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%\title[Horizonthal variation of the Petersson product]
%{Horizontal variation \\ of the $p$-adic Petersson product }

\title{Triple products of Coleman's families
and their periods \\ (a joint work with S.Boecherer)}

\author{A. A.  Panchishkin${}^*$\\
{\tt http://www-fourier.ujf-grenoble.fr/\~{}panchish}
\\
\small e-mail : panchish$@$mozart.ujf-grenoble.fr, 
FAX:  33 (0) 4 76 51 44 78}



\date{}




\maketitle

\footnotetext [1]{
A talk for the 8th Hakuba conference
 ``Periods and related topics from automorphic forms'', 
 September 25 - October 1, 2005
(11:30-12:30 on September 27)
}


\thispagestyle{empty}
%\noindent Institut Fourier, B.P.74, 38402 St.--Martin d'H\`eres, FRANCE

%\noindent 
%e-mail : panchish$@$mozart.ujf-grenoble.fr, 
%FAX:  33 (0) 4 76 51 44 78

%\special {red}
%\abstract
%\abstract{ Résumé}


%\noindent
%{\it To dear John Coates for his sixtieth birthday}

%\newpage
%\thispagestyle{empty}


\BS
\begin{abstract}
%\colour {red}
For a prime number $p\ge 5$,
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
we consider three classical cusp eigenforms  
\begin{align*}
 f_j(z)=\sum_{n=1}^\infty a_{n,j}e(nz)\in \Sr_{k_j}(N_j, \psi_j),\  (j=1, 2,3)
 %f_2(z)=\sum_{n=1}^\infty a_{n,2}e(nz),\ \ \ 
 %f_3(z)=\sum_{n=1}^\infty a_{n,3}e(nz),\ 
\end{align*}
of weights  $k_1, k_2, k_3$, of 
 conductors $N_1, N_2, N_3$,
 and of nebentypus characters  $\psi_j \bmod N_j$ ($j=1, 2, 3$).
%%%%%%%%%%%%%%%%%%%%%%
 
According to H.Hida \cite{Hi86} and R.Coleman \cite{CoPB},
 one can include each $f_j$ $(j=1, 2, 3)$
 (under suitable  assumptions on $p$ and on $f_j$)  into a
 {$p$-adic analytic family}
$$
k_j'\mapsto \{f_{j,k_j'}= \sum_{n=1}^\infty a_{n}(f_{j, k_j'})q^n\}
$$ 
 of cusp eigenforms $f_{j,k_j'}$ of weights $k_j'$ 
 in such a way that $f_{j,k_j}=f_j$ $(j=1,2,3)$,
  and that all their Fourier coefficients $a_n(f_{j, k_j'})$
  are given by 
 certain $p$-adic analytic functions
$k_j'\mapsto a_{n, j}(k_j')$.

The purpose of this talk is to describe a {\it four variable} $p$-adic
$L$ function attached to Garrett's triple product of three Coleman's families
$$
k_j'\mapsto \left \{f_{j,k_j'}= \sum_{n=1}^\infty a_{n,j}(k')q^n\right \} (j=1, 2, 3)
$$  
 of cusp eigenforms of three fixed slopes $\sigma_j=v_p(\alpha_{p, j}^{(1)}(k_j'))\ge 0$
 where $\alpha_{p,j}^{(1)} = \al_{p,j}^{(1)}(k_j')$
 is an eigenvalue (which depends on $k_j'$) of  Atkin's   operator $U=U_p$
 acting on Fourier expansions by
 $U(\sum_{n\ge 0}^\infty a_{n}q^n ) = \sum_{n \ge 0}^\infty a_{np} q^n$.

Let us consider the product of three eigenvalues:
$$
\lambda= \lambda(k_1', k_2', k_3')
= \alpha_{p, 1}^{(1)}(k'_1)\alpha_{p, 2}^{(1)}(k'_2)\alpha_{p, 3}^{(1)}(k_3')
$$
and assume that the slope of this product  
$$
\sigma= v_p(\lambda(k_1', k_2', k_3'))=
\sigma(k_1', k_2', k_3')=
\sigma_1+\sigma_2+\sigma_3
$$
 is {\it constant and positive} for all triplets $(k_1', k_2', k_3')$ 
 in an appropriate $p$-adic neighbourhood
 of the fixed triplet of weights $(k_1, k_2, k_3)$.
The each value  $\sigma_j$ is fixed throughout   this section.

% and that $k'>2\si+2$


{\violet
Our $p$-adic $L$-functions are  {\sf Mellin transforms} of certain
 measures with values in $\Ar$, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Ar=\Ar({\cal  B})$ denotes  an affinoid algebra associated
 with an affinoid space ${\cal  B}$
 as in \cite{CoPB}, where 
${\cal  B}={\cal  B}_1\times{\cal  B}_2\times{\cal  B}_3,
$
 is an affinoid neighbourhood
 around $(k_1, k_2, k_3)\in {\cal W}^3$ (with
 a given  integer $k$ and Dirichlet character $\psi \bmod N$).}
%%%%%%%%%%%%%%%%%%%%%%
We construct such a measure  from {\sf \red higher twists of classical
 Siegel-Eisenstein series}, 
 which produce distributions with values in
 certain Banach $\Ar$-modules $\Mr = \Mr(N;\Ar)$
 of triple modular forms with coefficients in the algebra $\Ar$.\end{abstract}
\ES

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