The biquadratic Guo-Jacquet and linear arithmetic fundamental lemmas for GL(4) with the unit Hecke function are proved by reduction to the coquadratic GL(2) case.
On the Linear AFL: The Non-Basic Case
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abstract
The linear Arithmetic Fundamental Lemma (AFL) conjecture compares intersection numbers on Lubin--Tate deformation spaces with derivatives of orbital integrals. It has been introduced for elliptic orbits in arXiv:1803.07553 and arXiv:2010.07365. In these cases, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the theory to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which is achieved by exploiting the connected-\'etale sequence. Our theory will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.
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A Proof of the Biquadratic Linear AFL for GL(4)
The biquadratic Guo-Jacquet and linear arithmetic fundamental lemmas for GL(4) with the unit Hecke function are proved by reduction to the coquadratic GL(2) case.