Selmer groups of twists of elliptic curves over K with K-rational torsion points

TL;DR

This paper extends previous results on Selmer groups of elliptic curve twists from rational points over Q to number fields of small degree, providing new theoretical insights and concrete examples.

Contribution

It generalizes a known result about Selmer groups to elliptic curves over small degree number fields with rational torsion points, including explicit examples.

Findings

Generalization of Frey's result to number fields of small degree

Identification of elliptic curves satisfying the new conditions

Provision of explicit examples from Zyw15

Abstract

We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples of elliptic curves coming from [Zyw15] that satisfy the conditions of our Theorem B.