# Séminaires : Séminaire Théorie des Nombres

 Equipe(s) : fa, tn, tga, Responsables : Marc Hindry, Bruno Kahn, Wieslawa Niziol, Cathy Swaenepoel Email des responsables : cathy.swaenepoel@imj-prg.fr Salle : Adresse : Description http://www.imj-prg.fr/tn/STN/stnj.html

 Orateur(s) G. Stevens - , Titre p-Adic L-Functions and the Evil Twin Date 28/04/2003 Horaire 14:00 à 16:00 Diffusion Résume In the classical theory of modular forms, the Eichler-Shimura theoremassociates parabolic cohomology classes to classical cusp forms.In attempting to extend the Eichler-Shimura theory to $p$-adic analyticfamilies of modular forms one encounters subtle problems of localfreenessand semi-simplicity of Hecke algebras acting on certain overconvergentcohomology spaces. To solve these problems, more concrete knowledgeisneeded about these cohomology spaces.This talk will give an analytic construction of a $p$-adic $L$-functionassociated to the "evil twin" of a classical ordinary eigenform.Adichotomy arises out of this construction: the criticalvalues of the $p$-adic $L$-function either (1) all vanish, or (2) agreewith the complex $L$-values up to standard multipliers. I willexplainhow this is related to semi-simplicity of the Hecke algebra actingontheoverconvergent cohomolgy. Moreover, I will describe joint workwithRobertPollack, in which we establish a sufficient condition for the latterconclusion in terms of modular forms modulo $p$. I willalso presentsome intriquing numerical data on the disribution of the zeroes ofthese$p$-adic $L$-functions that arose out of our work. Salle Adresse