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 singularity is resolved via an arithmetic Atiyah flop using formal Balloon-Ground stratification. Then we do intersections and modify derived fixed points on the resolution.By a local-global method and double induction, we prove these conjectures for F 0 unramified over Qp, including the arithmetic fundamental lemma for p > 2. We introduce the relative Cayley map and also establish Jacquet-Rallis transfers of vertex lattices. We also prove new modularity results for arithmetic theta series at maximal parahoric levels via modifications over Fq and C. Along the way, we study the complex and mod p geometry of Shimura varieties and special cycles through natural stratifications.