Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties

TL;DR

This paper investigates the intersection of the Torelli locus with Newton polygon strata in PEL-type Shimura varieties, providing new examples of Newton polygons for smooth curves and confirming intersections in special cases.

Contribution

It develops a clutching method to show non-empty intersections and answers Oort's question positively under certain conditions, expanding understanding of Newton polygons in characteristic p.

Findings

Non-empty intersection of Torelli locus with certain Newton strata.

Construction of infinitely many Newton polygons for smooth cyclic covers.

Proof that all Newton strata intersect the Torelli locus in specific Shimura varieties.

Abstract

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems which demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the twenty special PEL-type Shimura varieties found in Moonen's work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).