A Proof of the Biquadratic Linear AFL for GL(4)
Qirui Li
TL;DR
This work proves the biquadratic Guo–Jacquet Fundamental Lemma and the biquadratic linear Arithmetic Fundamental Lemma for GL$_4$ with the unit test function, by reducing the biquadratic problem to the coquadratic GL$_2$ case via a detailed study of pairs of quadratic embeddings and their coproduct $B_{K_1,K_2}$. The authors develop an explicit algebraic framework, using canonical elements $ extbf{w}$ and $ extbf{z}$ to encode embedding data and to relate analytic orbital integrals to arithmetic intersections on Lubin–Tate spaces; they also establish maximal-order reduction tactics to transfer results from GL$_2$ to GL$_4$ and provide evidence for higher-rank cases on special orbits. The results hold over both $p$-adic fields and local fields of positive characteristic and lay groundwork for global analogues of Gross–Zagier–type formulas in ramified scenarios. The paper thus advances the understanding of local transfer identities in the biquadratic setting and opens paths for extending arithmetic transfer principles to more general algebras and ramified contexts.
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.