A weighted stable trace formula I: Basic functions

TL;DR

This paper develops weighted stable trace formulas incorporating automorphic L-functions and basic functions, advancing the understanding of endoscopic and transfer phenomena in automorphic representation theory.

Contribution

It introduces a new weighted stable trace formula framework with basic functions, extending previous invariant trace formulas and proposing a weighted transfer conjecture.

Findings

Established endoscopic and stable trace formulas with weighted spectral terms.

Formulated a weighted Langlands-Shelstad transfer conjecture.

Connected trace formulas to automorphic L-functions and transfer principles.

Abstract

We establish endoscopic and stable trace formulas whose discrete spectral terms are weighted by automorphic $L$-functions, by the use of basic functions that are incorporated into the global spectral and geometric coefficients. This is a continuation of a previous work of the author which established the corresponding weighted invariant trace formula for noncompactly supported test functions. The meromorphic continuation of these weighted trace formulas would yield $r$-trace formulas, and can therefore be seen as precursors to them. Along the way, we formulate a weighted form of the Langlands-Shelstad transfer conjecture, generalizing the weighted fundamental lemma of Arthur.