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 

closed as unclear what you're asking by Alex B., Tilman, Stefan Kohl, Chris Godsil, Yemon Choi Jul 20 '14 at 16:22Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 


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 $l \iff \mathrm{ord}_{s=1} L(E/K,s) \leq \mathrm{rank} (E/K)$ 

