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Suppose $r\geq 0$ is a rank attainable by infinitely many elliptic curves over $\mathbb{Q}$. Let $T$ be one of the fifteen finite abelian groups in Mazur's theorem.

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(\mathbb{Q})\approx \mathbb{Z}^r\times T$?

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  • $\begingroup$ I think we don't understand the rank well enough to prove anything interesting. E.g. what is the set of $r$ attainable by infinitely many curves? $\endgroup$
    – WhatsUp
    Commented May 3, 2021 at 10:16
  • $\begingroup$ @WhatsUp I don't know what that set is but I assume $r$ is already a member of that set $\endgroup$
    – keiso
    Commented May 3, 2021 at 10:19
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    $\begingroup$ I found a relevant paper arxiv.org/abs/2003.00077 $\endgroup$
    – keiso
    Commented May 3, 2021 at 10:40

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Dujella's webpage contains relevant information:

Infinite families of elliptic curves with high rank and prescribed torsion

He also has pages describing rank records for individual curves and for curves defined over quadratic fields.

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