Ekedahl-Oort Types and Newton Polygons of Abelian Covers of $\mathbf{P}^1$ Branched at Three Points

Abstract

In this paper, we study the Newton polygons and Ekedahl-Oort types of reductions of abelian covers of the projective line branched at three points modulo a prime. We study the natural density of primes where these covers give supersingular and superspecial curves and show they appear much more often than expected. We also show that unlikely Newton polygons and Ekedahl-Oort types in the moduli space of curves appear frequently. Finally, we prove a theorem that provides evidence of Oort's Conjecture about Newton polygons in certain cases and gives new constructions of supersingular curves.