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The $p$-rank stratification of the moduli space of double covers of a fixed elliptic curve

Kevin Chang, Dušan Dragutinović, Steven R. Groen, Yuxin Lin, Natalia Pacheco-Tallaj, Deepesh Singhal

TL;DR

This work analyzes the $p$-rank stratification of the moduli spaces of genus $g$ curves that admit a degree-$2$ map to a fixed elliptic curve $E$ over an algebraically closed field of characteristic $p>2$. The authors develop a framework using the Deligne–Mumford stacks of double covers, including the reduced moduli spaces $\overline{M}_{E,g;n}^{red}$, to study degenerations via boundary clutching morphisms, and prove that the closed $p$-rank strata are pure and have the expected dimensions: $\dim V_f(\overline{\mathcal{B}}_{E,g}) = g-2+f-f_E$ and $\dim V_f(\overline{\mathcal{B}}_g) = g-2+f$. They further provide explicit constructions of smooth double covers with constrained $p$-rank, and investigate supersingular phenomena within bielliptic and $E$-cover loci, including dimension estimates and open questions for higher genus. The results have arithmetic significance, offering existence results for smooth supersingular bielliptic curves and new insights into Oort-type questions for supersingular abelian varieties via bielliptic maps.

Abstract

In this paper we investigate the $p$-rank stratification of the moduli space of curves of genus $g$ that admit a double cover to a fixed elliptic curve $E$ in characteristic $p>2$. We show that the closed $p$-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of $E$ of all the possible values of the $p$-rank on this moduli space.

The $p$-rank stratification of the moduli space of double covers of a fixed elliptic curve

TL;DR

This work analyzes the -rank stratification of the moduli spaces of genus curves that admit a degree- map to a fixed elliptic curve over an algebraically closed field of characteristic . The authors develop a framework using the Deligne–Mumford stacks of double covers, including the reduced moduli spaces , to study degenerations via boundary clutching morphisms, and prove that the closed -rank strata are pure and have the expected dimensions: and . They further provide explicit constructions of smooth double covers with constrained -rank, and investigate supersingular phenomena within bielliptic and -cover loci, including dimension estimates and open questions for higher genus. The results have arithmetic significance, offering existence results for smooth supersingular bielliptic curves and new insights into Oort-type questions for supersingular abelian varieties via bielliptic maps.

Abstract

In this paper we investigate the -rank stratification of the moduli space of curves of genus that admit a double cover to a fixed elliptic curve in characteristic . We show that the closed -rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of of all the possible values of the -rank on this moduli space.

Paper Structure

This paper contains 16 sections, 26 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Structure of this paper
  3. Moduli of double covers of $E$
  4. The boundary $\overline{M}_{E,g}-M_{E,g}$
  5. The reduced moduli spaces $M_{E,g;n}^{{\rm red}} \subset \overline{M}_{E,g;n}^{{\rm red}}$
  6. The $p$-rank stratification of $\mathcal{B}_{E, g}$
  7. The boundary of $\mathcal{B}_{E, g}$
  8. Computing the dimensions of the $p$-rank strata of $\overline{\mathcal{B}}_{E, g}$
  9. The $p$-rank stratification of the locus of bielliptic curves $\mathcal{B}_g$
  10. Applications
  11. Explicit smooth covers of small $p$-rank
  12. Supersingular double covers of $E$
  13. The geometry of $\overline{M}_{E,g}$
  14. Stable maps to Deligne-Mumford stacks
  15. The definition of $\overline{M}_{E,g}$ and $\overline{M}_{E,g;n}$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $g \geq 2$ and $p>2$. For any $f_E \leq f \leq g - 1 + f_E$, the locus $V_f(\overline{\mathcal{B}}_{E, g})$ is pure of dimension $g - 2 + f - f_E$. As a consequence, $V_f(\overline{\mathcal{B}}_g)$ is pure of dimension $g - 2 + f$, for any $0 \leq f \leq g$.

Theorems & Definitions (65)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Example 2.5: Description of the boundary of $\overline{M}_{E, 2}$
  • Corollary 2.6
  • proof
  • ...and 55 more