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Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups

Bo-Hae Im, Hansol Kim

TL;DR

The paper addresses the density of elliptic curves over a number field $K$ with prescribed torsion subgroups. By focusing on genus-zero modular curves $X_1(m,n)$, it shows that if a curve’s torsion over $K$ contains $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$ and $g_{m,n}=0$, then almost all such curves have torsion exactly this group, extending known results from $\mathbb{Q}$ to arbitrary $K$ in the trivial-torsion case. The approach translates torsion questions into Galois-group problems via primitive division polynomials $\Psi_N$ and uses Hilbert's Irreducibility Theorem, together with genus-zero parameterizations and quadratic-twist invariance, to establish density statements without requiring full torsion classifications over $K$. The work provides explicit parameterizations for genus-zero $(m,n)$ and clarifies when larger torsion can occur, with implications for understanding the distribution of torsion structures across number fields. Overall, the paper deepens the connection between modular-curve geometry, Galois theory, and density results for elliptic curves over number fields.

Abstract

Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let $\mathscr{S}_{m,n}$ be the set of elliptic curves $E/K$ defined over $K$ such that $E(K)_{\operatorname{tors}}\supseteq \mathscr{T}\cong \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$. We prove that if the genus of the modular curve $X_{1}(m,n)$ is $0$, then `almost all' $E\in \mathscr{S}_{m,n}$ satisfy that $E(K)_{\operatorname{tors}}= \mathscr{T}$, i.e., not larger than $\mathscr{T}$. In particular, if $m=n=1$, this result generalizes Duke's theorem over $\mathbb{Q}$ to arbitrary number fields $K$ for the trivial torsion subgroup.

Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups

TL;DR

The paper addresses the density of elliptic curves over a number field with prescribed torsion subgroups. By focusing on genus-zero modular curves , it shows that if a curve’s torsion over contains and , then almost all such curves have torsion exactly this group, extending known results from to arbitrary in the trivial-torsion case. The approach translates torsion questions into Galois-group problems via primitive division polynomials and uses Hilbert's Irreducibility Theorem, together with genus-zero parameterizations and quadratic-twist invariance, to establish density statements without requiring full torsion classifications over . The work provides explicit parameterizations for genus-zero and clarifies when larger torsion can occur, with implications for understanding the distribution of torsion structures across number fields. Overall, the paper deepens the connection between modular-curve geometry, Galois theory, and density results for elliptic curves over number fields.

Abstract

Let be a number field. For positive integers and such that , we let be the set of elliptic curves defined over such that . We prove that if the genus of the modular curve is , then `almost all' satisfy that , i.e., not larger than . In particular, if , this result generalizes Duke's theorem over to arbitrary number fields for the trivial torsion subgroup.
Paper Structure (8 sections, 32 theorems, 132 equations, 1 table)

This paper contains 8 sections, 32 theorems, 132 equations, 1 table.

Table of Contents

  1. Introduction
  2. Thin sets and Hilbert's Irreducibility Theorem
  3. $N$-Torsion subgroups and their associated Galois groups
  4. Quadratic Twists
  5. The proofs of the main results
  6. The trivial torsion subgroup
  7. Parameterizing elliptic curves whose $j$-invariant $\ne 0,1728$ and which satisfy Condition $\mathcal{P}\left(m,n\right)$ for $\left(m,n\right) \in T_{g=0} \setminus \left\{\left(1,1\right)\right\}$
  8. Proof of the main theorem

Key Result

Theorem 1.1

For an elliptic curve $E/{\mathbb{Q}}$ defined over ${\mathbb{Q}}$, the torsion subgroup $E({\mathbb{Q}})_{\operatorname{tors}}$ of $E({\mathbb{Q}})$ is isomorphic to one of the following groups: Conversely, each group listed above can be realizable as a torsion subgroup $E({\mathbb{Q}})_{\operatorname{tors}}$ for infinitely many (non-isomorphic) elliptic curves $E$ defined over ${\mathbb{Q}}$.

Theorems & Definitions (68)

  • Theorem 1.1: Mazur
  • Corollary 1.2
  • Definition 1.3
  • Theorem 1.4: Harron_Snowden
  • Corollary 1.5
  • Theorem 1.6: Duke, Zywina
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 58 more