Torsion groups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d}),$ $0<d<100$

TL;DR

This paper advances the classification of torsion subgroups of elliptic curves over quadratic fields with square-free integers less than 100, achieving a near-complete classification for most such fields.

Contribution

It provides a comprehensive classification for 49 quadratic fields and identifies potential torsion groups in the remaining cases, improving understanding of elliptic curve torsion structures over quadratic fields.

Findings

Complete classification for 49 quadratic fields.

Potential torsion group $ ext{Z}/16 ext{Z}$ in 11 remaining fields.

Progress towards full classification of torsion subgroups.

Abstract

We prove results towards classifying the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$, where $0<d<100$ is a square-free integer, and obtain a complete classification for 49 out of 60 such fields. Over the remaining 11 quadratic fields, we cannot rule out the possibility of the group $\mathbb{Z}/16\mathbb{Z}$ appearing as a torsion group of an elliptic curve.