Torsion groups of elliptic curves over quadratic fields

TL;DR

This paper develops methods to classify all torsion groups of elliptic curves over quadratic fields, identifies minimal discriminant fields for each group, and explores the relationship between torsion and rank, revealing differences from the rational case.

Contribution

It introduces techniques to determine possible torsion groups over quadratic fields and finds minimal discriminant fields for each, advancing understanding of elliptic curves over quadratic fields.

Findings

Classification of all possible torsion groups over quadratic fields.

Identification of minimal discriminant fields for each torsion group.

Analysis of the relationship between torsion and rank over quadratic fields.

Abstract

We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of it's discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over $\Q$. Finally we give some results concerning the number and density of fields with an elliptic curve with given torsion over them.