2006, VOLUME 12, NUMBER 3, PAGES 89-

Triple products of Coleman's families

A. A. Panchishkin


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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 fj(z) = ån=1¥ an,j e(nz) Î Skj(Nj, yj) (j = 1,2,3) of weights k1, k2, and k3, of conductors N1, N2, and N3, and of Nebentypus characters y j mod Nj. 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 kj → {fj,kj = ån=1¥ an,j(k) qn} of cusp eigenforms of three fixed slopes s j = vp(a p,j(1)(kj)) ³ 0, where a p,j(1) = a p,j(1)(kj) is an eigenvalue (which depends on kj) of Atkin's operator U = Up.

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Last modified: July 22, 2006