Crystalline Hodge cycles and Shimura curves in positive characteristics

TL;DR

This paper investigates the conditions under which certain curves in positive characteristic can be considered reductions of Shimura curves, focusing on crystalline Hodge cycles and their liftability in the moduli space of abelian fourfolds.

Contribution

It provides a new characterization of Shimura curves in positive characteristic via crystalline Hodge cycles and their liftability criteria.

Findings

Certain conditions on crystalline Hodge cycles imply liftability of curves to Shimura curves.

Established a link between crystalline Hodge cycles and the reduction of Shimura curves.

Identified criteria for lifting curves in moduli space of abelian fourfolds over fields of positive characteristic.

Abstract

In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e. a characterization of curves in positive characteristics which are reduction of Shimura curve over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian fourfolds over k, char k=p. We show that some conditions on the crystalline Hodge cycles over such a curve imply that this curve can be lifted to a Shimura curve.