The motivation to this question can be found in:


[About equivalent statements of the Birch and Swinnerton-Dyer Conjecture](https://mathoverflow.net/questions/125050/about-equivalent-statements-of-the-birch-and-swinnerton-dyer-conjecture?rq=1) 


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?



How the equality in the first case is equivalent to the inequality in the last case?
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.


