A weak form of beyond endoscopic decomposition for the stable trace formula of odd orthogonal groups

TL;DR

This paper demonstrates a weak form of beyond endoscopic decomposition for the stable trace formula of special odd orthogonal groups, extending Arthur's classification and analyzing specific $r$-stable trace formulas.

Contribution

It introduces a weak form of beyond endoscopic decomposition for the stable trace formula of odd orthogonal groups, building on Arthur's work and Langlands $L$-functions.

Findings

Cuspidal component satisfies weak beyond endoscopic decomposition

Analysis of $r$-stable trace formula for standard and second fundamental representations

Results depend on Arthur's classification and properties of Langlands $L$-functions

Abstract

We show that the cuspidal component of the stable trace formula of a special odd orthogonal group over a number field, satisfies a weak form of beyond endoscopic decomposition. We also study the $r$-stable trace formula, when $r$ is the standard or the second fundamental representation of the dual group. The results are consequences of Arthur's works on endoscopic classification of automorphic representations, together with known results concerning a class of Langlands $L$-functions for special odd orthogonal groups.