Exceptional elliptic curves over quartic fields

Abstract

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\mathbb{Z}/m\mathbb{Z}\oplus \mathbb{Z}/n\mathbb{Z}$, where $m|n$, be a torsion group such that the modular curve $X_1(m,n)$ is an elliptic curve. Let $K$ be a number field such that there is a positive and finite number of elliptic curves $E_T$ over $K$ having $T$ as a subgroup. We call such pairs $(E_T,K)$ \emph{exceptional}. It is known that there are only finitely many exceptional pairs when $K$ varies through all quadratic or cubic fields. We prove that when $K$ varies through all quartic fields, there exist infinitely many exceptional pairs when $T=\mathbb{Z}/14\mathbb{Z}$ or $\mathbb{Z}/15\mathbb{Z}$ and finitely many otherwise.