On the Arithmetic Fundamental Lemma in the minuscule case

TL;DR

This paper proves the arithmetic fundamental lemma conjecture in the minuscule case, linking orbital integral derivatives with intersection numbers on moduli spaces, advancing the understanding of the arithmetic Gan-Gross-Prasad conjecture.

Contribution

It provides the first proof of the conjecture in the minuscule case, a key step in the broader program connecting orbital integrals and intersection theory.

Findings

Confirmed the conjecture in the minuscule case

Established new connections between orbital integrals and intersection numbers

Contributed to the progress on the arithmetic Gan-Gross-Prasad conjecture

Abstract

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. We prove this conjecture in the minuscule case.