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Minimal torsion curves in geometric isogeny classes

Abbey Bourdon, Nina Ryalls, Lori D. Watson

TL;DR

This work introduces minimal torsion curves within fixed geometric isogeny classes of elliptic curves by analyzing the least degree of points on $X_1(N)$ arising from class members. For non-CM rational classes with $N= ext{ell}^k$, it provides a detailed characterization of odd-degree points and explicit divisibility bounds governed by the $ ext{ell}$-adic Galois representations, including several exceptional cases. In CM cases, it derives precise degree formulas depending on whether $ ext{ell}$ splits, is inert, or ramifies in the CM field $K$, tying the minimal degrees to class numbers and discriminants, with explicit conductor behavior. The results combine Serre’s Open Image Theorem, isogeny calculus, and residue-field analyses on modular curves, and are complemented by Magma code to construct and verify minimal torsion scenarios. Together, they illuminate how torsion phenomena across isogeny classes interact with level, Galois actions, and CM structure, yielding both general divisibility principles and precise optimal bounds.

Abstract

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. In the present work, we consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational $j$-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterization upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting.

Minimal torsion curves in geometric isogeny classes

TL;DR

This work introduces minimal torsion curves within fixed geometric isogeny classes of elliptic curves by analyzing the least degree of points on arising from class members. For non-CM rational classes with , it provides a detailed characterization of odd-degree points and explicit divisibility bounds governed by the -adic Galois representations, including several exceptional cases. In CM cases, it derives precise degree formulas depending on whether splits, is inert, or ramifies in the CM field , tying the minimal degrees to class numbers and discriminants, with explicit conductor behavior. The results combine Serre’s Open Image Theorem, isogeny calculus, and residue-field analyses on modular curves, and are complemented by Magma code to construct and verify minimal torsion scenarios. Together, they illuminate how torsion phenomena across isogeny classes interact with level, Galois actions, and CM structure, yielding both general divisibility principles and precise optimal bounds.

Abstract

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a -isogeny class of elliptic curves and , we wish to determine the least degree of a point on the modular curve associated to any . In the present work, we consider the cases where is rational, i.e., contains an elliptic curve with rational -invariant, or where consists of elliptic curves with complex multiplication (CM). If is a power of a single prime, we give a complete characterization upon restricting to points of odd degree, and also in the case where is CM. We include various partial results in the more general setting.
Paper Structure (31 sections, 30 theorems, 57 equations)

This paper contains 31 sections, 30 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. General Approach.
  3. Other Related Work.
  4. Code.
  5. Background and Notation
  6. Conventions.
  7. Galois Representations
  8. Elliptic Curves with an Isogeny
  9. Modular Curves
  10. Complex Multiplication
  11. Complex Multiplication
  12. Preliminary Results
  13. Fields of Definition for Isogenies
  14. General divisibility conditions
  15. Image of Galois Representations Under Isogeny
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a $\overline{\mathbb{Q}}$-isogeny class of elliptic curves with CM by an order in the imaginary quadratic field $K$. For a prime number $\ell$, the the least degree of a point on $X_1(\ell^k)$ associated to any $E \in \mathcal{E}$ is given in Propositions CaseSplit, CaseInert, a

Theorems & Definitions (60)

  • Theorem 1.1
  • Example 1
  • Example 2
  • Theorem 1.2
  • Theorem 1.4
  • Proposition 1.5
  • Remark 1.6
  • Theorem 2.1: Mazur mazur78, Serre Serre81, Bilu, Parent, and Rebolledo BPR13
  • Theorem 2.2: Furio, Lombardo FurioLombardo23
  • Proposition 2.3
  • ...and 50 more