Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups

TL;DR

This paper studies the distribution of elliptic curves over number fields with prescribed torsion subgroups, showing that most such curves have exactly the specified torsion, especially when associated modular curves have genus zero.

Contribution

It generalizes Duke's theorem to arbitrary number fields, establishing that elliptic curves with certain torsion subgroups almost always have exactly those subgroups when the modular curve has genus zero.

Findings

Almost all elliptic curves with prescribed torsion subgroup have exactly that subgroup.

The result applies to modular curves of genus zero, including the case of trivial torsion over any number field.

Generalizes Duke's theorem from $\\\\ ext{Q}$ to arbitrary 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.