We formulate Guo–Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

The linear arithmetic fundamental lemma (AFL) is a conjectural identity of intersection numbers on Lubin–Tate deformation spaces and derivatives of orbital integrals. It was introduced for elliptic orbits by Li, and Howard and Li. For elliptic orbits, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the conjecture 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 relies on an analysis of the connected-étale sequence. Our results will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.

Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.