Invariance Galoisienne des z\'eros centraux de fonctions L

TL;DR

This paper proves the Galois invariance of the vanishing at 1/2 of certain automorphic L-functions and epsilon factors for general linear groups over arbitrary number fields, using advanced cohomological and automorphic techniques.

Contribution

It establishes the Galois invariance of the central vanishing of L-functions for a broad class of automorphic representations without the assumption of totally real fields.

Findings

Galois invariance of vanishing at 1/2 of L-functions

Galois invariance of epsilon factors

Resolution of intertwining operator difficulties

Abstract

Nous d\'emontrons l'invariance Galoisienne de la propri\'et\'e d'annulation en $1/2$ des fonctions L standard ou de Rankin-Selberg pour certaines repr\'esentations automorphes cuspidales alg\'ebriques r\'eguli\`eres autoduales ou autoduales conjugu\'ees de groupes lin\'eaires sur un corps de nombres arbitraire. La d\'emonstration repose sur l'utilisation de la cohomologie pond\'er\'ee de Goresky-Harder-MacPherson et sur la construction de certaines repr\'esentations automorphes discr\`etes pour les groupes classiques comme r\'esidus de s\'eries d'Eisenstein. L'abandon de l'hypoth\`ese ``$F$ totalement r\'eel'' introduit de nouvelles difficult\'es concernant certains op\'erateurs d'entrelacement. Celles-ci sont r\'esolues gr\^ace \`a l'appendice, r\'edig\'e par J.-L. Waldspurger et l'un d'entre nous, d\'emontrant l'holomorphie et la non-annulation de certains op\'erateurs d'entrelacement normalis\'es. Nous d\'emontrons \'egalement l'invariance Galoisienne des facteurs epsilon correspondants, impliquant l'invariance Galoisienne de la parit\'e de l'ordre d'annulation en $1/2$ de ces fonctions $L$. -- We prove the invariance under the Galois group of the vanishing at $1/2$ of standard and Rankin-Selberg L-functions for certain self-dual or conjugate self-dual algebraic cuspidal automorphic representations for general linear groups over an arbitrary number field. The proof uses Goresky-Harder-MacPherson weighted cohomology and the construction of certain discrete automorphic representations for classical groups as residues of Eisenstein series. New difficulties appear concerning certain intertwining operators. These are solved in the appendix by J.-L. Waldspurger and O. Ta\"ibi proving the holomorphy and non-vanishing of these operators. We also prove the Galois invariance of epsilon factors, implying Galois invariance of the parity of the order at $1/2$ of L-functions.