Tensor decomposition of isocrystals characterizes Mumford curves

TL;DR

This paper characterizes Mumford curves in positive characteristic using tensor decompositions of isocrystals, providing criteria for lifting certain curves to Shimura curves in the moduli space of abelian varieties.

Contribution

It introduces a new approach to defining Shimura curves of Hodge type in positive characteristic through isocrystal tensor decompositions.

Findings

Tensor decomposition of isocrystals indicates liftability to Shimura curves.

In the generic ordinary case, certain curves can be lifted to Shimura curves.

Provides criteria for lifting curves in the moduli space of abelian varieties.

Abstract

We seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics via characterizing curves in positive characteristics which are reduction of Shimura curve over $\mathbb{C}$. In this paper, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties over $k, \text{char} k=p$. We show that in the generic ordinary case, some tensor decomposition of the isocrystal associated to the family imply that this curve can be lifted to a Shimura curve.