On the Linear AFL: The Non-Basic Case

TL;DR

This paper extends the linear AFL conjecture to all orbits and isogeny classes, reducing non-basic cases to basic ones using the connected-étale sequence, with implications for global arithmetic geometry.

Contribution

It generalizes the linear AFL to non-basic cases, providing a reduction method that broadens its applicability in arithmetic geometry.

Findings

Reduction of non-basic AFL cases to basic cases

Use of connected-étale sequence in the reduction

Relevance for global settings with non-elliptic orbits

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.