Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

TL;DR

This paper investigates the torsion subgroups of rational elliptic curves over the compositum of all cubic fields, establishing finiteness, classifying possible structures, and providing explicit parameterizations and invariants.

Contribution

It proves the finiteness of torsion subgroups over the compositum of all cubic fields and classifies all possible structures with explicit descriptions and parameterizations.

Findings

Torsion subgroup of E(Q(3^∞)) is finite.

20 possible torsion structures identified.

Explicit parameterizations and j-invariants provided.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not.