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Rational points on modular curves: parameterization and geometric explanations

Maarten Derickx, Sachi Hashimoto, Filip Najman, Ari Shnidman

Abstract

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM $\mathbb{Q}$-rational points on all modular curves in terms of the rational points on finitely many modular curves. Our proof refines Zywina's work to give a (conditional) parameterization of the images of adelic Galois representations of elliptic curves. In particular, we show that there are 41 $j$-invariants of elliptic curves whose associated Galois image does not vary in an infinite family. Using our explicit parameterization, we show that all rational points on all modular curves arise from the geometry of modular curves in a formal sense, confirming a philosophy of Mazur and Ogg.

Rational points on modular curves: parameterization and geometric explanations

Abstract

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM -rational points on all modular curves in terms of the rational points on finitely many modular curves. Our proof refines Zywina's work to give a (conditional) parameterization of the images of adelic Galois representations of elliptic curves. In particular, we show that there are 41 -invariants of elliptic curves whose associated Galois image does not vary in an infinite family. Using our explicit parameterization, we show that all rational points on all modular curves arise from the geometry of modular curves in a formal sense, confirming a philosophy of Mazur and Ogg.
Paper Structure (33 sections, 32 theorems, 47 equations, 2 tables)

This paper contains 33 sections, 32 theorems, 47 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Three interpretations of Mazur's Program B
  3. Parameterization
  4. Geometric characterization
  5. Discussion
  6. Relation to recent work
  7. Outline
  8. Definitions, conventions and notation
  9. Agreeable groups
  10. Toric operators and agreeable modular curves
  11. Twist-parameterized and twist-isolated curves
  12. Families of groups
  13. Twists of a subgroups of GL2Zhat by a character
  14. Twists of modular curves
  15. Examples
  16. ...and 18 more sections

Key Result

Theorem 2

Assume conj: GIC. For $i \in \{1,\ldots, 160\}$, let $(K_i,M_i)$ be the groups listed in Tables table:twist isolated and table:bigtableofgroups, corresponding to the $T_i$-cover $\mathcal{X}_{K_i} \to \mathcal{X}_{M_i}$, where $T_i = M_i/K_i$ and the $\overline{T}_i$-cover $\pi_i \colon X_{K_i} \to

Theorems & Definitions (92)

  • Conjecture 1: Galois Images Parameterization Conjecture
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 5
  • Definition 2.1
  • Definition 2.2: Level
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Modular maps
  • ...and 82 more