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Modular curves of prime-power level with infinitely many quadratic points

Michael Cerchia, Rakvi

Abstract

We completely determine the $1085$ open subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level that satisfy $-I \in H$ and $\operatorname{det}(H)=\widehat{\mathbb{Z}}^{\times}$ for which the corresponding modular curve $X_H$ has infinitely many quadratic points. When $g(X_H)\geq 2$ this is equivalent to determining all the hyperelliptic modular curves of prime-power level and all the bielliptic modular curves of prime-power level that admit a degree two map to a positive rank elliptic curve. From the moduli perspective, this means that there are exactly 1085 subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level for which there are infinitely many elliptic curves $E/K$ over quadratic extensions such that $ρ_E(G_k)$ is conjugate to a subgroup of $H$.

Modular curves of prime-power level with infinitely many quadratic points

Abstract

We completely determine the open subgroups of of prime-power level that satisfy and for which the corresponding modular curve has infinitely many quadratic points. When this is equivalent to determining all the hyperelliptic modular curves of prime-power level and all the bielliptic modular curves of prime-power level that admit a degree two map to a positive rank elliptic curve. From the moduli perspective, this means that there are exactly 1085 subgroups of of prime-power level for which there are infinitely many elliptic curves over quadratic extensions such that is conjugate to a subgroup of .

Paper Structure

This paper contains 32 sections, 18 theorems, 16 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation and Previous Work
  3. Strategy and Outline
  4. Comments on code
  5. Acknowledgments
  6. Background and preliminary results
  7. Subgroups of $\operatorname{GL}_2({\widehat{\mathbb{Z}}})$ and Congruence Subgroups
  8. Modular functions
  9. Modular Curves
  10. Jacobian Decomposition
  11. Gonality
  12. Enumerating subgroups to check for hyperelliptic and bielliptic modular curves
  13. Genus 0
  14. Genus 1
  15. The remaining seven genus one cases
  16. ...and 17 more sections

Key Result

Theorem 1

Let $H \subseteq \operatorname{GL}_2({\widehat{\mathbb{Z}}})$ be open with $-I \in H$ and $\operatorname{det}(H)={\widehat{\mathbb{Z}}}^\times$, and assume that the level of $H$ is a prime power. If $X_H$ has infinitely many quadratic points, then the level of $H$ is at most 131.

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • proof
  • Remark 5
  • Remark 6
  • Theorem 7
  • proof
  • Proposition 8
  • ...and 35 more