# Newton polygons arising from 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 \(\mu \)-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 \(g \ge 6\), the Newton polygon with *p*-rank \(g-6\) with slopes 1 / 6 and 5 / 6.

## Keywords

Curve Cyclic cover Jacobian Abelian variety Shimura variety PEL-type Moduli space Reduction*p*-Rank Supersingular Newton polygon

*p*-Divisible group Kottwitz method Dieudonné module Ekedahl–Oort type

## Mathematics Subject Classification

Primary 11G18 11G20 11M38 14G10 14G35 Secondary 11G10 14H10 14H30 14H40 14K22## Notes

### Acknowlegements

This project began at the *Women in Numbers 4* workshop at the Banff International Research Station. Pries was partially supported by NSF grant DMS-15-02227. We thank Liang Xiao, Xinwen Zhu, and Rong Zhou for discussions about the appendix and thank Liang Xiao for the detailed suggestions on the writing of the appendix. We would like to thank the referee for many helpful comments.

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