Constructing families of 3-Selmer companions

TL;DR

This paper constructs families of elliptic curves parameterized by integers, where each pair consists of non-isogenous 3-Selmer companion curves, expanding the known examples and understanding of Selmer companions.

Contribution

It introduces new infinite families of non-isogenous 3-Selmer companion elliptic curves parameterized by integers, with explicit conditions.

Findings

Constructed several families of 3-Selmer companion elliptic curves

Each family is parameterized by an integer t with simple congruence conditions

Provides explicit examples of non-isogenous Selmer companions

Abstract

Mazur and Rubin introduced the notion of $n$-Selmer companion elliptic curves and gave several examples of pairs of non-isogenous Selmer companions. We construct several pairs of families of elliptic curves, each parameterised by $t\in\ZZ$, such that the two curves in a pair corresponding to a given $t$ are non-isogenous $3$-Selmer companions, possibly provided that $t$ satisfies a simple congruence condition.