Growth of torsion of elliptic curves with odd-order torsion over quadratic cyclotomic fields

TL;DR

This paper investigates how the torsion subgroup of elliptic curves over quadratic cyclotomic fields grows in quadratic extensions, focusing on specific torsion structures, quadratic twists, and isogenies to understand their behavior and limitations.

Contribution

It provides a detailed analysis of torsion growth, quadratic twists, and isogenies for elliptic curves over quadratic cyclotomic fields, highlighting new bounds and classifications.

Findings

Torsion structures can grow in controlled ways in quadratic extensions.

Maximum growth of torsion occurs in specific quadratic extensions.

Certain isogenies over $K$ influence torsion growth patterns.

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 $E(K)[2] \approx C_1$ we investigate how a given torsion structure can grow in a quadratic extension and the maximum number of extensions in which it grows. We also study the torsion structures which occur as the quadratic twist of a given torsion structure. In order to achieve this we examine $N$-isogenies defined over $K$ for $N=15,20,21,24,27,30,35$.