Selmer companion curves

TL;DR

This paper introduces the concept of n-Selmer companions for elliptic curves over number fields, providing conditions for their existence and examples of such pairs, enhancing understanding of Selmer groups in number theory.

Contribution

It establishes sufficient conditions for elliptic curves to be n-Selmer companions and provides explicit examples of non-isogenous pairs, expanding the classification of elliptic curve relationships.

Findings

Identified criteria for n-Selmer companionship.

Constructed examples of non-isogenous elliptic curve pairs.

Enhanced understanding of Selmer group behavior under quadratic twists.

Abstract

We say that two elliptic curves E_1, E_2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character \chi of K, there is an isomorphism between the n-Selmer groups Sel_n(E_1^\chi/K) and Sel_n(E_2^\chi/K) of the quadratic twists E_1^\chi, E_2^\chi. We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenous pairs of companions.