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The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken.

Question: Is any result in that direction known to hold for number fields?

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    $\begingroup$ If I'm not mistaken, the finiteness of this group implies the parity conjecture $r_{alg}\equiv r_{an}\pmod 2$, but not the full BSD conjecture. $\endgroup$
    – Sylvain JULIEN
    Aug 23, 2016 at 16:21
  • $\begingroup$ @Sylvain JULIEN: source for that? $\endgroup$
    – The Thin Whistler
    Aug 23, 2016 at 16:22
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    $\begingroup$ That's the problem. I read something along these lines in a special issue of "Pour la science" several years ago, but my memory is so bad that I can't remember the precise statement. $\endgroup$
    – Sylvain JULIEN
    Aug 23, 2016 at 16:32
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    $\begingroup$ found it, thanks. arxiv.org/pdf/1009.5389v1.pdf $\endgroup$
    – The Thin Whistler
    Aug 23, 2016 at 16:33
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    $\begingroup$ There's a version 2 - arxiv.org/abs/1009.5389 Jeez I wish people wouldn't post links to ArXiv temporary PDF files instead of the abstract page (which is easy to find using the paper number). Sorry to sound grumpy, but that is one of my red buttons, and academics should know better! $\endgroup$
    – John R Ramsden
    Aug 24, 2016 at 17:14

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Firstly, the functional field result your state is due to Tate in his Bourbaki talk. In fact he proves that the finiteness of the $p$-primary part of Sha is enough for $p$ different from the characteristic.

For elliptic curves over a number field, the finiteness of Sha (over larger fields) gives us the parity on the BSD conjecture, by work of Dokchitsers.

The analogue of Tate's method carried over to the number field case is what Iwasawa theory for elliptic curves attempts to do. The corresponding result is considerably weaker. Here an example of what one can prove:

Let $E/\mathbb{Q}$. If we can find a prime $p$ of good ordinary reduction (just to make the statement cleaner) such that

  • The $p$-primary of Sha of $E/\mathbb{Q}$ is finite,
  • The canonical $p$-adic height is non-degenerate on $E(\mathbb{Q})$, and
  • The orders of vanishing of the $p$-adic $L$-function and the complex $L$-function agree.

Then the order of vanishing of the complex $L$-function is equal to the rank of $E(\mathbb{Q})$.

As for the precise formula of the leading term, one only gets the formula for the $p$-adic $L$-function, subject to the first two conditions.

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