Birch and Swinnerton-Dyer conjecture

Remarks

The refined form of the conjecture predicts the leading coefficient. $$\lim_{s \to 1} \frac{L(E, s)}{(s - 1)^r} = \frac{\Omega_E \cdot \mathrm{Reg}_E \cdot \#\text{Ш}(E/\Q) \cdot \prod_{p \mid N} c_p}{(\# E(\Q)_{\mathrm{tors}})^2},$$ where $\Omega_E$ is the real period, $\mathrm{Reg}_E$ the regulator of the canonical height pairing, $\text{Ш}(E/\Q)$ the Tate–Shafarevich group (conjectured to be finite), and $c_p$ the Tamagawa numbers. Status: the conjecture is open; it is one of the Clay Millennium Prize Problems. Known results: if the order of vanishing of $L(E, s)$ at $s = 1$ is $0$ or $1$, then the rank equals it and $\text{Ш}(E/\Q)$ is finite. This follows from the Gross–Zagier theorem together with the work of Kolyvagin. Coates and Wiles had earlier proved that if $E$ has complex multiplication and $L(E, 1) \ne 0$, then $E(\Q)$ is finite. Birch and Swinnerton-Dyer formulated the conjecture in the 1960s after computer experiments on the products $\prod_{p \le x} \#\tilde{E}(\F_p)/p$.