Triple products of Coleman's families

A. A. Panchishkin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 3, pp. 89-100.

Résumé

We discuss modular forms as objects of computer algebra and as elements of certain $p$-adic Banach modules. We discuss a problem-solving approach in number theory, which is based on the use of generating functions and their connection with modular forms. In particular, the critical values of various $L$-functions of modular forms produce nontrivial but computable solutions of arithmetical problems. Namely, for a prime number we consider three classical cusp eigenforms $$ f_j(z)=\sum_{n=1}^\infty a_{n,j}e(nz)\in\mathcal S_{k_j}(N_j,\psi_j)\quad (j=1, 2,3) $$ of weights $k_1$, $k_2$, and $k_3$, of conductors $N_1$, $N_2$, and $N_3$, and of Nebentypus characters $\psi_j\bmod N_j$. 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 $$ k_j\mapsto\biggl\{f_{j,k_j}=\sum_{n=1}^\infty a_{n,j}(k)q^n\biggr\} $$ of cusp eigenforms of three fixed slopes $\sigma_j=v_p\bigl(\alpha_{p, j}^{(1)}(k_j)\bigr)\ge0$, where $\alpha_{p,j}^{(1)}=\alpha_{p,j}^{(1)}(k_j)$ is an eigenvalue (which depends on $k_j$) of Atkin's operator $U=U_p$.

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