Newton Polygons Arising for Special Families of Cyclic Covers of the Projective Line

TL;DR

This paper computes Newton polygons and Ekedahl--Oort types for special families of cyclic covers of the projective line, revealing new examples and patterns in the characteristic p reductions of related Shimura varieties.

Contribution

It provides explicit calculations of Newton polygons and Ekedahl--Oort types for 20 special families, producing new examples and expanding understanding of Jacobians in characteristic p.

Findings

Most Newton polygons appear on the open Torelli locus.

Produced multiple new Newton polygons and Ekedahl--Oort types.

Identified specific Newton polygons for various genera under certain conditions.

Abstract

By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the $\mu$-ordinary Ekedahl--Oort type, occurring in the characteristic $p$ reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl--Oort types of Jacobians of smooth curves in characteristic $p$. Under certain congruence conditions on $p$, these include: the supersingular Newton polygon for genus $5,6,7$; fourteen new non-supersingular Newton polygons for genus $5-7$; eleven new Ekedahl--Oort types for genus $4-7$ and, for all $g \geq 6$, the Newton polygon with $p$-rank $g-6$ with slopes $1/6$ and $5/6$.