On the arithmetic Siegel--Weil formula for orthogonal Shimura varieties over a general totally real field

Abstract

Let $\mathrm{F}/\mathbb{Q}_p$ be a finite extension with $p\ne 2$, and let $V$ be a quadratic space over $\mathrm{F}$ of dimension at least two admitting a self-dual lattice. We prove the local arithmetic Siegel-Weil formula for GSpin Rapoport-Zink spaces formulated by Li and Zhang. We use the global-to-local strategy of Mihatsch and Zhang and its application by Luo. Using a generalization of the prismatic realization functor of Imai, Kato, and Youcis, we construct and study derived special cycles on integral models of SO Shimura varieties and their Rapoport-Zink uniformizations. We derive the local intersection formula from the modularity of special cycles on the generic fiber. As a first application, we prove the corresponding semi-global arithmetic Siegel-Weil identity for orthogonal Shimura varieties over a totally real field $F\ne\mathbb{Q}$.