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

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 $μ$-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 $5/6$.