Generalized Serre--Tate Ordinary Theory

TL;DR

This paper extends Serre--Tate theory to abelian varieties with additional structures, including crystalline Hodge cycles, revealing new phenomena and applying to integral models of certain Shimura varieties.

Contribution

It generalizes Serre--Tate theory to include crystalline Hodge cycles and explores new phenomena in this broader context.

Findings

New phenomena in generalized Serre--Tate theory

Existence results for integral models of specific Shimura varieties

Application to abelian varieties with crystalline Hodge cycles

Abstract

We study a generalization of Serre--Tate theory of ordinary abelian varieties and their deformation spaces. This generalization deals with abelian varieties equipped with additional structures. The additional structures can be not only an action of a semisimple algebra and a polarization, but more generally the data given by some ``crystalline Hodge cycles'' (a $p$-adic version of a Hodge cycle in the sense of motives). Compared to Serre--Tate ordinary theory, new phenomena appear in this generalized context. We give an application of this theory to the existence of ``good'' integral models of those Shimura varieties whose adjoints are products of simple, adjoint Shimura varieties of $D_l^{\bf H}$ type with $l\ge 4$.