A linear AFL for quaternion algebras

TL;DR

This paper proves new identities related to linear algebraic groups over quaternion division algebras, confirming key conjectures in specific cases involving Hasse invariant 1/2.

Contribution

It establishes fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras, verifying important transfer conjectures.

Findings

Verification of transfer conjecture for Hasse invariant 1/2

Proof of arithmetic transfer conjecture in specific cases

New identities for linear groups over quaternion algebras

Abstract

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjeture and the arithmetic transfer conjecture from arXiv:2307.11716 in cases of Hasse invariant 1/2.