Séminaires : Séminaire Théorie des Nombres

Equipe(s) : tn,
Responsables :Ziyang Gao, Marc Hindry, Bruno Kahn, João Pedro P. dos Santos
Email des responsables : ziyang.gao@imj-prg.fr
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Orateur(s) G. Stevens - ,
Titre p-Adic L-Functions and the Evil Twin
Date28/04/2003
Horaire14:00 à 16:00
Diffusion
RésumeIn 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.
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