A Proof of the Biquadratic Linear AFL for GL(4)

TL;DR

This paper proves the biquadratic Guo--Jacquet Fundamental Lemma and the biquadratic linear Arithmetic Fundamental Lemma for GL(4), using a reduction to the coquadratic case of GL(2), applicable over p-adic and positive characteristic fields.

Contribution

It introduces a reduction technique from the biquadratic case of GL(4) to the coquadratic case of GL(2), establishing the conjectures for all orbits and extending results to certain GL(2n) cases.

Findings

Proved biquadratic Guo--Jacquet FL for GL(4).

Established biquadratic AFL for GL(4) with the unit test function.

Extended results to special orbits in GL(2n).

Abstract

We prove both the biquadratic Guo--Jacquet Fundamental Lemma (FL) and the biquadratic linear Arithmetic Fundamental Lemma (AFL) for GL(4) with the unit test function. Our approach relies on a detailed study of pairs of quadratic embeddings, which ultimately enables a reduction from the biquadratic case of GL(4) to the coquadratic case of GL(2). We further identify conditions under which the biquadratic case can be derived from the coquadratic case, and show that this reduction allows us to establish the conjectures for all orbits in GL(4). As an additional consequence, we also prove the biquadratic FL for the identity test function in certain special families of orbits in GL(2n). All results hold over both p-adic fields and local fields of positive characteristic.