3
$\begingroup$

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 $E(\mathbb{Q})\approx \mathbb{Z}^r\times T$?

$\endgroup$
3
  • $\begingroup$ 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 May 3 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 May 3 at 10:19
  • 1
    $\begingroup$ I found a relevant paper arxiv.org/abs/2003.00077 $\endgroup$ – keiso May 3 at 10:40
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.