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 $$f_j(z)=\sum_{n=1}^\infty a_{n,j}e(nz)\in \Sr_{k_j}(N_j, ψ_j),\ (j=1, 2,3)$$ of weights k_1, k_2, k_3, of conductors N_1, N_2, N_3, and of nebentypus characters ${\psi }_j\mathrm{mod}{N}_j$. 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 $α_{p,j}^{(1)} = \al_{p,j}^{(1)}(k_j{})$ is an eigenvalue (which depends on $k_j$) of Atkin's operator $U=U_p$ acting on Fourier expansions by $U({\sum }_{n\ge 0}^{\infty }{a}_n{q}^n)={\sum }_{n\ge 0}^{\infty }{a}_{np}{q}^n$.

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