The Birch and SwinnertonDyer Conjecture is well known in the current literature
http://en.wikipedia.org/wiki/Birch_and_SwinnertonDyer_conjecture
My question is about the possible equivalent statements of the Birch and SwinnertonDyer Conjecture
The Birch and SwinnertonDyer Conjecture is well known in the current literature http://en.wikipedia.org/wiki/Birch_and_SwinnertonDyer_conjecture My question is about the possible equivalent statements of the Birch and SwinnertonDyer Conjecture 


See e.g. Bloch's article http://link.springer.com/article/10.1007%2FBF01402274?LI=true 


Weaker version of BSD (Parity Conjecture): $$(1)^{\mathrm{rank}(E/K)} = w(E/K),$$ where $w(E/K)$ ( +1 or 1) is the global root number of $E/K$. Standard statement: BSD I: If $K$ is a number field and $E$ is an elliptic curve over $K$, then $$\mathrm{ord}_{s=1} L(E/K,s) = \mathrm{rank} (E/K), $$ where $\mathrm{rank}(E/K)$ := Analytic rank of $E$ over $K$ := MordellWeil rank of $E$ over $K$. BSD II: The order of $Ш$ is finite and the leading coeffcient of $L(E/K,s)$ at $s=1$ is given by $$\lim_{s \to 1} \frac{L(E/K,s)}{ (s1)^r} = \frac{R.Ш.C}{\sqrt{{\triangle}_K} {T}^2 },$$ where $r$ is the MordellWeil rank of $E/K$, $R$ is the regulator of $E/K$ (with respect to the NeronTate height pairing), $Ш$ is the order of the TateShafarevich group, $T$ is the order of the torsion group, $\triangle_K$ is the discriminant of $K$ and $C$ = $\prod_{v} c_{v} $ is the product of the local tamagawa numbers ($v$ varies over places of $K$). When $K$ is a function field over a finite field of +ve characteristic, $\mathrm{ord}_{s=1} L(E/K,s) = \mathrm{rank} (E/K) \iff Ш < \infty \iff Ш_l^{\infty} < \infty$ for some $l \iff \mathrm{ord}_{s=1} L(E/K,s) \leq \mathrm{rank} (E/K)$ 

