l-adic Monodromy and Shimura curves in positive characteristics

TL;DR

This paper investigates the geometric properties of Shimura curves in positive characteristic, focusing on their liftability via l-adic monodromy conditions, to better understand their structure and classification in algebraic geometry.

Contribution

It provides a new characterization of Shimura curves in positive characteristic using l-adic monodromy and conditions for their liftability from mod p to characteristic zero.

Findings

Certain l-adic monodromy conditions imply liftability of curves to Shimura curves.

Established criteria for when a curve in moduli space lifts to a Shimura curve.

Enhanced understanding of the reduction and lifting of Shimura curves in 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, in terms of geometry mod p, of curves in positive characteristics which are reduction of Shimura curves over the complex field. Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian varieties over k, char k=p. We show that some conditions on the l-adic monodromy over such a curve imply that this curve can be lifted to a Shimura curve.