Torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of the rational numbers

TL;DR

This paper investigates the torsion subgroups of rational elliptic curves over the compositum of all $D_4$ extensions of $Q$, establishing finiteness, classifying possible torsion structures, and linking them to $j$-invariants.

Contribution

It proves the finiteness of torsion subgroups over this field and classifies all possible torsion structures, connecting them to elliptic curve $j$-invariants.

Findings

Torsion subgroup of $E(Q(D_4^inite))$ is finite.

Exactly 24 possible torsion structures.

Complete classification of elliptic curves by torsion structure.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their $j$-invariants.