Abelian covers of ${\mathbb{P}}^1$ of $p$-ordinary Ekedahl-Oort type

Abstract

Given a family of abelian covers of ${\mathbb{P}}^1$ and a prime $p$ of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort type, and the Newton polygon, at $p$ of the curves in the family. In this paper, we investigate whether such lower bounds are sharp. In particular, we prove sharpness when the number of branching points is at most five and $p$ sufficiently large. Our result is a generalization under stricter assumptions of [2, Theorem 6.1] by Bouw, which proves the analogous statement for the $p$-rank, and it relies on the notion of Hasse-Witt triple introduced by Moonen in [9].