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Birch-Swinnerton-Dyer Conjecture: elliptic curves over Q

The Birch-Swinnerton-Dyer conjecture has two parts:

(a) The L-function L(E,s) has an analytic continuation to {\mathbb C}, and

\text{ord}_{s=1} L(E,s) = \text{Mordell-Weil rank of E} = \text{rk}\>E({\mathbb Q})/E({\mathbb Q})_{tors}.

(b) The leading coefficient of L(E,s) at s=1 is

\text{lead}_{s=1} L(E,s) = \frac{c\cdot\Omega \cdot R \cdot |Sha|}{|E({\mathbb Q})_{tors}|^2}

where c=\prod_v c_v is the product of local Tamagawa numbers, \Omega is the real period, R is the regulator (determinant of the N\'eron-Tate height pairing on E({\mathbb Q})/E({\mathbb Q})_{tors}.) and Sha is the (conjecturally finite) Tate-Shafarevich group.

Birch-Swinnerton-Dyer Conjecture: abelian varieties over number fields

Tate has extended the conjecture to all abelian varieties over numbers fields. Suppose A/K is an abelian variety and write A^t for its dual. Then the conjecture reads

(a) The L-function L(A/K,s) has an analytic continuation to {\mathbb C}, and

\text{ord}_{s=1} L(A/K,s) = \text{Mordell-Weil rank of A} = \text{rk}\>A({\mathbb Q})/A({\mathbb Q})_{tors}.

(b) The leading coefficient of L(A/K,s) at s=1 is

\text{lead}_{s=1} L(A/K,s) = \frac{C\cdot\Omega \cdot R \cdot |Sha|}{\sqrt{\Delta_K}\cdot|A({\mathbb Q})_{tors}|\cdot|A^t({\mathbb Q})_{tors}|}

with the ingredients defined as follows:

* R is the regulator, i.e. the determinant of the N\'eron-Tate height pairing

A({\mathbb Q})/A({\mathbb Q})_{tors} \times A^t({\mathbb Q})/A^t({\mathbb Q})_{tors} \longrightarrow {\mathbb Q}.

* Sha is the (conjecturally finite) Tate-Shafarevich group.

* \Delta_K discriminant of K.

To define \Omega (contribution from infinite places) and C (from finite places), fix an exterior form \omega on A/K. Then

\Omega = \prod_{v \text{real}} \int_{A(K_v)} |\omega| \prod_{v \text{complex}} 2\int_{A(K_v)} \omega\wedge \bar\omega. \quad\text{and}\quad C=\prod_{v \text{finite}} c_v ||\omega/\omega_{\text{N\'eron}}||_v.

* c_v is the local Tamagawa number, the number of connected components of A(K_v).

* \omega_{\text{N\'eron}} is the N\'eron differential at v.

* ||x||_v is the normalised v-adic absolute value of x (\>=\>|O_v/x| for x\in O_v).

Birch-Swinnerton-Dyer Conjecture: elliptic curves over Q

Birch-Swinnerton-Dyer Conjecture: abelian varieties over number fields

Known cases

References