Maximal parahoric arithmetic transfers, resolutions and modularity

TL;DR

This paper formulates and proves arithmetic transfer conjectures at maximal parahoric levels for unramified quadratic extensions, resolving singularities, establishing transfer maps, and deriving modularity results for arithmetic theta series.

Contribution

It introduces new conjectures and proves them using local-global methods, resolving singularities and establishing transfer maps at maximal parahoric levels.

Findings

Proved arithmetic transfer conjectures for unramified quadratic extensions.

Established Jacquet--Rallis transfers at maximal parahoric levels.

Derived modularity results for arithmetic theta series.

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.