Arithmetic transfer for inner forms of $GL_{2n}$

TL;DR

This paper formulates and confirms fundamental and arithmetic transfer conjectures for inner forms of $GL_{2n}$, specifically for division algebras with invariants 1/4 and 3/4, advancing understanding in this area.

Contribution

It introduces new conjectures for inner forms of $GL_{2n}$ and proves them in specific cases involving division algebras with certain invariants.

Findings

Confirmed conjectures for division algebras of invariant 1/4

Confirmed conjectures for division algebras of invariant 3/4

Established foundational results for arithmetic transfer in this context

Abstract

We formulate Guo--Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.