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

Abstract

We prove the Arithmetic Fundamental Lemma conjecture over a general $p$-adic field with odd residue cardinality $q\ge \dim V$. Our strategy is similar to the one used by the second author in his proof of the AFL over ${\mathbb{Q}}_p$, 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 our strategy, we also generalize results of Howard (2012) on CM cycle intersection and of Ehlen–Sankaran (2018) on Green function comparison from $\mathbb{Q}$ to general totally real base fields.