The Effect of Quadratic Base Change on Torsion of Elliptic Curves

TL;DR

This paper investigates how quadratic base change affects the torsion subgroup of elliptic curves over quadratic fields, providing algorithms and classifications for torsion growth, especially over quadratic cyclotomic fields.

Contribution

It introduces a fast algorithm to find quadratic extensions where torsion grows and classifies torsion growth over quadratic cyclotomic fields.

Findings

Determined torsion growth patterns over quadratic cyclotomic fields.

Developed an efficient algorithm for identifying quadratic extensions with torsion increase.

Complete classification of torsion growth for elliptic curves over quadratic fields.

Abstract

Let $K$ be a quadratic number field and let $E$ be an elliptic curve defined over $K$ such that $E[2] \not\subseteq E(K).$ In this paper, we study the effect of quadratic base change on $E(K)_{\text{tor}}.$ Moreover, for a given elliptic curve $E/K$ with prescribed torsion group over $K,$ (no restriction on its $2$-torsion part) we describe a fast algorithm to find all quadratic extensions $L/K$ in which $E(K)_{\text{tor}} \subsetneq E(L)_{\text{tor}}$ and describe $E(L)_{\text{tor}}$ in each such case. In particular, we determine the growth of $E(K)_{\text{tor}}$ upon quadratic base change when $K$ is any quadratic cyclotomic field, which completes the earlier work of the second author.