Global rigid inner forms and multiplicities of discrete automorphic representations

TL;DR

This paper constructs a Galois gerbe over a number field to connect local and global endoscopy, providing explicit formulas and pairings that advance understanding of automorphic representations and the Arthur-Selberg trace formula.

Contribution

It introduces a canonical Galois gerbe linking local and global endoscopy, and explicitly constructs transfer factors and pairings crucial for automorphic spectrum analysis.

Findings

Expressed the adelic transfer factor as a product of local factors

Constructed the pairing between L-packets and S-groups explicitly

Proved some of Arthur's conjectural expectations

Abstract

Given a number field we construct a canonical Galois gerbe over it and show that its cohomology provides a bridge between the refined local endoscopy introduced in arXiv:1304.3292 and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur-Selberg trace formula as a product of normalized local transfer factors, we give an explicit construction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group, and we give a proof of some expectations of Arthur.