Classification of torsion of elliptic curves over quartic fields

TL;DR

This paper classifies all possible torsion groups of elliptic curves over quartic fields, showing that all such groups either do not occur or occur infinitely often, by analyzing modular curves and their rational points.

Contribution

It provides a complete classification of torsion groups over quartic fields, proving the non-existence of sporadic groups and analyzing modular curves with new methods.

Findings

No sporadic torsion groups over quartic fields.

All torsion groups either do not appear or appear infinitely often.

Most cases handled with rank 0, Hecke sieve, or global methods.

Abstract

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and $E$ ranges over all elliptic curves over $K$. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves $E$. Proving this requires showing that numerous modular curves $X_1(m,n)$ have no non-cuspidal degree $4$ points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.