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Are there any known infinite families of elliptic curves over the rationals, that are proved to have rank $\geq 2$, without assuming finiteness of their Tate-Shafarevich group?

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    $\begingroup$ Sure, take an elliptic surface whose rank over the base is $2$. Then by Silverman's specialisation theorem, all but finitely many fibres have rank at least $2$. As far as I know the open question is to have infinitely many elliptic curves over $\mathbb{Q}$ whose rank is exactly 2 $\endgroup$
    – Chris Wuthrich
    Commented Apr 26, 2023 at 8:55
  • $\begingroup$ Thank you! Yes, what I really had in mind is the open question! So there is no known infinite family of elliptic curves of rank exactly two over the rationals. $\endgroup$
    – EAg
    Commented Apr 26, 2023 at 9:05

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