Triple products of Coleman's families

Abstract

We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. Problem-solving approach in number theory is discussed which is based on the use of generating functions and their links with modular forms. In particular, the critical values of various L-functions of modular forms produce non-trivial but computable solutions of arithmetical problems. Namely, for a prime number $p\ge 5$, we consider three classical cusp eigenforms $$f_j(z)=\sum_{n=1}^\infty a_{n,j}e(nz)\in \Sr_{k_j}(N_j, \psi_j),\ (j=1, 2,3)$$ 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 \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 $f_{j,k_j}=f_j$, 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 $$k_j{}\mapsto \left \{f_{j,k_j{}}= \sum_{n=1}^\infty a_{n,j}(k{}) q^n\right \}$$ 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 $U(\sum_{n\ge 0}^\infty a_{n}q^n) = \sum_{n \ge 0}^\infty a_{np} q^n$.