Une formule des traces pour les espaces sym\'etriques. Le cas de Guo-Jacquet

TL;DR

This paper develops a trace formula for symmetric spaces linked to involutions of D-modules over number fields, aiming to connect automorphic spectra, linear periods, and special L-values.

Contribution

It establishes a general trace formula for symmetric spaces associated with involutions, extending Arthur's trace formula to new algebraic and geometric contexts.

Findings

Spectral distributions asymptotic to truncated automorphic kernel integrals

Geometric distributions expressed as weighted orbital integrals

Procedure of descent for non-regular data to centralizers

Abstract

In the spirit of Arthur's trace formula, we establish a general trace formula for symmetric spaces associated with the variety of involutions of a finite $D$-module where $D$ is a division algebra central over a number field $F$. Such a formula should be useful for studying the automorphic spectrum of these symmetric spaces and the deep links between linear periods and special values of standard $L$-functions at their center of symmetry. Indeed, our formula yields an identity between spectral distributions, which generalize relative characters built on linear periods, and geometric distributions, which are an extension of relative orbital integrals. We show that the spectral distributions are, in a certain sense, asymptotic to truncated integrals of the components of the automorphic kernel associated with a cuspidal datum: this provides a handle on these distributions and has allowed, in a companion paper, to express some of these distributions in the form of a weighted relative character. The geometric distributions attached to "regular semi-simple" geometric data are expressed as weighted relative orbital integrals. In general, for non-regular geometric data, we introduce a procedure of descent to the centralizer, which allows us to express any geometric distribution in terms of the nilpotent contribution of infinitesimal trace formulas studied in previous papers.