Arithmetic transfers, modularity of arithmetic theta series and geometry of local-global Shimura varieties at parahoric levels

Abstract

Firstly, we introduce a method to establish the conjectured arithmetic modularity of arithmetic theta series on unitary Shimura varieties at general levels, which is based on modifications and uniformizations over vertical fibers. We carry out the method at maximal parahoric levels and obtain new arithmetic modularity results. We study the mod 𝑝 geometry of related Shimura varieties and Rapoport–Zink spaces via natural stratifications, in particular their irreducible components. We also formulate and prove some local modularity questions.

Secondly, we formulate some semi-Lie and group version arithmetic transfer conjectures at maximal parahoric levels, relating central derivatives of orbital integrals of explicit test functions to arithmetic intersection numbers of special cycles. The formulation involves a way to resolve the singularity of relevant moduli spaces via natural stratifications and modify derived cycles. They appear naturally in the context of W. Zhang’s relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture (and its 𝑝-adic analogs) for unitary groups, as well as Y. Liu’s work for Fourier– Jacobi cycles. We also obtain regular integral models for these Shimura varieties to do arithmetic intersection theory.

For any unramified quadratic extension of 𝑝-adic local fields (𝑝 > 2) and any maximal parahoric level, using the modification method towards arithmetic modularity, by a local-global method and double induction we establish these arithmetic transfer identities, under mild assumptions on the 𝑝-adic field. In particular, we establish the arithmetic fundamental lemma for all 𝑝-adic fields (𝑝 > 2). We introduce the relative Cayley map as a natural reduction tool.