Constructing families of 3-Selmer companions

Harry Spencer

Constructing families of 3-Selmer companions

Number Theory 2025-06-26 v2

Authors: 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 $t\in\ZZ$ , 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.

Keywords

elliptic curves Selmer companions 3-Selmer non-isogenous families of curves congruence condition

Cite

@article{arxiv.2502.01351,
  title  = {Constructing families of 3-Selmer companions},
  author = {Harry Spencer},
  journal= {arXiv preprint arXiv:2502.01351},
  year   = {2025}
}

Comments

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

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