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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?

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?

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?

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?

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.

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?