Constructing families of 3-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.

arXiv commentMain theorem updated with more families and many details expanded upon. Final version, accepted to appear in Res. Number Theory