{Borevich Z.I. Shafarevich I.R. (1985)}, 
\cite{Koblitz N. (1980)}.

As was noted at the end of \S 2, all absolute values of $\Q$
 are equivalent either to the usual Archimedean absolute value,
 or to the $p$--adic absolute value.
Thus any completion of $\Q$
 is either $\R$, or $\Q_p$, the field of
 $p$-adic numbers,\index{p-adic number@$p$-adic number}
 \index{Field!-- of p-adic numbers@Field!-- of $p$-adic numbers}
 i.e. the completion of the field of rational numbers $\Q$
 with respect to the $p$-adic absolute value.
Using the embeddings
 $\Q \hookrightarrow \R$ and $\Q \hookrightarrow \Q_p$
 (for all primes $p$)
 many arithmetical problems can be simplified.
An important example is given by the following
 {\it Minkowski--Hasse theorem}\index{Minkowski-Hasse theorem}
 \cite{Borevich Z.I. Shafarevich I.R. (1985)}, Ch.1.
%\cite{Cassels J.W.S. (1978)}, 
%\cite{Chevalley C. (1940)}:
 the equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (4.18)
\begin{align}\label{ii23.1}
 Q(x_1, x_2, \dots, x_n)=0, 
%\eqno(3.1)
\end{align}
 given by a quadratic form $Q(x_1, x_2, \dots, 
x_n)=\sum_{i,j}a_{ij}x_ix_j$,
 $a_{ij}\in \Q$ has a non--trivial solution in rational numbers,
 iff it is non--trivially solvable over $\R$ and over all $\Q_p$.
There are very effective tools for finding solutions in $\Q_p$.
These tools are somewhat analogous to those for $\R$
 such as the
 ``Newton - Raphson algorithm'',\index{Newton-Raphson 
algorithm}\index{Algorithm!Newton-Raphson --}
 which in the $p$--adic case becomes
 {\it Hensel's lemma}.\index{Hensel's lemma}
\end{frame}%[shrink=7]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
The simplest way to define the $p$--adic numbers is to consider 
expressions of the type
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (4.19)
\begin{align}\label{ii23.2}
 \al = a_m p^m + a_{m+1} p^{m+1} + \dots, 
%\eqno(3.2)
\end{align}
where $a_i\in \{0, 1, \dots. p-1\}$ are digits to the base $p$, and 
$m\in \Z$. It is convenient to write down $\al$ as a sequence of 
digits, infinite to the left:
$$
\al = \begin{cases}\cdots a_{m+1}a_m{\buildrel m-1 \ {\rm zeros } \over 
{\overbrace {000\dots 0}}_{(p)}},&\mbox{if }m\ge 0, \cr
\cdots a_1a_0.a_{-1} \cdots a_m{}_{(p)}, &\mbox{if }m<0.
\end{cases}
$$
These expressions form a field,
 in which algebraic operations are executed
 in the same way as for natural numbers $n=a_0+a_1p+\dots a_rp^r$,
 written as sequences of digits to the base $p$.
Consequently, this field contains all the natural numbers
 and hence all rational numbers.
\end{frame}
\begin{frame}
\frametitle{Expression of rational numbers as $p$-adic numbers}
For example,
$$
 -1
 =
 \frac {p-1}{1-p}
 =
 (p-1) + (p-1)p+(p-1)p^2+\cdots
 =
 \cdots (p-1) (p-1)_{(p)};
$$
$$
 \frac {-a_0}{ p-1}
 =
 a_0+a_0p+a_0p^2+\cdots
 =
 \cdots a_0a_0a_0{}_{(p)}.
$$
For $n\in \N$ the expression for $-n=n\cdot(-1)$ of type 
(\ref{ii23.2}) is 
obtained if we multiply the above expressions for $n$ and for $-1$.
Generally, for $\al \in \Q$ write $\al = c-\frac a b $, where $a,c\in 
\Z$, $b\in N$, $0\le a<b$, i.e. $a/b$ is a proper fraction.
Then by an elementary theorem of Euler, $p^{\ph(b)}-1=bu$, $u\in \N$.
Hence
$$
-\frac a b  = \frac {au}{ p^{\ph(b)}-1}  ,
$$
and $au<bu=p^r-1$, $r=\ph (b)$. Now let $au$  be written to the base 
$p$ as $a_{r-1}\cdots a_0{}_{(p)}$, then the expression 
of type (\ref{ii23.2}) 
for $\al$ is obtained as the sum of the expression for $c\in \N$ and 
$$
-\frac a b  = \cdots a_0{\buildrel r \ {\rm digits }\over {\overbrace 
{a_{r-1}\cdots a_0}}}{\buildrel r\ {\rm digits}\over {\overbrace 
{a_{r-1}\cdots a_0}}}{}_{(p)}.
$$
\end{frame}
\begin{frame}
For example, if $p=5$, 
$$
\frac 9 7  = 2-\frac 5 7  = 2 + \frac {5\cdot {2232}}  {1-5^6} \ \ c=2\ 
a=5,\ b=7,
$$
so that
$$
2232=32412_{(5)} = 3\cdot 5^4 + 2\cdot 5^3 + 4\cdot 5^2 + 1\cdot 5 + 
2,
$$
thus
$$
\frac 9 7  = \cdots  {\overbrace {324120}}{\overbrace 
{324120}}324122_{(5)}.
$$

It is easy to verify that the completion of $\Q$ with respect to the 
$p$--adic metric $|\cdot|_p$ can be identified with the described 
field of $p$--adic expansions (\ref{ii23.2}), where $|\al|_p=p^m$ for $\al$ as 
in (\ref{ii23.2}) with $a_m\not=0$ (see Koblitz N. (1980)).
\end{frame}
\begin{frame}

It is curious to compare the expansions (\ref{ii23.2}) infinite to the left 
with the ordinary expansions of real numbers $\al \in \R$, infinite 
to the right:
$$
\al = a_ma_{m-1} \cdots a_0.a_{-1}\cdots = a_m10^m + a_{m-1}10^{m-1} 
+ \cdots a_0 + a_{-1}10^{-1} + \cdots,
$$
where $a_i \in \bra {0,1,\cdots, 9},$ are digits, $a_m\not=0$. These 
expansions 
to any natural base lead to the same field $\R$. 
Also, a given $\al $ 
can possess various expressions of this type, e.g. $2.000\cdots = 
1.999\cdots$. However, in the $p$--adic case the expressions (\ref{ii23.2}) 
are uniquely determined by $\al$.
This fact provides additional comfort when calculating with
 $p$--adic numbers. 
%%   ADD: Computations with $p$-adic numbers on PARI-GP
\end{frame}





\subsection{Computation with PARI/GP}
\begin{frame}[fragile]
\frametitle{\small Computation with PARI/GP (see \cite{BBBCO})}
{\tiny
\begin{verbatim}  
gp > forprime(p=2,131,print("p="p,",""9/7="9/7+O(p^6)))
p=2,9/7=1 + 2 + 2^2 + 2^3 + 2^5 + O(2^6)
p=3,9/7=3^2 + 3^3 + 2*3^5 + O(3^6)
p=5,9/7=2 + 2*5 + 5^2 + 4*5^3 + 2*5^4 + 3*5^5 + O(5^6)
p=7,9/7=2*7^-1 + 1 + O(7^6)
p=11,9/7=6 + 9*11 + 7*11^2 + 4*11^3 + 9*11^4 + 7*11^5 + O(11^6)
p=13,9/7=5 + 9*13 + 3*13^2 + 9*13^3 + 3*13^4 + 9*13^5 + O(13^6)
p=17,9/7=11 + 14*17 + 4*17^2 + 7*17^3 + 2*17^4 + 12*17^5 + O(17^6)
p=19,9/7=4 + 8*19 + 5*19^2 + 16*19^3 + 10*19^4 + 13*19^5 + O(19^6)
p=23,9/7=21 + 9*23 + 16*23^2 + 19*23^3 + 9*23^4 + 16*23^5 + O(23^6)
p=29,9/7=22 + 20*29 + 20*29^2 + 20*29^3 + 20*29^4 + 20*29^5 + O(29^6)
p=31,9/7=19 + 26*31 + 8*31^2 + 13*31^3 + 4*31^4 + 22*31^5 + O(31^6)
p=37,9/7=33 + 15*37 + 26*37^2 + 31*37^3 + 15*37^4 + 26*37^5 + O(37^6)
p=41,9/7=13 + 29*41 + 11*41^2 + 29*41^3 + 11*41^4 + 29*41^5 + O(41^6)
p=43,9/7=32 + 30*43 + 30*43^2 + 30*43^3 + 30*43^4 + 30*43^5 + O(43^6)
p=47,9/7=8 + 20*47 + 13*47^2 + 40*47^3 + 26*47^4 + 33*47^5 + O(47^6)
p=53,9/7=24 + 45*53 + 37*53^2 + 22*53^3 + 45*53^4 + 37*53^5 + O(53^6)
p=59,9/7=35 + 50*59 + 16*59^2 + 25*59^3 + 8*59^4 + 42*59^5 + O(59^6)
p=61,9/7=10 + 26*61 + 17*61^2 + 52*61^3 + 34*61^4 + 43*61^5 + O(61^6)
p=67,9/7=30 + 57*67 + 47*67^2 + 28*67^3 + 57*67^4 + 47*67^5 + O(67^6)
p=71,9/7=52 + 50*71 + 50*71^2 + 50*71^3 + 50*71^4 + 50*71^5 + O(71^6)
p=73,9/7=43 + 62*73 + 20*73^2 + 31*73^3 + 10*73^4 + 52*73^5 + O(73^6)
p=79,9/7=69 + 33*79 + 56*79^2 + 67*79^3 + 33*79^4 + 56*79^5 + O(79^6)
p=83,9/7=25 + 59*83 + 23*83^2 + 59*83^3 + 23*83^4 + 59*83^5 + O(83^6)
p=89,9/7=14 + 38*89 + 25*89^2 + 76*89^3 + 50*89^4 + 63*89^5 + O(89^6)
p=97,9/7=29 + 69*97 + 27*97^2 + 69*97^3 + 27*97^4 + 69*97^5 + O(97^6)
p=101,9/7=59 + 86*101 + 28*101^2 + 43*101^3 + 14*101^4 + 72*101^5 + O(101^6)
p=103,9/7=16 + 44*103 + 29*103^2 + 88*103^3 + 58*103^4 + 73*103^5 + O(103^6)
p=107,9/7=93 + 45*107 + 76*107^2 + 91*107^3 + 45*107^4 + 76*107^5 + O(107^6)
p=109,9/7=48 + 93*109 + 77*109^2 + 46*109^3 + 93*109^4 + 77*109^5 + O(109^6)
p=113,9/7=82 + 80*113 + 80*113^2 + 80*113^3 + 80*113^4 + 80*113^5 + O(113^6)
p=127,9/7=92 + 90*127 + 90*127^2 + 90*127^3 + 90*127^4 + 90*127^5 + O(127^6)
p=131,9/7=20 + 56*131 + 37*131^2 + 112*131^3 + 74*131^4 + 93*131^5 + O(131^6)
\end{verbatim}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=7]
\frametitle{Topology of $p$-adic numbers}
The field $\Q_p$ is a {\it complete metric space} with the topology 
generated by the ``open discs'':
$$
U_a(r) = \bra {x\ | \ |x-a|<r}, \ \ \ (x,\ a\in \Q_p,\ r>0)
$$
(or ``closed discs'' $D_a(r) = \bra {x\ | \ |x-a|\le r},$).
From the topological point of view,
 the sets $U_a(r)$ and $D_a(r)$ are both open and closed in $\Q_p$.

An important topological property of $\Q_p$ is its
 {\it local compactness}:
 all discs of finite radius are compact.
The easiest way to show this is to consider any sequence
 $\bra {\al_n},_{n=1}^\infty$ of elements $\al_n \in D_a(r)$
 and to construct a limit point.
Such a point may be found step--by--step using the
 {\it $p$--adic digits}\index{Digit!p-adic@Digit!$p$-adic--}
 \index{p-adic!-- digit@$p$-adic!-- digit}
 \index{Ring!-- of p-adic integers@Ring!-- of $p$-adic integers}
 (\ref{ii23.2}).
One knows that the number of digits ``after the point''
 is bounded on any finite disc.
In particular, the disc
$$
 \Z_p 
 =
 D_0(1)
 =
 \bra {x\ |\ |x|_p\le 1},
 =
 \bra { x=a_0 + a_1 p + a_2 p^2 + \cdots },
$$
 is a compact topological ring,
 whose elements are called
 $p$--adic integers.\index{Integer!p-adic --@Integer!$p$-adic --}
 $\Z_p$ is the closure of $\Z$ in $\Q_p$.
The ring $\Z_p$ is local,
 i.e. it has only one maximal ideal $p\Z_p = U_0(1)$
 with residue field $\Z_p/p\Z_p = \F_p$.
The set of invertible elements (units) of $\Z_p$ is
$$
 \Z_p^\times
 =
 \Z_p \backslash p\Z_p
 =
 \bra {x\ |\ |x|_p = 1},
 =
 \bra {x = a_0 + a_1 p + a_2 p^2+ \cdots\ |\ a_0\not=0},.
$$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Applications of $p$--adic Numbers to Solving 
Congruences.}%\label{ii23.2.}
The first appearances of $p$--adic numbers,
 in papers by Hensel,
 were related to the problem of finding solutions to congruences 
modulo $p^n$. An application of this method by his student H.Hasse
 to the theory of quadratic forms
 has lead to an elegant reformulation of this theory,
 without the use of considerations over the residue rings $\Z/p^n\Z$.
These considerations are tiring because of the zero--divisors in 
$\Z/p^n\Z$. From the above presentation of $\Z_p$ as the projective 
limit 
$$
\lim_{\longleftarrow\atop n}\Z/p^n\Z
$$ 
 it follows that
 for $f(x_1, \dots, x_n) \in \Z_p [x_1, \dots, x_n]$,
 the congruences
$$
f(x_1, \dots, x_n) \equiv 0 (\bmod\ p^n)
$$
 are solvable for all $n\ge 1$ iff the equation
$$
f(x_1, \dots, x_n) = 0
$$
 is solvable in $p$--adic integers.
Solutions in $\Z_p$
 can be obtained using the following $p$--adic version of the
 {\it ``Newton - Raphson algorithm''}.\index{Newton-Raphson 
algorithm}\index{Algorithm!Newton-Raphson --}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%   Theorem 4.17
%***
\end{frame}
\begin{frame}[shrink]
\begin{theorem}[Hensel's Lemma]\index{Hensel's Lemma}
Let $f(x)\in \Z_p[x]$ be a polynomial in one variable $x$,  $f^\prime 
(x) \in \Z_p [x]$ its formal derivative, and suppose that for some 
$\al_0 \in \Z_p$ the initial condition
%***
\begin{align}\label{ii23.6}
 |f(\al_0)/ f^\prime (\al_0)^2|_p < 1 
%\eqno(3.6)
\end{align}
is satisfied. 

Then there exists a unique $\al\in \Z_p$ such that
$$
 f(\al) = 0,\ \ |\al - \al_0|<1.
$$
\end{theorem}



We prove this by induction using the sequence of ``successive 
approximations'':
$$
\al_n = \al_{n-1} - \frac {f(\al_{n-1})}{ f^\prime (\al_{n-1})} .
$$
Taking into account the formal Taylor expansion of $f(x)$ at 
$x=\al_{n-1}$
 one shows that this sequence is Cauchy,
 and its limit $\al$ has all the desired properties (cf. %\cite{Cassels J.W.S. Frölich A. (eds.) (1967)}, 
\cite{Borevich Z.I. Shafarevich I.R. (1985)}, 
\cite{Se70}).

For example, if $f(x) = x^{p-1}-1$,
 then any $\al_0 \in \bra {1, 2, \dots, p-1}, $
 satisfies the condition $|f(\al_0)|_p<1$
At the same time
 $f^\prime(\al_0) = (p-1)\al_0^{p-2} \not \equiv 0\bmod\ p$,
 hence the initial condition (\ref{ii23.6}) is satisfied.
The root $\al$ coincides then with the uniquely defined
 Teichm\"uller representative of $\al_0$: $\al = \w (\al_0)$.

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Continous  and  $p$-adic analytic functions. Mahler's criterion. Newton polygons.} 
\begin{frame}[shrink=3]
\frametitle{\ncours\small Continuous and analytic functions over a non-Archimedean field}
%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%------------------------PARAGRAPHE1. 2.1.-----------------------------------


Let $K$ be a closed subfield of the Tate field $\C_p$. 
For a subset
$W \subset K$ we consider continuous functions $f: W \rightarrow \C_p$. The
standard examples of continuous functions are provided by polynomials, by
rational functions (at points where they are finite), and also by locally
constant functions. 
If $W$ is compact then for any continuous function
$f : W \rightarrow \C_p$ and for any $\varepsilon>0$ there exists a polynomial
$h(x) \in \C_p[x]$ such that $| f(x)-h(x) |_p < \varepsilon$ for all
$x \in W$. If $f(W) \subset L$ for a closed subfield $L$ of $\Cp$ then $h(x)$
can be chosen so that $h(x) \in L[x]$ (see \cite{Koblitz N. (1980)}, \cite{Washington L.C. (1982)}).

\

Interesting examples of continuous $p$-adic functions are provided by
interpolation of functions, defined on certain subsets, such as $W = \ZZ$ or
$\NN$ with $K = \Qp$. Let $f$ be any function on non-negative integers with
values in $\Qp$ or in some (complete) $\Qp$-Banach space. In order to extend
$f(x)$ to all $x \in \Zp$ we can use the interpolation polynomials
  $$ \binom{x}{n} = \frac{x(x-1)\cdots (x-n+1)}{n!}.  $$
Then we have that $\binom{x}{n}$ is a polynomial of degree $n$ of $x$, which
for $x \in \ZZ$, $x \geq 0$ gives the binomial coefficient. If $x \in \Zp$
then $x$ is close (in the $p$-adic topology) to a positive integer, hence the
value of $\binom{x}{n}$ is also close to an integer, therefore
$\binom{x}{n} \in \Z_p$.

\end{frame}
\begin{frame}[shrink=3]
\frametitle{Mahler's criterion}
The classical Mahler's interpolation theorem says that any continuous function
$f : \Zp \rightarrow \Qp$ can be written in the form (see \cite{Hi91},
\cite{Washington L.C. (1982)}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.15)
  \begin{equation} \label{EQfser}
     f(x) = \sum_{n=0}^{\infty} a_n \binom{x}{n},
  \end{equation}
with $a_n \rightarrow 0$ ($p$-adically) for $n \rightarrow \infty$. For a
function $f(x)$ defined for $x \in \ZZ$, $x \geq 0$ one can write formally
  $$ f(x) = \sum_{n=0}^{\infty} a_n \binom{x}{n},  $$
where the coefficients can be founded from the system of linear equations
  $$ f(n) = \sum_{m=0}^n a_m \binom{n}{m},  $$
that is
  $$ a_m = \sum_{j=0}^m (-1)^{m-j} \binom{m}{j} f(j).  $$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}  
The series for $f(x)$ is always reduced to a finite sum for each $x \in \ZZ$,
$x \geq 0$. If $a_n \rightarrow 0$ then this series is convergent for all
$x \in \Zp$. As was noticed above, the inverse statement is also valid
(``{\it Mahler's criterion}\index{Mahler's criterion}''). If convergence
of $a_n$ to zero is so fast that the
series defining the coefficients of the $x$-expansion of $f(x)$ also converge,
then $f(x)$ can be extended to an
{\it analytic function}\index{analytic function}. 
Unfortunately, for an arbitrary sequence $a_n$ with $a_n \rightarrow 0$
the attempt to use (\ref{EQfser}) for continuation of $f(x)$ out of the subset
$\Zp$ in $\Cp$ may fail. However, in the sequel we mostly consider anlytic
functions, that are defined as sums of power series.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Distributions, measures, and the abstract Kummer congruences. 
The Kubota and Leopoldt $p$-adic $L$-functions  and the Iwasawa theory} 
\begin{frame}[shrink=3]
\frametitle{ %\ncours \small 
Distributions and measures.} 
Let us consider a commutative associative ring $R$, an $R$-module $\Ar$ and
a profinite (i.e. compact and totally disconnected) topological space $Y$.
Then $Y$ is a projective limit of finite sets:
$$
 Y = \lim_{\stackrel{\longleftarrow}{I}} Y_i  
$$
where $I$ is a (partially ordered) inductive set and for $i \geq j$, $i$,
$j \in I$
there are surjective homomorphisms $\pi_{i,j} : Y_i \rightarrow Y_j$ with the
condition $\pi_{i,j} \circ \pi_{j,k} = \pi_{i,k}$ for $i \geq j \geq k$. The
inductivity of $I$ means that for any $i,j \in I$ there exists $k \in I$ with
the condition $k \geq i$, $k \geq j$. By the universal property we have that
for each $i \in I$ a unique map $\pi_i : Y \rightarrow Y_i$ is defined,
which satisfies the property $\pi_{i,j} \circ \pi_i = \pi_j$ (for each
$i,j \in I$).
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
\begin{frame}[shrink=3] 
Let $\Step(Y,R)$ be the $R$-module consisting of all $R$-valued locally
constant functions $\phi : Y \rightarrow R$.
%DEFINITION 1.5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{definition}
  A distribution on $Y$ with values in a $R$-module $\Ar$ is a $R$-linear
  homomorphism
    $$ \mu : \Step(Y,R) \longrightarrow \Ar.  $$
\end{definition}

For $\varphi \in \Step(Y,R)$ we use the notations
$$
 \mu(\varphi) = \int_Y \varphi \dd \mu = \int_Y \varphi(y) \dd \mu(y).  
$$
Each distribution $\mu$ can be defined by a system of functions
$\mu^{(i)} : Y_i \rightarrow \Ar$, satisfying the following finite-additivity
condition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.25)
  \begin{equation} \label{EQmuj}
     \mu^{(j)}(y) = \sum_{x \in \pi_{i,j}^{-1}(y)} \mu^{(i)}(x) ~~~~
                    (y \in Y_j,~~x \in Y_i).
  \end{equation}
In order to construct such a system it suffices to put
  $$ \mu^{(i)}(x) = \mu(\de_{i,x}) \in \Ar ~~~~ (x \in Y_i),  $$
where $\de_{i,x}$ is the characteritic function of the inverse image
$\pi_i^{-1}(x) \subset Y$ with respect to the natural projection
$Y \rightarrow Y_i$. For an arbitrary function $\varphi_j : Y_j \rightarrow R$
and $i \geq j$ we define the functions
  $$ \varphi_i = \varphi_j \circ \pi_{i,j},~~~~ \varphi = \varphi_j \circ
     \pi_j, ~~~~ \varphi \in \Step(Y,R), ~~~~ \varphi_i : Y_i 
     \stackrel{\pi_{i,j}}{\longrightarrow} Y_j \longrightarrow R.  $$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
\begin{frame}
A convenient criterion of the fact that a system of functions
$\mu^{(i)} : Y_i \rightarrow \Ar$ satisfies the finite additivity condition
(\ref{EQmuj}) (and hence is associated to some distribution) is given by the
following condition
({\it compatibility criterion}\index{compatibility criterion}):
for all $j \in I$, and
$\varphi_j : Y_j \rightarrow R$ the value of the sums
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (1.26)
  \begin{equation} \label{EQmuphi}
     \mu(\varphi) = \mu^{(i)}(\varphi_i) = \sum_{y \in Y_i} \varphi_i(y)
                    \mu^{(i)}(y),
  \end{equation}
is independent of $i$ for all large enough $i \geq j$. When using
(\ref{EQmuphi}), it suffices to verify the condition (\ref{EQmuphi}) for some
``basic'' system of functions. For example, if
$$
 Y = G = \lim_{\stackrel{\longleftarrow}{i}} G_i  
$$
is a profinite abelian group, and $R$ is a domain containing all roots of
unity of the order dividing the order of $Y$ (which is a ``supernatural
number'') then it suffices to check the condition (\ref{EQmuphi}) for all
characters of finite order $\chi : G \rightarrow R$, since their
$R \otimes \QQ$ -linear span coincides with the whole ring
$\Step(Y, R \otimes \QQ)$ by the orthogonality properties for characters
of a finite group (see \cite{Kat78}, 
\cite{Mazur B. Swinnerton--Dyer H.P.F. (1974)}).
\end{frame}
\begin{frame}%[shrink=5]
\frametitle{Example: Bernoulli distributions} 
Let $M$ be a positive integer, $f : \ZZ \rightarrow \CC$ is a periodic function with the
period $M$ (i.e. $f(x+M) = f(x)$, $f : \ZZ/M\ZZ \rightarrow \CC$). The
generalized Bernoulli number 
(see \cite{Borevich Z.I. Shafarevich I.R. (1985)} ) 
$B_{k,f}$ is
defined as $k!$ times the coefficient by $t^k$ in the expansion in $t$
of the following rational quotient
$$
 \sum_{a=0}^{M-1} \frac{f(a) t e^{at}}{e^{Mt}-1},  
$$
that is,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.27)
  \begin{equation} \label{EQBkf}
     \sum_{k=0}^{\infty} \frac{B_{k,f}}{k!} t^k =
     \sum_{a=0}^{M-1} \frac{f(a) t e^{at}}{e^{Mt}-1}.
  \end{equation}
  
\end{frame}
\begin{frame}%[shrink=5]  
Now let us consider the profinite ring
  $$ Y = \ZZ_S = \lim_{\stackrel{\longleftarrow}{M}} 
         \left( \ZZ/M\ZZ \right)  $$
$(S(M) \subset S)$, the projective limit being taken over the set of all
positive integers $M$ with support $S(M)$ in a fixed finite set $S$ of prime
numbers. Then the periodic function $f : \ZZ/M\ZZ \rightarrow \CC$ with
$S(M) \subset S$ may be viewed as an element of $\Step(Y,\CC)$. We claim
that there exists a distribution $E_k : \Step(Y,\CC) \rightarrow \CC$
which is uniquely determined by the condition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.28)
  \begin{equation} \label{EQEkBk}
     E_k(f) = B_{k,f} \mbox{  for all  } f \in \Step(Y,\CC).
  \end{equation}
In order to prove the existence of this distribution we use the above criterion
(\ref{EQmuphi}) and check that for every $f \in \Step(Y,\CC)$ the right hand
side in (\ref{EQEkBk}) (i.e. $B_{k,f}$) does not depend on the choice of a
period $M$ of the function $f$. 
It follows directly from the definition
(\ref{EQBkf}); however we give here a different proof which is based on an
interpretation of the  numbers $B_{k,f}$ as certain special values of
$L$-functions.
\end{frame}
\begin{frame}[shrink=5]
For a function $f : \ZZ/M\ZZ \rightarrow \CC$ let
  $$ L(s,f) = \sum_{n=1}^{\infty} f(n) n^{-s}  $$
be the corresponding $L$-series which is absolutely convergent for all $s$
with $\Re(s) > 1$ and admits an analytic continuation over all
$s \in \CC$. For this series we have that
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.29)
  \begin{equation} \label{EQLf}
     L(1-k,f) = - \frac{B_{k,f}}{k}.
  \end{equation}
For example, if $f \equiv 1$ is the constant function with the period
$M=1$ then we have that
$$ \zeta(1-k) = -\frac{B_k}{k}, ~~~~
     \sum_{k=0}^{\infty} \frac{B_k}{k!} t^k = \frac{t}{e^t-1},  
     $$
$B_k$ being the Bernoulli number. The formula (\ref{EQLf}) is established by
means of the contour integral discovered by Riemann.
% (see \cite{Lan76}). This
formula apparently implies the desired independence of $B_{k,f}$ on the choice
of $M$. We note also that if $K \subset \CC$ is an arbitrary subfield, and
$f(Y) \subset K$ then we have from the formula (\ref{EQBkf}) that
$B_{k,f} \in K$ hence the distribution $E_k$ is a $K$-valued
distribution on $Y$.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse
%'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%--------------------------PARAGRAPHE 1.3.2.----------------------------------
\subsection{Measures}
\begin{frame}[shrink=5]
\frametitle{Measures} 
Let $R$ be a topological ring, and $\Cr(Y,R)$ be the topological module
of all $R$-valued functions on a profinite set $Y$.

%DEFINITION 1.6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{definition}
  A measure on $Y$ with values in the topological $R$-module $\Ar$ is a
  continuous homomorphism of $R$-modules
    $$ \mu : \Cr(Y,R) \longrightarrow \Ar.  $$
\end{definition} 
\pause
The restriction of $\mu$ to the $R$-submodule $\Step(Y,R) \subset \Cr(Y,R)$
defines a distribution which we denote by the same letter $\mu$, and the
measure $\mu$ is uniquely determined by the corresponding distribution since
the $R$-submodule $\Step(Y,R)$ is dense in $\Cr(Y,R)$. The last statement
expresses the well known fact about the uniform continuity of a continuous
function over a compact topological space.

\end{frame}%[shrink=2]
\begin{frame}[shrink=2]

Now we consider any closed subring $R$ of the Tate field $\Cp$,
$R \subset \Cp$, and let $\Ar$ be a complete $R$-module with topology given by
a norm $|\cdot|_{\Ar}$ on $\Ar$ compatible with the norm $|\cdot|_p$ on $\Cp$
so that the following conditions are satisfie: \\
$\bullet$ for $x \in \Ar$ the equality $|x|_{\Ar}=0$ is equivalent to $x=0$, \\
$\bullet$ for $a \in R$, $x \in \Ar$: $|ax|_{\Ar} = |a|_p|x|_{\Ar}$, \\
$\bullet$ for all $x,y \in \Ar$: $|x+y|_{\Ar} < \max(|x|_{\Ar},|y|_{\Ar})$. \\
Then the fact that a distribution (a system of functions
$\mu^{(i)} : Y_i \rightarrow \Ar$) gives rise to a $\Ar$-valued measure on
$Y$ is equivalent to the condition that the system $\mu^{(i)}$ is bounded,
i.e. for some constant $B>0$ and for all $i \in I$, $x \in Y_i$
the following uniform estimate holds:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.30)
  \begin{equation}
     |\mu^{(i)}(x)|_{\Ar} < B.
  \end{equation}
This criterion is an easy consequence of the non-Archimedean property
  $$ |x+y|_{\Ar} \leq \max (|x|_{\Ar},|y|_{\Ar })  $$
of the norm $|\cdot|_{\Ar}$ (see \cite{Ma73}, \cite{Vis76}).
In particular if $\Ar = R = \Or_p = \{ x \in \Cp ~~\mid~~ |x|_p \leq 1 \}$ is
the subring of integers in the Tate field $\Cp$ then the set of $\Or_p$-valued
distributions on $Y$ coincides with $\Or_p$-valued measures (in fact, both
sets are $R$-algebras with multiplication defined by convolution.
%, see section \ref{SECIwasawa}).
\end{frame}%[shrink=2]
\begin{frame}[shrink]
\frametitle{\ncours \normalsize The abstract Kummer congruences and the $p$-adic Mellin transform}
A useful criterion for the existence of a measure with given properties is:
%PROPOSITION 3.2.2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{prop}[The abstract Kummer congruences] \label{PROPabKum}
  (see \cite{Kat78}). Let $\{ f_i \}$ be a system of continuous
  functions $f_i \in \Cr(Y,\Or_p)$
  in the ring $\Cr(Y,\Or_p)$ of all continuous functions on the compact
  totally disconnected group $Y$ with values in the ring of integers $\Or_p$
  of $\Cp$ such that $\Cp$-linear span of $\{ f_i \}$ is dense in $\Cr(Y,\Cp)$.
  Let also $\{ a_i \}$ be any system of elements $a_i \in \Or_p$. Then the
  existence of an $\Or_p$-valued measure $\mu$ on $Y$ with the property
    $$ \int_Y f_i d\mu = a_i  $$
  is equivalent to the following congruences, for an arbitrary choice of
  elements $b_i \in \Cp$ almost all of which vanish
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.31)
    \begin{equation}  \label{EQabKum}
      \sum_i b_i f_i(y) \in p^n \Or_p \mbox{ for all } y \in Y
      \mbox{ implies } \sum_i b_i a_i \in p^n \Or_p.
    \end{equation}
\end{prop}
%REMARK 1.8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{rema}
  Since $\Cp$-measures are characterised as bounded $\Cp$-valued distributions,
  every $\Cp$-measures on $Y$ becomes a $\Or_p$-valued measure after
  multiplication by some non-zero constant.
\end{rema}
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=2]
\medskip{\it Proof of proposition \ref{PROPabKum}.}
  The necessity is obvious since
    $$ \begin{array}{rcccl}
         \di{\sum_i b_i a_i}  &  =  &  \di{\int_Y (p^n \Or_p-
         \mbox{valued function})d\mu}  &  =  &  \\  &  =   &
         \di{p^n \int_Y (\Or_p- \mbox{valued function})d\mu}  &  \in  &
         p^n \Or_p.
       \end{array}  $$
  In order to prove the sufficiency we need to construct a measure $\mu$ from
  the numbers $a_i$. For a function $f \in \Cr_(Y, \Or_p)$ and a positive
  integer $n$ there exist elements $b_i \in \Cp$ such that only a finite
  number of $b_i$ does not vanish, and
    $$ f - \sum_i b_i f_i \in p^n \Cr(Y, \Or_p),  $$
  according to the density of the $\Cp$-span of $\{ f_i \}$ in $\Cr(Y,\Cp)$.
  By the assumption (\ref{EQabKum}) the value $\sum_i a_i b_i$ belongs to
  $\Or_p$ and is well defined modulo $p^n$ (i.e. does not depend on the choice
  of $b_i$). Following N.M. Katz (\cite{Kat78}), we denote this value by
  ``$\int_Y f d\mu \bmod p^n$ ''. Then we have that the limit procedure
    $$ \int_Y f d\mu = \lim_{n \rightarrow \infty}
       \mbox{``$\di{\int_Y f d\mu \bmod p^n}$ ''}
       \in \lim_{\stackrel{\longleftarrow}{n}} \Or_p/p^n\Or_p = \Or_p,  $$
  gives the measure $\mu$.
%\end{proof}
%'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%-----------------------------PARAGRAPHE 1.3.3.-------------------------------
\subsection{The $S$-adic Mazur measure}
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink]
\frametitle{Mazur's measure}

Let $c > 1$ be a positive integer coprime to
  $$ M_0 = \prod_{q \in S} q  $$
with $S$ being a fixed set of prime numbers. Using the criterion of the
proposition \ref{PROPabKum} we show that the $\QQ$ -valued distribution defined
by the formula
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.33)
  \begin{equation}  \label{EQmesEk}
    E_k^c(f) = E_k(f) - c^k E_k(f_c), ~~~~ f_c(x) = f(cx),
  \end{equation}
turns out to be a measure where $E_k(f)$ are defined by (\ref{EQEkBk}),
$f \in \Step(Y,\Qp)$ and the field $\QQ$ is viewed as a subfield of $\Cp$.
 
Define the generelized Bernoulli polynomials $B_{k,f}^{(M)}(X)$ as
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.33)
 \begin{equation}  \label{EQpolyBer}
    \sum_{k=0}^{\infty} B_{k,f}^{(M)}(X) \frac{t^k}{k!} =
    \sum_{a=0}^{M-1} f(a) \frac{t e^{(a+X)t}}{e^{Mt}-1} ,
  \end{equation}
and the generalized sums of powers
  $$ S_{k,f}(M) = \sum_{a=0}^{M-1} f(a) a^k.  $$

\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=5]
Then the definition (\ref{EQpolyBer}) formally implies that
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.34)
  \begin{equation}  \label{EQsumpolyBer}
    \frac{1}{k} [ B_{k,f}^{(M)}(M) -  B_{k,f}^{(M)}(0)] = S_{k-1,f}(M),
  \end{equation}
and also we see that
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.35)
 \begin{equation}  \label{EQBerbino}
    B_{k,f}^{(M)}(X) = \sum_{i=0}^k \binom{k}{i} B_{i,f} X^{k-i} =
    B_{k,f} + kB_{k-1,f}X + \cdots + B_{0,f}X^k.
  \end{equation}
The last identity can be rewritten symbolically as
  $$ B_{k,f}(X) = (B_f + X)^k.  $$

The equality (\ref{EQsumpolyBer}) enables us to calculate the (generalized)
sums of powers in terms of the (generalized) Bernoulli numbers. In particular
this equality implies that the Bernoulli numbers $B_{k,f}$ can be obtained by
the following $p$-adic limit procedure (see \cite{Lang S. (1976)}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.36)
  \begin{equation}  \label{EQBerlim}
    B_{k,f} = \lim_{n \rightarrow \infty} \frac{1}{Mp^n} S_{k,f}(Mp^n)
              ~~~~(\mbox{a }p\mbox{-adic limit}),
  \end{equation}
where $f$ is a $\Cp$-valued function on $Y = \ZZ_S$. Indeed, if we replace $M$
in (\ref{EQsumpolyBer}) by $Mp^n$ with growing $n$ and let $D$ be the common
denominator of all coefficients of the polynomial $B_{k,f}^{(M)}(X)$. Then we
have from (\ref{EQBerbino}) that
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.37)
  \begin{equation}  \label{EQBermod}
    \frac{1}{k} \left[ B_{k,f}^{(Mp^n)}(M) - B_{k,f}^{(M)}(0) \right]
    \equiv B_{k-1,f}(Mp^n) \pmod{\frac{1}{kD} p^2n}.
  \end{equation}
The proof of (\ref{EQBerlim}) is accomplished by division of (\ref{EQBermod})
by $Mp^n$ and by application of the formula (\ref{EQsumpolyBer}).
\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=2]

Now we can directly show that the distribution $E_k^c$ defined by
(\ref{EQmesEk}) are in fact bounded measures. If we use (\ref{EQabKum}) and
take the functions $\{f_i\}$ to be all of the functions in $\Step(Y,\Or_p)$.
Let $\{b_i\}$ be a system of elements $b_i \in \Cp$ such that for all $y \in Y$
the congruence
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.38)
 \begin{equation}
    \sum_i b_i f_i(y) \equiv 0 \pmod{p^n}
  \end{equation}
holds. Set $f = \sum_i b_i f_i$ and assume (without loss of generality) that
the number $n$ is large enough so that for all $i$ with $b_i \neq 0$ the
congruence
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (1.39)
 \begin{equation}
    B_{k,f_i} \equiv \frac{1}{M p^n} S_{k,f_i} (M p^n) \pmod{p^n}
  \end{equation}
is valid in accordance with (\ref{EQBerlim}). Then we see that
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.40)
 \begin{equation}
    B_{k,f} \equiv (Mp^n)^{-1} \sum_i \sum_{a=0}^{Mp^n-1} b_i f_i(a) a^k
    \pmod{p^n},
  \end{equation}
hence we get by definition (\ref{EQmesEk}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1.41)
  \begin{eqnarray}     E_k^c(f)  & = &\label{EQmesEkBer}  B_{k,f} - c^k B_{k,f_c}  \\   & \equiv  &
    (Mp^n)^{-1} \sum_i \sum_{a=0}^{Mp^n-1} b_i
    \left[f_i(a) a^k - f_i(ac)(ac)^k \right] \pmod{p^n}.  \nonumber
  \end{eqnarray}
  

\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=5] 
Let $a_c \in \{ 0, 1, \cdots, Mp ^n-1 \}$, such that $a_c \equiv ac
\pmod{Mp^n}$, then the map $a \longmapsto a_c$ is well defined and acts as a
permutation of the set $\{0, 1, \cdots, Mp^n-1\}$, hence (\ref{EQmesEkBer}) is
equivalent to the congruence
%%%%%%%%%%%%%%%%%%%%%%%%%%%%    (1.42)
  \begin{equation}  \label{EQmesEkBermod}
    E_k^c(f) = B_{k,f} - c^k B_{k,f_c} \equiv \sum_i \frac{a_c^k - (ac)^k}
               {Mp^n} \sum_{a=0}^{Mp^n-1} b_i f_i(a) a^k \pmod{p^n}.
  \end{equation}
Now the assumption (\ref{EQBermod}) formally inplies that
$E_k^c(f) \equiv 0 \pmod{p^n}$, completing the proof of the abstact
congruences and the construction of measures $E_k^c$.

%REMARK 1.9  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{rema}
  The formula $(\ref{EQmesEkBer})$ also implies that for all $f \in \Cr(Y,\Cp)$
  the following holds
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.43)  
   \begin{equation}  \label{EQEkf}
      E_k^c(f) = k E_1^c(x_p^{k-1}f)
    \end{equation}
  where $x_p : Y \longrightarrow \Cp \in \Cr(Y,\Cp)$ is the composition of
  the projection $Y \longrightarrow \Zp$ and the embedding $\Zp
  \hookrightarrow \Cp$.
\end{rema}

Indeed if we put $a_c = ac + Mp^nt$ for some $t \in \ZZ$ then we see that
  $$ a_c^k - (ac)^k = (ac + Mp^nt)^k - (ac)^k \equiv kMp^nt(ac)^{k-1}
                      \pmod{(Mp^n)^2},  $$
and we get that in (\ref{EQmesEkBermod}):
  $$ \frac{a_c^k - (ac)^k}{Mp^n} \equiv k(ac)^{k-1} \frac{a_c - ac}{Mp^n}
     \pmod{Mp^n}.  $$
The last congruence is equivalent to saying that the abstract Kummer
congruences (\ref{EQabKum}) are satisfied by all functions of the type
$x_p^{k-1} f_i$ for the measure $E_1^c$ with $f_i \in \Step(Y,\Cp)$
establishing the identity (\ref{EQEkf}).
\end{frame}%[shrink=2]
%\iffalse
%*************************************************************************
%---------------------------SECTION 1.4.------------------------------------
%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%-------------------------PARAGRAPHE 1.4.1.----------------------------------
\subsection{Iwasawa algebra and the non-Archimedean Mellin transform}
%\label{SECIwasawa}
%\begin{frame}%[shrink=3]
%\frametitle{
%Distributions, measures, and the abstract Kummer congruences.
%$p$-adic $L$-functions,  Iwasawa algebra,  Mellin transform. 
% and relations with modular forms}


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

\begin{frame}[shrink=5]
\frametitle{\small The domain of definition  of the  non-Archimedean zeta functions}

In the classical case the set on which zeta functions are defined is the set of
complex numbers $\CC$ which may be viewed equally as the set of all
continuous characters (more precisely, quasicharacters) via the following
isomorphism:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.44)  
  \begin{eqnarray}
    \CC  &  \stackrel{\sim}{\longrightarrow}  &  \Hom_{\cont} (\RR_+^{\times},
                                                 \CC^{\times})  \\  \nonumber
    s    &  \longmapsto                       &  (y \longmapsto y^s)
  \end{eqnarray}

\pause
 The construction which associates to a function $h(y)$ on $\RR_+^{\times}$
(with certain growth conditions as $y \rightarrow \infty$ and
$y \rightarrow 0$) the following integral
$$
 L_h(s) = \int_{\RR_+^{\times}} h(y) y^s \frac{dy}{y}  
$$
(which converges probably not for all values of $s$) is called
the {\it Mellin transform}\index{Mellin transform}.

\end{frame}%[shrink=2]
%\iffalse
\begin{frame}[shrink=5]
\frametitle{}
For example, if $\zeta(s) = \sum_{n \geq 1} n^{-s}$ is the Riemann
zeta function, then the function $\zeta(s) \Ga(s)$ is the Mellin transform
of the function $h(y) = 1/(1-e^{-y})$:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.45)  
  \begin{equation}  \label{EQtransMell}
    \zeta(s) \Ga(s) = \sum_0^{\infty} \frac{1}{1-e^{-y}} y^s \frac{dy}{y},
  \end{equation}
so that the integral and the series are absolutely convergent for $\Re(s) > 1$.
For an arbitrary function of type
  $$ f(z) = \sum_{n=1}^{\infty} a(n) e^{2i\pi n z}  $$
with $z = x + iy \in \HH$ in the upper half plane $\HH$ and with the growth
condition $a(n) = \Or(n^c)$ ($c>0$) on its Fourier coefficients, we see that
the zeta function
$$ 
L(s,f) = \sum_{n=1}^{\infty} a(n) n^{-s} ,  
$$
essentially coincides with the Mellin transform of $f(z)$, that is
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.46)  
  \begin{equation}  \label{EQgaLMel}
    \frac{\Ga(s)}{(2\pi)^s} L(s,f) = \int_0^{\infty} f(iy) y^s \frac{dy}{y}.
  \end{equation}
\end{frame} 
%[shrink=2]

%\iffalse
\begin{frame}%[shrink=5]  
  
Both sides of the equality (\ref{EQgaLMel}) converge absolutely for
$\Re(s) > 1+c$. The identities (\ref{EQtransMell}) and (\ref{EQgaLMel}) are
immediately deduced from the well known integral representation for
the gamma-function
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.47)  
  \begin{equation}  \label{EQgam}
    \Ga(s) = \int_0^{\infty} e^{-y} y^s \frac{dy}{y},
  \end{equation}
where $\frac{dy}{y}$ is a measure on the group $\RR_+^{\times}$ which is
invariant under the group translations (Haar measure). The integral
(\ref{EQgam}) is absolutely convergent for $\Re(s) > 0$ and it can be
interpreted as the integral of the product of an additive character
$y  \mapsto e^{-y}$ of the group $\RR^{(+)}$ restricted to
$\RR_+^{\times}$, and of the multiplicative character $y \mapsto y^s$,
integration is taken with 
respect to the Haar measure $\dd y/y$ on the group $\RR_+^{\times}$.
\end{frame} 
%[shrink=2]

%\iffalse
\begin{frame}
In the theory of the non-Archimedean integration one considers the group
$\ZZ_S^{\times}$ (the group of units of the $S$-adic completion of the ring of
integers $\ZZ$) instead of the group $\RR_+^{\times}$, and the Tate field
$\Cp = \hat{\Qb}_p$ (the completion of an algebraic closure of $\Qp$)
instead of the complex field $\CC$. The domain of definition of the $p$-adic
zeta functions is the $p$-adic analytic group
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.48)  
  \begin{equation}
    X_S = \Hom_{\cont}(\ZZ_S^{\times},\Cp^{\times}) = X(\ZZ_S^{\times}),
  \end{equation}
where:
$$
 \ZZ_S^{\times} \cong \oplus_{q \in S} \ZZ_q^{\times},  
$$
and the symbol
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.49)  
  \begin{equation}
    X(G) = \Hom_{\cont}(G,\Cp^{\times})
  \end{equation}
denotes the functor of all $p$-adic characters of a topological group $G$
(see \cite{Vis76}).
\end{frame}
%[shrink=2]
%\iffalse
\subsection{The analytic structure of $X_S$ and
$p$-adic Mellin transform}
\begin{frame}[shrink=8]
\frametitle{The analytic structure of $X_S$}
Let us consider in detail the structure of the topological group $X_S$.
Define
  $$ U_p = \{ x \in \Zp^{\times} ~~\mid~~ x \equiv 1 \pmod{p^{\nu}} \},  $$
where $\nu = 1$ or $\nu = 2$ according as $p > 2$ or $p = 2$. Then we have the
natural decomposition
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.50)  
  \begin{equation}  \label{EQdecdeXS}
    X_S = X \left( (\ZZ/p^{\nu}\ZZ)^{\times} \times \prod_{q \neq p}
          \ZZ_q^{\times} \right) \times X(U_p).
  \end{equation}
The analytic dstructure on $X(U_p)$ is defined by the following isomorphism
(which is equivalent to a special choice of a local parameter):
  $$ \varphi : X(U_p) \stackrel{\sim}{\longrightarrow} T =
     \{ z \in \Cp^{\times} ~~\mid~~ |z-1|_p < 1 \},  $$
where $\varphi(x) = x(1+p^{\nu})$, $1+p^{\nu}$ being a topoplogical generator
of the multiplicative group $U_p \cong \Zp$. An arbitrary character
$\chi \in X_S$ can be uniquely represented in the form $\chi = \chi_0 \chi_1$
where $\chi_0$ is trivial on the component $U_p$, and $\chi_1$ is trivial
on the other component
$$
 (\ZZ/p^{\nu}\ZZ)^{\times} \times \prod_{q\neq p} \ZZ_q^{\times}.  
 $$ 
\end{frame}
%\iffalse
\begin{frame}[shrink=5]  
The character $\chi_0$ is called the
{\it tame component}\index{tame component}, and $\chi_1$ the
{\it wild component}\index{wild component} of the character $\chi$. We
denote by the symbol $\chi_{(t)}$ the
(wild) character which is uniquely determined by the condition
  $$ \chi_{(t)}(1+p^{\nu}) = t  $$
with $t \in \Cp$, $|t|_p < 1$.

In some cases it is convenient to use another local coordinate $s$ which is
analogous to the classical argument $s$ of the Dirichlet series:
  $$ \begin{array}{rcl}
       \Or_p  &  \longrightarrow  &  X_S  \\
       s      &  \longmapsto      &  \chi^{(s)} ,
     \end{array}  $$
where $\chi^{(s)}$ is given by
$\chi^{(s)}((1+p^{\nu})^{\al}) = (1+p^{\nu})^{\al s} =
                                 \exp(\al s \log(1+p^{\nu}))$.
The character $\chi^{(s)}$ is defined only for such $s$ for which the series
$\exp$ is $p$-adically convergent (i.e. for $|s|_p < p^{\nu-1/(p-1)}$).
In this domain of values of the argument we have that $t = (1+p^{\nu})^s-1$.
But, for example, for $(1+t)^{p^n} = 1$ there is certainly no such value of
$s$ (because $t \neq 1$), so that the $s$-coordonate parametrizes a
smaller neighborhood of the trivial character than the $t$-coordinate (which
parametrizes all wild characters) (see \cite{Ma73}, \cite{Man76}).
\end{frame}
%[shrink=2]
%\iffalse
%\subsection{The analytic structure of $X_S$}
\begin{frame}[shrink=5]
\frametitle{$p$-adic analytic functions on $X_S$}
Recall that an analytic function $F : T \longrightarrow \Cp$ 
($T = \{ z \in \Cp^{\times} ~~\mid~~ |z-1|_p < 1 \}$), is defined as the
sum of a series of the type $\sum_{i \geq 0} a_i (t-1)^i$ ($a_i \in \Cp$),
which is assumed to be absolutely convergent for all $t \in T$. The notion of
an analytic function is then obviously extended to the whole group $X_S$ by
shifts. The function
  $$ F(t) = \sum_{i=0}^{\infty} a_i (t-1)^i  $$
is bounded on $T$ iff all its coefficients $a_i$ are universally bounded. This
last fact can be easily deduced for example from the basic properties of the
Newton polygon of the series $F(t)$ (see \cite{Koblitz N. (1980)}, \cite{Vis76}). 
If we apply to these series the Weierstrass preparation theorem
(see  \cite{Koblitz N. (1980)}, \cite{Manin Yu.I. (1971)}), % and theorem \ref{THEOWeier})
we see that in this case the function $F$ has only a finite number of zeroes
on $T$ (if it is not identically zero). 
\end{frame} 
%[shrink=2]

%\iffalse
\begin{frame}

In particular, consider the torsion
subgroup $X_S^{\tors} \subset X_S$. This subgroup is discrete in $X_S$ and its
elements $\chi \in X_S^{\tors}$ can be obviously identified with primitive
Dirichlet characters $\chi \mod M$ such that the support $S(\chi) = S(M)$ of
the conductor of $\chi$ is containded in $S$. This identification is provided
by a fixed embedding denoted
$$
 i_p : \Qb^{\times} \hookrightarrow \Cp^{\times}  
$$
if we note that each character $\chi \in X_S^{\tors}$ can be factored through
some finite factor group $(\ZZ/M\ZZ)^{\times}$:
$$
 \chi : \ZZ_S^{\times} \rightarrow (\ZZ/M\ZZ)^{\times}
     \rightarrow  \Qb^{\times} \stackrel{i_p}{\hookrightarrow}
     \Cp^{\times},  
$$
and the smallest number $M$ with the above condition is the conductor of
$\chi \in X_S^{\tors}$.

The symbol $x_p$ will denote the composition of the natural projection
$\ZZ_S^{\times} \rightarrow \Zp^{\times}$ and of the natural embedding
$\Zp^{\times} \rightarrow \Cp^{\times}$, so that $x_p \in X_S$ and all
integers $k$ can be considered as the characters $x_p^k : y \longmapsto y^k$.
\end{frame}
%[shrink=2]
%\iffalse
%\subsection{The analytic structure of $X_S$}
\begin{frame}[shrink=5]
Let us consider a bounded $\Cp$-analytic function $F$ on $X_S$. The above
statement about zeroes of bounded $\Cp$-analytic functions implies now that
the function $F$ is uniquely determined by its values $F(\chi_0 \chi)$, where
$\chi_0$ is a fixed character and $\chi$ runs through all elements
$\chi \in X_S^{\tors}$ with possible exclusion of a finite number of
characters in each analyticity component of the decomposition
(\ref{EQdecdeXS}). This condition is satisfied, for example, by the set of
characters $\chi \in X_S^{\tors}$ with the $S$-complete conductor
(i.e. such that $S(\chi) = S$), and even for a smaller set of characters,
for example for the set obtained by imposing the additional
assumption that the character $\chi^2$ is not trivial (see \cite{Ma73},
\cite{Man76}, \cite{Vis76}).
\end{frame}
%[shrink=2]
%\iffalse
\subsection{The analytic structure of $X_S$}
\begin{frame}[shrink=5]
%[shrink=2]
%\iffalse
\frametitle{$p$-adic Mellin transform}
Let $\mu$ be a (bounded) $\Cp$-valued measure on $\ZZ_S^{\times}$. Then the
{\it non-Archimedean Mellin transform}\index{non-Archimedean Mellin transform}
of the measure $\mu$ is defined by
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.51)  
  \begin{equation}  \label{EQMeltrans}
    L_{\mu}(x) = \mu(x) = \int_{\ZZ_S^{\times}} x \dd \mu, ~~~~ (x \in X_S), 
  \end{equation}
which represents a bounded $\Cp$-analytic function
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.52)  
  \begin{equation}
    L_{\mu} : X_S \longrightarrow \Cp.
  \end{equation}
Indeed, the boundedness of the function $L_{\mu}$ is obvious since all
characters $x \in X_S$ take values in $\Or_p$ and $\mu$ also is bounded. The
analyticity of this function expresses a general property of the integral
(\ref{EQMeltrans}), namely that it depends analytically on the parameter
$x \in X_S$. However, we give below a pure algebraic proof of this fact
which is based on a description of the Iwasawa algebra. This description
will also imply that every bounded $\Cp$-analytic function on $X_S$ is
the Mellin transform of a certain measure $\mu$.
\end{frame}
%\iffalse
%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%----------------------------PARAGRAPHE 1.4.4.-------------------------------
\subsection{The Iwasawa algebra}
\begin{frame}
\frametitle{The Iwasawa algebra}

Let $\Or$ be a closed subring in
$\Or_p = \{ z \in \Cp ~~\mid~~ |z|_p \leq 1 \}$,
$$
 G = \lim_{\stackrel{\longleftarrow}{i}} G_i, ~~~~ (i \in I),  
$$
a profinite group. 
Then the canonical homomorphism
$G_i \stackrel{\pi_{ij}} {\longleftarrow} G_j$
induces a homomorphism of the corresponding group rings
$$ 
\Or[G_i] \longleftarrow \Or[G_j].  
$$
Then the {\it completed group ring}\index{completed group ring} $\Or[[G]]$
is defined as the projective limit
$$
 \Or[[G]] = \lim_{\stackrel{\longleftarrow}{i}} \Or[[G_i]],
     ~~~~(i \in I).  
$$
Let us consider also the set ${\rm Dist}(G,\Or)$ of all $\Or$-valued distributions
on $G$ which itself is an $\Or$-module and a ring with respect to
multiplication given by the
{\it convolution of distributions}\index{convolution of distributions},
which is defined in terms of families of functions
  $$ \mu_1^{(i)}, \mu_2^{(i)} : G_i \longrightarrow \Or,  $$
%$
(see the previous section) as follows:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.53)  
  \begin{equation}
    (\mu_1 \star \mu_2)^{(i)}(y) = \sum_{y=y_1y_2} \mu_1^{(i)}(y_1)
    \mu_2^{(i)}(y_2), ~~~~ (y_1,y_2 \in G_i)
  \end{equation}
Recall also that the $\Or$-valued distributions are identified with
$\Or$-valued measures. Now we describe an isomorphism of $\Or$-algebras
$\Or[[G]]$ and ${\rm Dist}(G,\Or)$. In this case when $G=\Zp$ the algebra
$\Or[[G]]$ is called the {\it Iwasawa algebra}\index{Iwasawa algebra}.

%THEOREM 1.10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{theorem}  \label{THEOisomIsa}
  (a) Under the same notation as above there is the canonical isomorphism of
  $\Or$-algebras
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.54)  
    \begin{equation}
      ;{\rm Dist}(G,\Or) \stackrel{\sim}{\longrightarrow} \Or[[G]].
    \end{equation}

  (b) If $G = \Zp$ then there is an isomorphism
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.55)  
    \begin{equation}  \label{EQisomOGOX}
      \Or[[G]] \stackrel{\sim}{\longrightarrow} \Or[[X]],
    \end{equation}
  where $\Or[[X]]$ is the ring of formal power series in $X$ over $\Or$. 
The
  isomorphism (\ref{EQisomOGOX}) depends on a choice of the topological
  generator of the group $G = \Zp$.
\end{theorem}
\end{frame}
%\iffalse
\begin{frame}[shrink=5]

%'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%---------------------------PARAGRAPHE 1.4.5.---------------------------------

We noticed above that the theorem \ref{THEOisomIsa} would imply a description
of $\Cp$-analytic bounded functions on $X_S$ in terms of measures. Indeed,
these functions are defined on analyticity components of the decomposition
(\ref{EQdecdeXS}) as certain power series with $p$-adically bounded
coefficients, that is, power series, whose coefficients belong to $\Or_p$ after
multiplication by some constant from $\Cp^{\times}$. Formulas for
coefficients of these series can be also deduced from the proof of the theorem.
However, we give a more direct computation of these coefficients in terms of
the corresponding measures. Let us consider the component $aU_p$ of the set
$\ZZ_S^{\times}$ where
  $$ a \in (\ZZ/p^{\nu}\ZZ)^{\times} \times \prod_{q\neq} \ZZ_q^{\times},  $$
and let $\mu_a(x) = \mu(ax)$ be the corresponding measure on $U_p$ defined by
restriction of $\mu$ to the subset $aU_p \subset \ZZ_S ^{\times}$. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\iffalse
\begin{frame}[shrink=5]
Consider the
isomorphism $U_p \cong \Zp$ given by:
  $$ y = \ga^x ~~(x \in \Zp, y \in U_p),  $$
with some choice of the generator $\ga$ of $U_p$ (for example, we can take
$\ga = 1+p^{\nu}$). Let $\mu'_a$ be the corresponding measure on $\Zp$. 
Then
this measure is uniquely determined by values of the integrals
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  (1.56)  
  \begin{equation}  \label{EQintmu}
    \int_{\Zp} \binom{x}{i} d\mu'_a(x) = a_i,
  \end{equation}
with the interpolation polynomials $\binom{x}{i}$, since the $\Cp$-span of the
family
  $$ \left\{ \binom{x}{i} \right\} ~~~~ (i \in \ZZ, i \geq 0)  $$
is dense in $\Cr(\Zp, \Or_p)$ according to Mahler's interpolation theorem
for continuous functions on $\Zp$).
Indeed, from the basic properties of the interpolation polynomials
it follows that
  $$ \sum_i b_i \binom{x}{i} \equiv 0 \pmod{p^n} ~~~~ (\mbox{for all }
     x \in \Zp) \Longrightarrow b_i \equiv 0 \pmod{p^n}.  $$
We can now apply the abstract Kummer congruences (see proposition
\ref{PROPabKum}), which imply that for arbitrary choice of numbers
$a_i \in \Or_p$ there exists a measure with the property (\ref{EQintmu}).

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

%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
%-------------------------------PARAGRAPHE 1.4.6.----------------------------

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

\begin{enumerate}

\item Modular forms,  $L$-functions
and congruences between  modular forms 

\item A traditional method of $p$-adic interpolation
  and the method of canonical projection of modular distributions

 

 \item The use of the Eisenstein series and of the Rankin-Selberg method

The Eisenstein measure by  N.M.Katz, \dots Convolutions of Eisenstein distributions with other distributions

\item {Examples of construction
of $p$-adic $L$-functions}

%Rankin-Selberg $L$-functions, Hecke $L$-functions, 
%symmetric powers, Garrett's triple products ...


% \item Distributions on (the closure of) factorizable functions.
% 
% $p$-adic integration on adelic groups (to develop). Euler systems and $p$-adic 
% $L$-functions (ideas of Laurent Berger)

\item Families of modular forms and $L$-functions. %Problem of Coleman-Mazur
 
\end{enumerate}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=3]
\frametitle{Modular forms as a  tool in arithmetic} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabular}{ll}
%\hskip-0.3cm
\pause
 \begin{tabular}{l}  
We view modular forms as:\\
\ 

1) {\blue $q$-power series}
\\ 
$\ds f=\sum_{n=0}^\infty a_n q^n\in {\mathbb C}[[ q]]$
and as
\\ \pause
 \begin{tabular}{l}
2) {\blue holomorphic functions}
\\  
on the
 upper half plane
 \\
$ \HH = \{ z\in {\mathbb C}\ |\ \Im\ z>0\}$\\
 \end{tabular}
 \end{tabular}
\hskip0.3cm
\pause
 \begin{tabular}{|l} 
where 
$q=\exp(2\pi i z)$, \\
$z\in \HH$, 
and define 
\\
$L$-function \\ 
\blue
$\ds L(f, s,\chi)=\sum_{n=1}^\infty \chi(n)a_n n^{-s}$ \\
for a
Dirichlet character 
\\
$\chi:(\Z/N\Z)^*\to {\mathbb C}^*$
\black
%\\
(its Mellin transform)
 \end{tabular}
 \end{tabular}
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pause

\

{\large\blue A famous example: the Ramanujan function 
$\tau (n)$}
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
 \begin{tabular}{ll} 
 \begin{tabular}{l|}  
The function 
$\Delta$ (of the variable $z$)
\\
is defined by the formal expansion 
\\
$\Delta=\sum_{n=1}^\infty  \tau (n) q^n$
\\ 
$=
q
\prod_{m=1}^\infty (1-q^m)^{24}$
\\ 
$= q - 24q^2 + 252q^3 + \cdots$
\\
is a  cusp form of weight $k=12$
\\
for the group $\Ga = \SL_2(\Z)$).
%%%%%%%%%%%%%%%%%%%%%%%%%
 \end{tabular} 
% \hskip-0.3cm 
&\begin{tabular}{l} 
$\tau (1)=1,  \tau (2)=-24$,\\
$  \tau (3) =252, \tau(4)=-1472$\\
$\tau (m) \tau (n) =\tau (mn)$ \\  for $(n,m)=1$, \\
$|\tau(p)|\le 2p^{11/2}$  \\ 
for all primes $p$ .
 \end{tabular}
 \end{tabular}
\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}%[shrink=3]
\frametitle{Classical modular forms}%\label{ii43.2.} 
are introduced
 as certain holomorphic functions on the
 upper half plane\index{Upper half plane}
 $ \mathbb H = \{ z\in \mathbb C\ |\ \Im\ z>0\}$,
%\end{frame}
%\end{document} 
which can be regarded as a homogeneous space
 for the group $G(\R) = \GL_2(\R)$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.83)
\begin{align}\label{ii43.1}
 {\mathbb H}
 =
 \GL_2(\mathbb R) / \Or (2)\cdot Z,
 %\eqno (3.1)
\end{align}
 where $Z = \{ \mat x,0,0,x,\ | x\in \R^\times\}$
 is the center of $G(\R)$ and $\Or (2)$ is the orthogonal group.
The group $\GL_2^+(\R)$
 of matrices $\ga = \mat a_\ga, b_\ga, c_\ga, d_\ga,$ with positive 
determinant
 acts on ${{\mathbb H}}$ by
 fractional linear
 transformations;\index{Fractional linear transformation}
 on cosets (\ref{ii43.1}) this action transforms
 into the natural action by group shifts. 
Let $\Gamma$ be a
 subgroup of finite index in the modular group 
$\SL_2(\Z)$.
\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=3]
\frametitle{Definition of a modular form}
A holomorphic function $f : {{\mathbb H}} \to \C$
 is called a
 modular form\index{Modular form}
 of (integral) weight $k$ with respect to $\Ga$
 iff the conditions a) and b) are satisfied:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.84)
%\medskip
\begin{itemize}
\item{a)} {\it Automorphy condition}
\begin{align}\label{ii43.2}
 f((a_\ga z + b_\ga)/(c_\ga z + d_\ga))
 =
 (c_\ga z + d_\ga)^k f(z)
% \eqno(3.2)
\end{align}
 for all elements $\ga \in \Ga$;

\item{b)} {\it Regularity at cusps}:
$f$ is regular at
 cusps\index{Cusp}
 $z \in \Q \cup i\infty$
 (the cusps can be viewed as fixed points of parabolic elements of 
$\Ga$);
 this means that for each element $\si = \mat a, b, c, d, \in 
\SL_2(\Z)$
 the function $(cz + d)^{-k}f\left(\frac {az + b} {cz + d} \right )$
 admits a Fourier expansion
 over non--negative powers of $q^{1/N} = e(z/N)$
 for a natural number $N$.
One writes traditionally 
$$
 q = e(z) = \exp (2\pi i z).
$$
A modular form
$$
f(z) = \sum _{n=0}^\infty a(n) e(nz/N)
$$
 is called a
 cusp form\index{Cusp!-- form}
 if $f$ vanishes at all cusps
 (i.e. if the above Fourier expansion
 contains only positive powers of $q^{1/N}$),
 see 
\cite{Lang S. (1976)}, 
\cite{Ma-Pa05}
\end{itemize}
\end{frame}

\begin{frame}[shrink=3]

The complex vector space of all modular (resp. cusp) forms
 of weight $k$ with respect to $\Ga$
 is denoted by $\Mc_k(\Ga)$ (resp.  $\Sc_k(\Ga)$).

A basic fact from the theory of modular forms
 is that the spaces of modular forms are finite dimensional.
Also, one has $\Mc_k(\Ga)\Mc_l(\Ga) \subset \Mc_{k + l}(\Ga)$.
The direct sum
 $$
 \Mc (\Ga)
 =
 \bigoplus_{k=0}^\infty \Mc _k(\Ga)
 $$
 turns out to be a graded algebra over $\C$
 with a finite number of generators.

An example of a modular form with respect to $\SL_2(\Z)$ of weight $k 
\ge 4$
 is given by the
 {\it Eisenstein series}\index{Eisenstein series}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   (6.85)
\begin{align}\label{ii43.3}
 G_k(z)
 =
 \sum_{m_1, m_2 \in \Z}\!\!\!\!{}^{\displaystyle {}'} 
(m_1+ m_2 z)^{-k}
% \eqno(3.3)
\end{align}
 (prime denoting
 $(m_1, m_2) \not= (0, 0)$).
For these series the automorphy condition (\ref{ii43.2})
 can be deduced straight from the definition.
One has $G_k(z) \equiv 0$ for odd $k$ and 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    (6.86)
\begin{align}\label{ii43.4}
 G_k(z)
 =
 \frac{ 2(2\pi i)^k}{ (k-1)!}
 \left[ - \frac {B_k} {2k}  + \sum_{n=1}^\infty \si_{k-1}(n) e(nz) 
\right ],
% \eqno(3.4)
\end{align}
 where $\si_{k-1} (n) = \sum_{d|n}d^{k-1}$
 and $B_k$ is the $k^{\rm th}$
 {\it Bernoulli number}.\index{Bernoulli number}

The graded algebra $\Mc (\SL_2(\Z))$
 is isomorphic to the polynomial ring
 of the (independent) variables $G_4$ and $G_6$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink=3]
\frametitle{Examples} 
%Eisenstein series and Congruences for the Ramanujan  function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recall that $B_k$ denote the Bernoullli numbers defined by the development
$$
\fraco x, e^x-1, =\sum_{k=0}^\infty B_k\fraco x^k, k!, 
$$
(Numerical table: 
\begin{align*}
&
B_0=1, \ B_1=-\fraco 1, 2,,  \ B_2=\fraco 1, 6, ,  \ B_3=B_5=\cdots=0, \ 
B_4=-\fraco 1, 30,, \ B_6= \fraco 1, 42, , \cr &\ B_8=-\fraco 5, 66, ,\  
B_{12}=\fraco 691, 2730,\ B_{14} =-\fraco 7, 6, ,\ B_{16}=\fraco 3617, 510,\ 
B_{18} = -\fraco 43867, 798, \ , \dots).
\end{align*}
One has 
$$
\z(k)=-\fraco (2\pi i)^k, 2, \ , 
G_k(z)= \fraco (2\pi i)^k, (k-1)!, 
\left[ -\fraco B_k, 2 k, + \sum_{n=1}^\infty
\si_{k-1}(n)q^n\right].
$$ 
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*}
&
E_4(z)=1+240\sum_{n=1}^\infty\si_{3}(n)q^n \in \Mc_4(\SL(2, \Z)),\cr &
E_6(z)=1-504\sum_{n=1}^\infty\si_{5}(n)q^n \in \Mc_6(\SL(2, \Z)),\cr &
E_8(z)=1+480\sum_{n=1}^\infty\si_{7}(n)q^n\in \Mc_8(\SL(2, \Z)),\cr &
E_{10}(z)=1-264\sum_{n=1}^\infty\si_{9}(n)q^n\in \Mc_{10}(\SL(2, \Z)),\cr &
E_{12}(z)=1+\fraco 65520, 691, \sum_{n=1}^\infty\si_{11}(n)q^n
\in \Mc_{12}(\SL(2, \Z)),\cr &
E_{14}(z)=1 - 24\sum_{n=1}^\infty\si_{13}(n)q^n\in \Mc_{14}(\SL(2, \Z)).
\end{align*}
\medskip
\noindent
{\sl Proof } see in  \cite{Se70}.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile=singleslide,shrink=20]
\frametitle{Fast computation of the Ramanujan function:}
 Put
$ \ds
h_k:=\sum_{n=1}^\infty \sum_{d|n}d^{k-1}q^n
=\sum_{d=1}^\infty \frac {d^{k-1}q^d}{1-q^d}.
$ 
The classical fact is that 
$\De=(E_4^3-E_6^2)/1728$ where
$E_4= 1+240h_4$ and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$ E_6= 1 - 504h_6$. 
%\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide,shrink=20]
%\frametitle

\

{\large\blue Computing with PARI-GP} 
 see
\rm \cite{BBBCO}, 
The PARI/GP number theory system),
{\tt http://pari.math.u-bordeaux.fr}

$ \ds
h_k:=\sum_{n=1}^\infty \sum_{d|n}d^{k-1}q^n
=\sum_{d=1}^\infty \frac {d^{k-1}q^d}{1-q^d}
\Longrightarrow
$ 
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{}
\begin{verbatim}
gp > h6=sum(d=1,20,d^5*q^d/(1-q^d)+O(q^20))
gp > h4=sum(d=1,20,d^3*q^d/(1-q^d)+O(q^20)
gp > Delta=((1+240*h4)^3-(1-504*h6)^2)/1728
\end{verbatim}

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

\small

\begin{verbatim}
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 
+ 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 
- 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 
+ 987136*q^16 - 6905934*q^17+ 2727432*q^18 + 10661420*q^19
 + O(q^20)
\end{verbatim}
\end{frame}
%\end{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile=singleslide,shrink=5]
\frametitle{Congruence of Ramanujan}
$\ds
 \tau (n) \equiv \sum_{d|n}d^{11} \ \bmod \ 691:
$

\small
\begin{verbatim}
gp > (Delta-h12)/691
%10 = -3*q^2 - 256*q^3 - 6075*q^4 - 70656*q^5 - 525300*q^6
 - 2861568*q^7 - 12437115*q^8 - 45414400*q^9
 - 144788634*q^10 - 412896000*q^11 - 1075797268*q^12
 - 2593575936*q^13 - 5863302600*q^14 - 12517805568*q^15
 - 25471460475*q^16 - 49597544448*q^17
 - 93053764671*q^18 - 168582124800*q^19 + O(q^20)
\end{verbatim}

{\large \blue More programs of computing  $\tau(n)$ (see \cite{Sloane})}
\begin{verbatim}
 PROGRAM   	

(MAGMA) M12:=ModularForms(Gamma0(1), 12); t1:=Basis(M12)[2]; 
PowerSeries(t1[1], 100); Coefficients($1);

(PARI) a(n)=if(n<1, 0, polcoeff(x*eta(x+x*O(x^n))^24, n))

(PARI) {tau(n)=if(n<1, 0, polcoeff(x*(sum(i=1, (sqrtint(8*n-7)+1)\2,
(-1)^i*(2*i-1)*x^((i^2-i)/2), O(x^n)))^8, n));} 
gp > tau(6911)
%3 = -615012709514736031488
gp > ##
  ***   last result computed in 3,735 ms.
\end{verbatim}
%\end{tabular}

\end{frame}

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


\ifslide
{\section*{Bibliography
%Références
}

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

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\begin{beamercolorbox}{section in foot} 
\vskip-10pt
%%%%%%%%%%%%%%%%%%%
\fbox{
\insertpagenumber}
%\insertframenumber}
{}
\end{beamercolorbox}%
}
\fi
%%%%%%%%%%%%%%%%%%%%%

\ifslide
{\large\blue %Références
Bibliography
}
\fi


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





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


\iffalse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem[Am-Ve]{Ami-Vel74} Y. {\sc Amice}, J. {\sc Velu},
        {\it Distributions $p$-Adiques associ\'ees aux S\'eries de Hecke},
        Journ\'ees Arithm\'etiques de Bordeaux
 (Conf. Univ. Bordeaux (1974)),
        Ast\'erisque N{${}^\circ$} 24/25, Soc. Math. France, Paris (1975), 
p 119-131.
\fi
\beamertemplatebookbibitems

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


\bibitem[BBBCO]{BBBCO} 
{\sc Batut, C., 
Belabas, D., 
Bernardi, H., 
Cohen, H., 
Olivier, M.}:
The PARI/GP number theory system.
{\tt http://pari.math.u-bordeaux.fr}


\bibitem[BS85]{Borevich Z.I. Shafarevich I.R. (1985)} 
%[38]. 
Borevich, Z.I., Shafarevich, I.R. (1985): 
Number Theory. (in Russian).
3rd ed. Nauka,  Moscow (1985).
English transl.: New York/London: Academic Press, 1966.
% Zbl.592.12001, Zbl.121,42.
 
 
 
 \beamertemplatearticlebibitems
\bibitem[Boe-Pa2006]{Boe-Pa6}
{\sc B\"ocherer}, S.,  {\sc Panchishkin},  A.A., 
{\em Admissible $p$-adic measures attached to triple products of elliptic cusp forms},
%accepted in Documenta Math. in March 2006 (a special volume  dedicated to John Coates).
Extra volume : John H.Coates' Sixtieth Birthday (2006), 77-132.
\iffalse
\bibitem[Boe-Schm]{Boe-Sch}
{\sc B\"ocherer, S.},  and {\sc Schmidt, C.-G.}, 
{\em $p$-adic measures attached to Siegel modular forms}, 
Ann. Inst. Fourier 50, \Numero 5, 1375-1443 (2000).
\fi

\beamertemplatebookbibitems
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Chand70]{Chand70} 
%[55]. 
Chandrasekharan, K. Arithmetical functions.
Berlin--Heidelberg--New York,  Springer--Verlag (1970). 
%Zbl.217,316.
\beamertemplatearticlebibitems




\bibitem[Co]{Co}
{\sc Coates, J.} {\em
On $p$--adic $L$--functions.} Sem. Bourbaki, 40eme annee,
1987-88, n${}^\circ$ 701, Asterisque (1989) 177--178.

\bibitem[Co-PeRi]{Co-PeRi}
{\sc Coates, J.} and {\sc Perrin-Riou, B.}, 
{\em On $p$-adic $L$-functions attached to 
motives over ${\Q}$},
Advanced Studies in Pure Math. 17, 23--54 (1989)


\bibitem[Co75]{Coh75} 
H.{\sc Cohen},
        {\it Sums Involving the Values at Negative Integers
 of $L$-Functions of Quadratic Characters},
        Math. Ann. 217 (1975), p. 271-285.

\bibitem[CoPB]{CoPB} R. {\sc Coleman},
{\it $p$-adic Banach spaces and families of modular forms}. 
Invent. Math. 127, \Numero 3, 417-479 (1997)

\bibitem[Co-Ma]{Co-Ma} R. {\sc Coleman}, B. {\sc Mazur},
        {\it The eigencurve. Galois representations 
in arithmetic algebraic geometry}
(Durham, 1996), 1--113,
London Math. Soc. Lecture Note Ser., 254,

\bibitem[CST98]{CST98} R. {\sc Coleman}, G. {\sc Stevens}, J. {\sc Teitelbaum},
        {\it Numerical experiments on families of $p$-adic modular forms}, in:
        Computational perspectives in Number Theory, ed. by 
         D.A. Buell, J.T. Teitelbaum, Amer. Math. Soc., 143-158 (1998).


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


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

\bibitem[Colm98]{Colm98} 
P.{\sc Colmez},
{\it Fonctions $L$ $p$-adiques},
 S\'eminaire Bourbaki,
51 \`eme ann\'ee, 1998--99, 
 \Numero  851. Novembre 1998.
Cambridge Univ. Press, Cambridge, 1998.


\iffalse

\bibitem[Colm03]{Colm03}
{\sc P.~Colmez}
{\em La conjecture de Birch et Swinnerton-Dyer $p$-adique},
Séminaire Bourbaki, exposé n${}^\circ$ 919, juin 2003.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\fi

\beamertemplatebookbibitems
\bibitem[CourPa]{CourPa}
{\sc Courtieu,}M.,  {\sc Panchishkin },A.A.,
{\em Non-Archimedean $L$-Functions and Arithmetical Siegel Modular Forms},
Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)
\beamertemplatearticlebibitems

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

\iffalse
\bibitem[DI98]{DI98}  
{\sc Duke} W. , {\sc  Imamoglu},  \"O.
{\em  Siegel modular forms of small weight.} Math. Annalen 310 (1998),
p. 73-82.
\fi


%\iffalse
\beamertemplatebookbibitems
\bibitem[Fa-Ch90]{Faltings G.  Chai Ching-Li (1990)} G. {\sc  Faltings}, Ching-Li
{\sc  Chai},
        {\it  Degeneration of abelian varieties.}
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 22. Berlin etc.: Springer-Verlag. xii, 316 p. (1990). 
\beamertemplatearticlebibitems

\bibitem[Gou-Ma]{Gou-Ma} F.{\sc Gouv\^ea}, B.{\sc Mazur},
        {\it On the density of modular representations}, in:
        Computational perspectives in Number Theory, ed. by 
         D.A. Buell, J.T. Teitelbaum, Amer. Math. Soc., 127--142 (1998).

\bibitem[HaH]{HaH} Khoai {\sc Ha Huy},
        {\it $p$-adic interpolation and the Mellin-Mazur transform },
        Acta  Math. Viet. 5,  77-99 (1980).
        
        
\bibitem[Ha1]{Har81} M. {\sc Harris},
        {\it The Rationality of Holomorphic Eisenstein Series},
        Invent. Math. 63,  305-310 (1981).

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%\fi
\end{document}

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