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

TL;DR

This paper investigates the properties of abelian covers of the projective line with three branch points, focusing on their reductions modulo primes, and reveals frequent occurrence of special Newton polygons and Ekedahl-Oort types, supporting conjectures and providing new supersingular curve constructions.

Contribution

It provides new insights into the distribution of Newton polygons and Ekedahl-Oort types for abelian covers, including evidence supporting Oort's Conjecture and methods for constructing supersingular curves.

Findings

Supersingular and superspecial curves occur more often than expected.

Unlikely Newton polygons and Ekedahl-Oort types appear frequently in the moduli space.

Theorem supporting Oort's Conjecture and new supersingular curve constructions.

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.