Affordable Access

$p$-adic $L$-functions for modular forms

Publication Date
  • Law
  • Mathematics


p-adic L-functions for modular forms COMPOSITIO MATHEMATICA SHAIHARAN p-adic L-functions formodular forms Compositio Mathematica, tome 62, no 1 (1987), p. 31-46. <> © Foundation Compositio Mathematica, 1987, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 31 p-adic L-functions for modular forms SHAI HARAN Department of Mathematics, The Hebrew University, Jerusalum, Israel Received 5 May 1986; accepted 7 May 1986 We construct ’many variabled’ adic L-functions for modular forms over arbitrary number field k. We take for our form a weight 2 Hecke eigenform (on GL (2), of level 03930(a)) and for simplicity assume it is cuspidal at infinity. is a finite set of primes away from the level of our form, and (if we want boundedness) is such that for p ~ we can choose a root p. of the p ’th Euler polynomial that is a p-unit. The -adic L-function is given by a measure on the Galois group of the maximal unramified-outside-9’ abelian extension of k; the measure obtained by playing the modular symbol game in an adelic setting. We prove that the -çe-adic L-function interpolates the critical values of the classical zeta function of the twists of our form by finite characters of conductor supported at Y, and that it satisfies a similar functional equation. The gist of the p-adic continuation is the proof that a certain module in which our distribution takes its values is finitely generated, and the idea is to give this module a geometric interpretation as periods

There are no comments yet on this publication. Be the first to share your thoughts.