Serre-Tate theory for Shimura varieties of Hodge type

TL;DR

This paper extends Serre-Tate theory to the formal neighborhoods of points in the μ-ordinary locus of Hodge type Shimura varieties, revealing a shifted cascade structure and density of CM points, with implications for lifts of abelian varieties.

Contribution

It introduces a shifted cascade structure for the formal neighborhood of μ-ordinary points in Hodge type Shimura varieties and characterizes CM points and lifts in this context.

Findings

Formal neighborhood has a shifted cascade structure.

CM points are dense in the formal neighborhood.

The identity section corresponds to a special lift of the abelian variety.

Abstract

We study the formal neighbourhood of a point in $\mu$-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a shifted cascade. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms in analogy with the Serre-Tate canonical lift of an ordinary abelian variety.