# Remarks on the arithmetic fundamental lemma

## TL;DR

This paper offers a new, simplified proof of W. Zhang's arithmetic fundamental lemma in the minuscule case by providing an easier method to evaluate the arithmetic intersection number, confirming the conjecture.

## Contribution

It introduces a more straightforward approach to evaluate the intersection number, leading to a new proof of the AFL conjecture in the minuscule case.

## Key findings

- Confirmed the AFL conjecture in the minuscule case
- Provided a simplified evaluation method for intersection numbers
- Enhanced understanding of the arithmetic intersection theory

## Abstract

W. Zhang's arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport-Zink space. In the minuscule case, Rapoport-Terstiege-Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.