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Is there any known example today of elliptic curves of rank $r\geq 2$ with finite Tate-Shafarevich group ? As far as I have searched the internet, I did not find such a statement. Is someone aware of such an example ?

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  • $\begingroup$ You mean where the rank has been proven to be finite? Probably not? $\endgroup$
    – Mathmo123
    Jun 23 at 11:41
  • $\begingroup$ @Mathmo123 rank proven to be finite for example. $\endgroup$ Jun 23 at 13:23
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    $\begingroup$ Yeah, I'm pretty sure that there are no known examples. Hopefully, someone more knowledgable can weigh in, but my understanding is that all the proofs of finiteness of Sha, even in individual examples, rely on known cases of BSD. Specifically, there are ways to compute $\mathrm{Sha}[p]$ for any given $p$, but that still leaves an infinite computation to prove finiteness. $\endgroup$
    – Mathmo123
    Jun 23 at 15:09
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    $\begingroup$ I don't think the state of the art has changed too much in the 10 years since this question was asked, except that Cremona's database has been subsumed into the LMFDB. We know virtually nothing about Sha. For example, for any given $p$, it is an open question whether there exists even a single elliptic curve with $\mathrm{Sha}(E)[p]\ne 0$. $\endgroup$
    – Mathmo123
    Jun 23 at 15:11
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    $\begingroup$ Perhaps we can mention a theorem of Kato (although it’s more than 10 years old – see Colmez’s Bourbaki talk, number 919) according to which, given an elliptic curve $E$ and a (large enough? good reduction? I can’t find the condition right now) prime $p$, the $p^{\infty}$-part of the Tate-Shafarevich group is finite if the $p$-adic $L$-function of the elliptic curve vanishes with the (exact) correct order at $s=1$. $\endgroup$
    – Aphelli
    Jun 23 at 16:40

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