Darren Schmidt

Affiliation: University of Florida

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

Abstract: Let $X$ be a curve of genus $g$ that is an abelian cover of the projective line branched at three points. I implemented an algorithm in SageMath that computes the Newton polygon and Ekedahl-Oort type of $X$. For a fixed genus $g$, I compute the natural densities of primes $p$ such that there is a curve of genus $g$ that is an abelian cover of $\mathbf{P}^1$ branched at three points that is supersingular, superspecial, or has an unlikely Ekedahl-Oort type or unlikely Newton polygon. These computations show that supersingular curves, superspecial curves, and unlikely Ekedahl-Oort types/Newton polygons occur much more frequently than expected for such a specific family of curves. Using patterns we found in these computations, we prove results that give examples of supersingular curves for arbitrarily large genus, provide evidence for a conjecture of Oort, and determine the limsup of the supersingular density.

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