Integral canonical models for Spin Shimura varieties
Keerthi Madapusi Pera
TL;DR
This work constructs regular integral canonical models for Shimura varieties attached to Spin groups at odd primes by embedding into larger self-dual lattices and realizing a relative PEL-type moduli problem. Central to the approach is the Kuga–Satake construction, which provides a polarized abelian scheme that encodes the Spin data via a canonical tensor in $H^{\otimes(2,2)}$, allowing descent to integral bases and a moduli interpretation of divisors defined by special endomorphisms. Local models and the framework of Vasiu–Zink healthiness underpin the regularity and extension properties, enabling the construction of integral canonical models $\mathscr{S}_K$ with regular compactifications and, in many cases, refined resolutions to handle singular loci. The paper also develops cycles defined by special endomorphisms, studies their geometric properties (e.g., flatness, lci-ness, and irregular loci), and constructs a theory of smooth integral canonical models for non-maximal lattices. Applications include the Tate conjecture for K3 surfaces in odd characteristic and Kudla’s program linking arithmetic intersections to modular forms, highlighting the arithmetic richness of Spin Shimura varieties.
Abstract
We construct regular integral canonical models for Shimura varieties attached to Spin groups at (possibly ramified) odd primes. We exhibit these models as schemes of 'relative PEL type' over integral canonical models of larger Spin Shimura varieties with good reduction. Work of Vasiu-Zink then shows that the classical Kuga-Satake construction extends over the integral model and that the integral models we construct are canonical in a very precise sense. We also construct good compactifications for our integral models. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla's program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.