Birch and Swinnerton-Dyer conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$, given by a minimal Weierstrass model

$$E:\; y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6},\qquad a_{i}\in\mathbb{Z}.$$

By the Mordell–Weil theorem the group of rational points is finitely generated,

$$E(\mathbb{Q})\;\cong\;\mathbb{Z}^{r}\oplus E(\mathbb{Q})_{\mathrm{tors}},\qquad r\in\mathbb{Z}_{\ge 0}.$$

For a prime $p\nmid N$ set $a_{p}=p+1-\#E(\mathbb{F}_{p})$, where $\#E(\mathbb{F}_{p})$ is the number of points of the reduction of $E$ modulo $p$ (including the point at infinity). For $p\mid N$ set $a_{p}=1$, $-1$, or $0$ according as $E$ has split multiplicative, nonsplit multiplicative, or additive reduction at $p$. Define

$$L(E,s)=\prod_{p\mid N}\left(1-a_{p}p^{-s}\right)^{-1}\;\prod_{p\nmid N}\left(1-a_{p}p^{-s}+p^{1-2s}\right)^{-1},$$

which converges for $\operatorname{Re}(s)>3/2$ and, by the modularity of elliptic curves over $\mathbb{Q}$, extends to an entire function on $\mathbb{C}$; in particular $\operatorname{ord}_{s=1}L(E,s)$ is defined.

Define the following invariants of $E$.

1. The real period

$$\Omega_{E}=\int_{E(\mathbb{R})}\left|\frac{dx}{2y+a_{1}x+a_{3}}\right|,$$

the integral of the absolute value of the invariant differential of the minimal model over the real locus.

2. The regulator $\mathrm{Reg}(E/\mathbb{Q})=\det\big(\langle P_{i},P_{j}\rangle\big)_{1\le i,j\le r}$, where $P_{1},\dots,P_{r}$ generate $E(\mathbb{Q})/E(\mathbb{Q})_{\mathrm{tors}}$ and $\langle\ ,\ \rangle$ is the Néron–Tate height pairing; $\mathrm{Reg}(E/\mathbb{Q})=1$ when $r=0$.

3. For each prime $p$ the Tamagawa number $c_{p}=[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]$, where $E_{0}(\mathbb{Q}_{p})$ is the subgroup of points with nonsingular reduction; $c_{p}=1$ for $p\nmid N$, so $\prod_{p}c_{p}$ is a finite product.

4. The Tate–Shafarevich group

$$\mathrm{Sha}(E/\mathbb{Q})=\ker\Big(H^{1}\big(\mathbb{Q},E\big)\longrightarrow \prod_{v}H^{1}\big(\mathbb{Q}_{v},E\big)\Big),$$

the product running over all places $v$ of $\mathbb{Q}$.

Then:

(A) $\operatorname{ord}_{s=1}L(E,s)=r$.

(B) $\mathrm{Sha}(E/\mathbb{Q})$ is finite and

$$\lim_{s\to 1}\frac{L(E,s)}{(s-1)^{r}}=\frac{\#\mathrm{Sha}(E/\mathbb{Q})\cdot\Omega_{E}\cdot\mathrm{Reg}(E/\mathbb{Q})\cdot\prod_{p}c_{p}}{\big(\#E(\mathbb{Q})_{\mathrm{tors}}\big)^{2}}.$$