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<p>Mazur and Rubin introduced the notion of <i>n</i>-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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_647_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\in \mathbb {Z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, such that the two curves in a pair corresponding to a given <i>t</i> are non-isogenous 3-Selmer companions, possibly provided that <i>t</i> satisfies a simple congruence condition.</p>

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Constructing families of 3-Selmer companions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月4日

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  • Harry Spencer

摘要

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 \(t\in \mathbb {Z}\) t ∈ Z , 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.