Congruence relations for Shimura varieties associated to unitary groups

TL;DR

This paper establishes the congruence relation for the mod-p reduction of Shimura varieties linked to unitary groups GU(n-1,1) when p is inert and n is odd, expanding understanding beyond previous cases.

Contribution

It proves the congruence relation for these Shimura varieties in cases where n is odd and p is inert, addressing a gap left by prior work that relied on the density of the ordinary locus.

Findings

Proved the congruence relation for odd n and inert p cases.

Analyzed the supersingular locus in these Shimura varieties.

Extended the understanding of mod-p reductions of unitary Shimura varieties.

Abstract

We prove the congruence relation for the mod-p reduction of Shimura varieties associated to a unitary similitude group GU(n-1,1), when p is inert and n odd. When n is even, this result was obtained by T. Wedhorn and O. B\"ultel using the density of the ordinary locus in the p-isogeny space. This condition fails in our case. A key element is the understanding of the supersingular locus, for which we refer to the two articles by I. Vollaard and T. Wedhorn. The proof makes extensive use of elementary algebraic geometry, but also some deeper results.