Title : Arithmetic Fundamental Lemmas

Abstract : In my talk, I will give an introduction to arithmetic fundamental lemmas (AFL). These are certain identities between two very different kinds of quantities of which the first is algebraic and the second analytic: On the algebraic side, they involve intersection numbers of cycles on integral models of Shimura varieties. Such intersection numbers are closely related to heights and height pairings which are important invariants of varieties over number fields. On the analytic side, they involve derivatives of orbital integrals that are closely related to derivatives of L-functions. The main aim of my talk is to explain all these notions and their connections.

Arithmetic fundamental lemmas were first conjectured by W. Zhang in 2010 in the Gan--Gross--Prasad setting. He later gave a proof of his conjecture in 2019. In between and after that, AFL identities were discovered and proved in other settings as well. During my talk, I will also sketch the current status of these developments