Torsion of elliptic curves in the compositum of Dihedral fields

Abstract

Let [Formula: see text] be an elliptic curve. Let [Formula: see text] denote the compositum of all fields with Galois group [Formula: see text] over [Formula: see text]. We classify the possible torsion subgroups of [Formula: see text] over [Formula: see text] for [Formula: see text] not divisible by 3 or 4, where [Formula: see text] is the dihedral group on a regular [Formula: see text]-gon. We also obtain upper bounds for the torsion of [Formula: see text] over [Formula: see text] for various semi-direct products [Formula: see text] by way of classifications of torsion over [Formula: see text].