On the Arithmetic Fundamental Lemma through Lie algebras

TL;DR

This paper proves the equivalence of the Arithmetic Fundamental Lemma (AFL) conjecture to a Lie algebra version in non-degenerate cases, simplifying the proof of AFL for n=3 and advancing understanding of the conjecture.

Contribution

It establishes the equivalence of the AFL conjecture with a Lie algebra version and provides a simplified proof for the case n=3.

Findings

AFL conjecture is equivalent to a Lie algebra version in non-degenerate cases

Simplified proof of AFL for n=3

Reduction to Lie algebra is effective for proving AFL

Abstract

We prove that the Arithmetic Fundamental Lemma conjecture of Wei Zhang is equivalent to a similar conjecture, but for Lie algebras, in the case of non-degenerate intersection. We use this result to give a simplified proof of the AFL for $n=3$. The idea for the reduction to the Lie algebra is due to Wei Zhang.