Une formule des traces pour les espaces symétriques. Le cas de Guo-Jacquet

TL;DR

The paper develops a comprehensive relative trace formula for symmetric spaces associated to involutions on a central division algebra, extending Arthur’s framework to a wide class of groups $G= ext{Aut}_D(D^n)$. It constructs a convergent, truncated kernel $K^T_f$ and provides a precise spectral side $J_ ext{χ}^T( ext{η},f)$ and geometric side $J_ ext{o}^T( ext{η},f)$, establishing the equality of spectral and geometric distributions in the tallies of the trace formula. The work includes deep structural results: a descent to centralizers describing nilpotent contributions, a robust theory for semi-simple regular geometric data, and explicit treatment of the case $G= ext{GL}_2(D)$, illustrating the mechanics of descent, truncation, and covariance. These developments illuminate links between automorphic spectra, linear periods, and special values of $L$-functions, and furnish tools for factorizing periods in terms of local data, with potential Jacquet–Langlands and Ichino–Ikeda-type applications.

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.