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

View as HTML     View as gif image

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 $$y _j mod N_j. The purpose of this paper is to describe a four-variable $p$-adic $L$-function attached to Garrett's triple product of three Coleman's families $k$_j → {f_j,kj = å_n=1^¥ a_n,j(k) qⁿ} of cusp eigenforms of three fixed slopes $$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