Fonctions L de paires pour les groupes classiques

TL;DR

This paper compares two definitions of L-functions for classical groups, showing that the Langlands-based L-function has fewer poles than the endoscopic transfer-based one, and relates poles to Eisenstein series and theta lifts under certain conditions.

Contribution

It provides a comparison between two methods of defining L-functions for classical groups and establishes relationships between their poles, Eisenstein series, and theta lifts.

Findings

The Langlands-based L-function has fewer poles than the endoscopic transfer-based L-function.

Poles of the L-functions are linked to poles of Eisenstein series.

For special cases, poles are connected to theta lifts.

Abstract

Let $\pi$ be a square integrable representation of a classical group and let $\rho$ be a cuspidal representation of a general linear group. We can define in two different ways an L-function $L(\rho \times \pi,s)$: first we can use the Langlands parametrization at each places which is now available, thanks to Arthur's work, and secondly we can transfer $\pi$ to a general linear group, using the twisted endoscopy as established by Arthur. In this paper, we compare the two definitions and we prove, as expected, that the first one has less poles that the second one. Assuming that $\pi$ is cuspidal, we link the poles of the first L-function to the poles of the Eisensteins series and when $\rho$ is a quadratic character and when the groupe is a special orthogonal group, we also link theses poles with the theta lifts. We have some hypothesis at the archimedean places.