Vol. 12, No. 4, 2018

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$(\varphi,\Gamma)$-modules de de Rham et fonctions $L$ $p$-adiques

Joaquín Rodrigues Jacinto

Vol. 12 (2018), No. 4, 885–934
Abstract

On développe une variante des méthodes de Coleman et Perrin-Riou permettant, pour une représentation galoisienne de de Rham, construire des fonctions L p-adiques à partir d’un système compatible d’éléments globaux. On obtient de la sorte des fonctions analytiques sur un ouvert de l’espace des poids contenant les caractères localement algébriques de conducteur assez grand. Appliqué au système d’Euler de Kato, cela fournit des fonctions L p-adiques pour les courbes elliptiques à mauvaise réduction additive et, plus généralement, pour les formes modulaires supercuspidales en p. En dimension 2, nous prouvons une équation fonctionnelle pour nos fonctions L p-adiques.

We develop a variant of Coleman and Perrin-Riou’s methods giving, for a de Rham p-adic Galois representation, a construction of p-adic L-functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the p-adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato’s Euler system, this gives p-adic L-functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at p. In the case of dimension 2, we prove a functional equation for our p-adic L-functions.

Keywords
$p$-adic $L$-functions, modular forms, $(\varphi,\Gamma)$-modules, $p$-adic Hodge theory
Mathematical Subject Classification 2010
Primary: 11S40
Secondary: 11R23, 11S37
Milestones
Received: 18 February 2017
Revised: 10 January 2018
Accepted: 23 February 2018
Published: 11 July 2018
Authors
Joaquín Rodrigues Jacinto
University College London
United Kingdom