On canonicity for integral models of Shimura varieties with hyperspecial level

TL;DR

This paper introduces a new approach to defining and constructing integral canonical models of Shimura varieties, extending previous work to include exceptional cases and large primes, with a focus on non-emptiness of various strata.

Contribution

It provides a novel definition and construction method for integral canonical models of Shimura varieties, applicable to exceptional cases and large primes, using the notion of an aperture.

Findings

Unified proof of non-emptiness of all Newton, Ekedahl--Oort, and central leaves.

Generalization of Tate's full faithfulness theorem for p-divisible groups.

Characterization of maps into the canonical model from normal, flat, and excellent schemes.

Abstract

We give a new definition -- and in some cases, a new construction -- of integral canonical models of Shimura varieties that uses the notion of an aperture appearing in work of Gardner--Madapusi on some conjectures of Drinfeld. This applies to Shimura varieties of pre-abelian type at odd primes of hyperspecial level, recovering and extending previous work of Kisin, Kim--Madapusi and Imai--Kato--Youcis, but also to exceptional Shimura varieties for large enough primes. The characterization in the exceptional case is \emph{a priori} different from the one recently shown by Bakker--Shankar--Tsimerman, and recovers many of their results, such as the existence of prime-to-$p$ Hecke operators, the non-emptiness of the $\mu$-ordinary stratum and the theory of the canonical lift. In fact, we give a uniform proof of the non-emptiness of \emph{all} possible Newton strata, and of the non-emptiness of Ekedahl--Oort strata and central leaves as well. An important ingredient in the proofs is a generalization of Tate's full faithfulness theorem for $p$-divisible groups to the context of apertures. This leads to a mapping property for the integral canonical model that characterizes maps into it from all normal, flat and excellent schemes over $\mathbb{Z}_{(p)}$.