Triple products of Coleman's families
A. A. Panchishkin University of Grenoble 1 — Joseph Fourier
Abstract:
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
$U=U_p$.
UDC:
511.334
Fulltext:
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