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

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 $4/5$.