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Constructing families of 3-Selmer companions · arXivDesk

2502.01351Feb 3, 2025Main theorem updated with more families and many details expanded upon. Final version, accepted to appear in Res. Number Theory

Constructing families of 3-Selmer companions

Harry Spencer

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∈, 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.

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