Exceptional elliptic curves over quartic fields

TL;DR

This paper investigates the finiteness of elliptic curves with specific torsion groups over quartic fields, discovering infinite exceptional pairs for certain torsion groups and finiteness for others.

Contribution

It extends the classification of exceptional elliptic curves with prescribed torsion over quartic fields, identifying cases with infinitely many such pairs.

Findings

Infinite exceptional pairs for T=Z/14Z and Z/15Z.

Finitely many exceptional pairs for other torsion groups.

Generalization of finiteness results from quadratic and cubic to 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 $T=\Z/m\Z \oplus \Z/n\Z$, 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 $T=\Z/14\Z$ or $\Z/15\Z$ and finitely many otherwise.