FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 3, PAGES 89-

**Triple products of Coleman's families**

A. A. Panchishkin

Abstract

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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) = å_{n=1}^{¥} a_{n,j} e(nz) Î S_{kj}(N_{j}, y_{j})
($j\; =\; 1,2,3$) of
weights $k$_{1}, $k$_{2},
and $k$_{3}, of
conductors $N$_{1}, $N$_{2},
and $N$_{3}, and of
Nebentypus characters $$y _{j} mod
N_{j}.
The purpose of this paper is to describe a four-variable
$p$-adic
$L$-function
attached to Garrett's triple product of three Coleman's families
$k$_{j} →
{f_{j,kj} = å_{n=1}^{¥}
a_{n,j}(k) q^{n}}
of cusp eigenforms of three fixed slopes $$s
_{j} = v_{p}(a
_{p,j}^{(1)}(k_{j})) ³ 0, where $$a
_{p,j}^{(1)} = a
_{p,j}^{(1)}(k_{j}) is an
eigenvalue (which depends on $k$_{j}) of Atkin's
operator $U\; =\; U$_{p}.

Location: http://mech.math.msu.su/~fpm/eng/k06/k063/k06306h.htm

Last modified: July 22, 2006