On the Hodge-Newton filtration for p-divisible groups of Hodge type

TL;DR

This paper extends Katz's Hodge-Newton filtration concept from p-divisible groups to F-crystals associated with unramified local Shimura data of Hodge type, with applications to deformation theory and Shimura varieties.

Contribution

It generalizes Katz's Hodge-Newton filtration to a broader class of F-crystals from Hodge type Shimura data and applies this to deformation theory and congruence relations.

Findings

Established a canonical Hodge-Newton filtration for these F-crystals.

Generalized Serre-Tate deformation theory for Hodge type Shimura data.

Applied deformation theory to study congruences on Shimura varieties.

Abstract

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.