On the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field

TL;DR

This paper proves the arithmetic fundamental lemma conjecture over general odd residue $p$-adic fields, extending previous results from $Q_p$ to broader base fields by leveraging modularity of divisor generating series.

Contribution

The authors generalize the proof of the AFL to arbitrary $p$-adic fields with odd residue cardinality, using modularity and extending key intersection and Green function results.

Findings

Proof of AFL over general $p$-adic fields with odd residue cardinality.

Extension of intersection results to totally real base fields.

Generalization of Green function comparison from $Q$ to totally real fields.

Abstract

We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$ (arXiv:1909.02697), but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out the strategy, we also generalize results of Howard (arXiv:1303.0545) on CM-cycle intersection and of Ehlen--Sankaran (arXiv:1607.06545) on Green function comparison from $\mathbb{Q}$ to general totally real base fields.