Abelian Covers of ℙ1 of <i>p</i> -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 non-trivial bounds for the Ekedahl–Oort types, and the Newton polygons, at prime $p$ for the curves in the family. In this paper, we investigate whether such 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 [ 12].