About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]

Question

The Birch and Swinnerton-Dyer Conjecture is well known in the current literature

http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

My question is about the possible equivalent statements of the Birch and Swinnerton-Dyer Conjecture

Answer 1

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

Answer 2

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 $K$ := Mordell-Weil 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 $$\lim_{s \to 1} \frac{L(E/K,s)}{ (s-1)^r} = \frac{R.|Ш|.C}{\sqrt{{\triangle}_K} {|T|}^2 },$$ where $r$ is the Mordell-Weil rank of $E/K$, $R$ is the regulator of $E/K$ (with respect to the Neron-Tate height pairing), $|Ш|$ is the order of the Tate-Shafarevich 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 $l \iff \mathrm{ord}_{s=1} L(E/K,s) \leq \mathrm{rank} (E/K)$