High rank elliptic curves with prescribed torsion group over quadratic fields

TL;DR

This paper constructs high-rank elliptic curves with all possible torsion groups over quadratic fields, establishing new record ranks for most torsion configurations.

Contribution

It provides explicit examples of high-rank elliptic curves for each torsion group over quadratic fields, advancing the understanding of their distribution.

Findings

Existence of elliptic curves with rank >= 2 for all torsion groups except possibly Z/15Z.

Construction of explicit examples for most torsion groups.

Current record ranks for these torsion groups are presented.

Abstract

There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank >= 2.