Maximal parahoric arithmetic transfers, resolutions and modularity

Abstract

For any unramified quadratic extension of $p$-adic local fields $F/F_0$ $(p>2)$, we formulate several arithmetic transfer conjectures at any maximal parahoric level, in the context of Zhang's relative trace formula approach to the arithmetic Gan--Gross--Prasad conjecture. The formulation involves a way to resolve the singularity of relevant moduli spaces via natural stratifications and modify derived fixed points. By a local-global method and double induction, we prove these conjectures for $F_0$ unramified over $\mathbb Q_p$, including the arithmetic fundamental lemma for $p>2$. Moreover, we prove new modularity results for arithmetic theta series at parahoric levels via a method of modification over $\mathbb F_q$ and $\mathbb C$. Along the way, we study the complex and mod $p$ geometry of Shimura varieties and special cycles. We introduce the relative Cayley map and also establish Jacquet--Rallis transfers at maximal parahoric levels.