Growth of torsion of elliptic curves with full 2-torsion over quadratic cyclotomic fields

TL;DR

This paper investigates how the torsion subgroup of elliptic curves with full 2-torsion over quadratic cyclotomic fields grows in quadratic extensions, classifies possible torsion structures, and analyzes their behavior under quadratic twists.

Contribution

It provides a detailed classification of torsion growth and structures for elliptic curves over specific quadratic cyclotomic fields in quadratic extensions.

Findings

Determines torsion growth patterns in quadratic extensions

Classifies torsion structures arising from quadratic twists

Identifies maximum extensions where torsion grows

Abstract

Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_2 \oplus C_2$ we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.