MAXIMAL PARAHORIC ARITHMETIC TRANSFERS, RESOLUTIONS AND MODULARITY

For any unramified quadratic extension of p-adic local fields F/F0 (p odd), 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 when F0/Qp is unramified and the arithmetic fundamental lemma for any F0. We introduce the relative Cayley map and also establish explicit Jacquet-Rallis transfers at maximal parahoric levels. Moreover, we prove new modularity results for arithmetic theta series with levels via a method of modification over fibers. Along the way, we study the complex and mod p geometry of Shimura varieties and special cycles.