A weighted invariant trace formula

TL;DR

This paper introduces a new invariant trace formula weighted by automorphic L-functions, extending previous results to include the refined and invariant expansions without removing nontempered contributions.

Contribution

It develops a novel approach to the r-trace formula by establishing an invariant trace formula with weighted spectral terms, extending prior work on the coarse and refined expansions.

Findings

Extended the continuity results of coarse to refined expansions

Established an invariant trace formula with weighted spectral terms

Incorporated basic functions at unramified places

Abstract

This paper begins a new approach to the $r$-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic $L$-functions. This involves extending the results of Finis, Lapid, and M\"uller on the continuity of the coarse expansion of Arthur's noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.