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

Darren Schmidt

TL;DR

We study the distribution of $p$-divisible group invariants for Jacobians of abelian covers of $P^1$ branched at three points in characteristic $p$, using CM from the associated Fermat cover to compute $H^1_{dR}$-level invariants and to define unlikely Newton polygons and Ekedahl-Oort types. A central contribution is a slope theorem that constructs Newton polygons with prescribed slopes for cyclic covers via the Shimura-Taniyama formula, complemented by extensive computational and density results showing that supersingular and superspecial curves, as well as unlikely PN/EO types, occur more frequently than naively expected. The paper also provides infinite families of supersingular curves and evidence toward Oort's conjecture in certain cases, along with constructions of Newton polygons with large denominators. By combining algorithmic exploration with CM/moduli-theoretic methods, the work advances understanding of how $p$-divisible group invariants distribute in families of abelian covers and yields explicit curve constructions with prescribed Newton polygons.

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.

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

TL;DR

We study the distribution of -divisible group invariants for Jacobians of abelian covers of branched at three points in characteristic , using CM from the associated Fermat cover to compute -level invariants and to define unlikely Newton polygons and Ekedahl-Oort types. A central contribution is a slope theorem that constructs Newton polygons with prescribed slopes for cyclic covers via the Shimura-Taniyama formula, complemented by extensive computational and density results showing that supersingular and superspecial curves, as well as unlikely PN/EO types, occur more frequently than naively expected. The paper also provides infinite families of supersingular curves and evidence toward Oort's conjecture in certain cases, along with constructions of Newton polygons with large denominators. By combining algorithmic exploration with CM/moduli-theoretic methods, the work advances understanding of how -divisible group invariants distribute in families of abelian covers and yields explicit curve constructions with prescribed Newton polygons.

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.
Paper Structure (19 sections, 20 theorems, 46 equations)

This paper contains 19 sections, 20 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Computational Results
  3. Theoretical Results
  4. Abelian Covers
  5. Ramification and Inertia Types
  6. Cyclic Covers
  7. Newton Polygons and Ekedahl-Oort Types
  8. Non-cyclic Covers
  9. Computing Newton Polygons and Ekedahl-Oort Types
  10. Computational Results
  11. Density Computations
  12. Unlikely Newton Polygons and Ekedahl-Oort Types
  13. Unlikely Newton Polygons
  14. A New Newton Polygon
  15. Quadratic Excesses and Newton Polygons of Covers
  16. ...and 4 more sections

Key Result

Theorem 1.6

Let $\ell > 3$ be a prime and $n \geq 1$ be coprime to $\ell$. Define $g := (\ell-1)/2$ and suppose $\alpha$ is the number of quadratic residues modulo $\ell$ contained in the intervals $(0,\ell/4)$, $(\ell/3,\ell/2)$, and $(2\ell/3, 3\ell/4)$. If $p$ is a prime with order $g$ in $\left (\mathbf{Z}/

Theorems & Definitions (54)

  • Definition 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Conjecture 1.4: oort_denominators
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Proposition 1.10
  • Proposition 1.11
  • ...and 44 more