p-Adic L-Functions and the Evil Twin

(Un exposé de Glenn Stevens au séminaire de théorie des nombres de Chevaleret le 28 avril 2003)

Résumé :

In the classical theory of modular forms, the Eichler-Shimura theorem
associates parabolic cohomology classes to classical cusp forms.
In attempting to extend the Eichler-Shimura theory to $p$-adic analytic
families of modular forms one encounters subtle problems of local
freeness
and semi-simplicity of Hecke algebras acting on certain overconvergent
cohomology spaces.  To solve these problems, more concrete knowledge is
needed about these cohomology spaces.

This talk will give an analytic construction of a $p$-adic $L$-function
associated to the "evil twin" of a classical ordinary eigenform.  A
dichotomy arises out of this construction: the critical
values of the $p$-adic $L$-function either (1) all vanish, or (2) agree
with the complex $L$-values up to standard multipliers.  I will explain
how this is related to semi-simplicity of the Hecke algebra acting on
the
overconvergent cohomolgy.  Moreover, I will describe joint work with
Robert
Pollack, in which we establish a sufficient condition for the latter
conclusion in terms of modular forms modulo $p$.   I will also present
some intriquing numerical data on the disribution of the zeroes of these
$p$-adic $L$-functions that arose out of our work.