Torsion subgroups of rational elliptic curves over the compositum of all extensions of generalized $D_4$-type

Harris B. Daniels

Published 2017-10-14Version 1

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 subdirect product of a finite number of transitive subgroups of $D_4$. In this article we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibility 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.