Integral canonical models for Spin Shimura varieties

TL;DR

This paper constructs integral canonical models for Spin Shimura varieties at ramified primes, extending the Kuga-Satake construction and providing compactifications, with applications to the Tate conjecture and Kudla's program.

Contribution

It develops regular integral models for Spin Shimura varieties at ramified primes, extending classical constructions and establishing their canonical nature.

Findings

Models are of 'relative PEL type' over larger Spin Shimura varieties.

Extension of the Kuga-Satake construction over the integral models.

Construction of good compactifications for the integral models.

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.