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The motivation to this question can be found in:

About equivalent statements of the Birch and Swinnerton-Dyer Conjecture

My question is about the last equivalences:

$\mathrm{ord}_{s=1} L(E/K,s) = \mathrm{rank} (E/K) \iff |Ш| < \infty \iff |Ш_l^{\infty}| < \infty$ for some $l \iff \mathrm{ord}_{s=1} L(E/K,s) \leq \mathrm{rank} (E/K)$

How the equality in the first case is equivalent to the inequality in the last case? Is this true for curves over rationals?

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This follows (for $A/K$ an Abelian variety over a function field or even for $K/\mathbf{F}_q$ finitely generated) since one always has $r_{\text{an}} \geq r$ ($r_{\text{an}}$ the analytic order $\mathrm{ord}_{s=1}L(A/K,s)$, and $r$ the rank of $A(K)$).

I proved this in my PhD thesis, which will be available in a few months.

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  • $\begingroup$ @ Timo Keller: But it is not true for curves over rationals. $\endgroup$
    – Safwane
    Commented Jul 20, 2014 at 12:40
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    $\begingroup$ @Shpigle: In the thread "About equivalent statements of the Birch and Swinnerton-Dyer Conjecture", it is said that "When $K$ is a function field over a finite field of +ve characteristic," $\endgroup$
    – user19475
    Commented Jul 20, 2014 at 13:05
  • $\begingroup$ @ Timo Keller: Can you please indicate to me the title of your PhD thesis or the possible form for a paper citation. $\endgroup$
    – Safwane
    Commented Oct 2, 2014 at 17:25
  • $\begingroup$ @Shpigle: Please wait a few days. $\endgroup$
    – user19475
    Commented Oct 5, 2014 at 7:11
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    $\begingroup$ @ Timo Keller: Thank you very much . I will read it now. $\endgroup$
    – Safwane
    Commented Oct 21, 2014 at 16:56

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