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.

106 pages, accepted version, to appear in Duke Mathematical Journal

Cited by in corpus (1)

• Modularity of arithmetic special divisors for unitary Shimura varieties (with an appendix by Yujie Xu)