p-adic Banach modules of arithmetical modular forms and triple products of Coleman's families

Abstract

For a prime number $p\ge 5$, we consider three classical cusp eigenforms $f_j(z)$ of weights $k_1, k_2, k_3$, of conductors $N_1, N_2, N_3$, and of nebentypus characters $\psi_j \bmod N_j$. According to H.Hida and R.Coleman, one can include each $f_j$ into a {$p$-adic analytic family} $k_j \mapsto \{f_{j,k_j}\}$ of cusp eigenforms $f_{j,k_j}$ of weights $k_j$ in such a way that $f_{j,k_j}=f_j$, 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 paper is to describe a four variable $p$-adic $L$-function attached to Garrett's triple product of three Coleman's families $k_j \mapsto \{f_{j,k_j}\}$ 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 $U(\sum_{n\ge 0}^\infty a_{n}q^n) = \sum_{n \ge 0}^\infty a_{np} q^n$. We consider the $p$-adic weight space $X$ containing all $(k{}_j, \psi_j)$. Our $p$-adic $L$-functions are 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 $(k_1, k_2, k_3)\in X^3$ (with a given integers $k_j$ and fixed Dirichlet characters $\psi_j \bmod N$). We construct such a measure from 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 $\Ar$.