Newton polygons of cyclic covers of the projective line branched at three points

TL;DR

This paper uses the Shimura-Taniyama method to compute Newton polygons of Jacobians of cyclic covers of the projective line with three branch points, producing new examples in characteristic p.

Contribution

It applies the Shimura-Taniyama method to cyclic covers of the projective line, generating new Newton polygons for Jacobians in positive characteristic.

Findings

Constructed supersingular Newton polygons for genera 4 to 11.

Produced nine non-supersingular Newton polygons with p-rank 0.

Identified Newton polygons with slopes 1/5 and 4/5 for all g ≥ 5.

Abstract

We review the Shimura-Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce multiple new examples of Newton polygons that occur for Jacobians of smooth curves in characteristic $p$. Under certain congruence conditions on $p$, these include: the supersingular Newton polygon for each genus $g$ with $4 \leq g \leq 11$; nine non-supersingular Newton polygons with $p$-rank $0$ with $4 \leq g \leq 11$; and, for all $g \geq 5$, the Newton polygon with $p$-rank $g-5$ having slopes $1/5$ and $4/5$.