A computational proof of the linear Arithmetic Fundamental Lemma of GL$_4$

TL;DR

This paper provides a computational proof of the linear Arithmetic Fundamental Lemma for GL4 over unramified quadratic extensions, confirming a key conjecture in the Langlands program for this case.

Contribution

It offers the first explicit computational proof of the linear AFL for GL4, expanding the understanding of fundamental lemmas in higher rank cases.

Findings

Confirmed the linear AFL for GL4 with the unit element in the spherical Hecke algebra.

Validated the conjecture for unramified quadratic extensions with residue characteristic not equal to 2.

Established measure normalization consistent with hyperspecial subgroups.

Abstract

Let $K/F$ be an unramified quadratic extension of non-Archimedian local fields with residue character not equals to 2. We prove the linear Arithmetic Fundamental Lemma for GL$_4$ with the unit element in the spherical Hecke Algebra. In this article, all measures are normalized by its hyperspecial subgroup.