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OpenNumber Theory·1965

Birch and Swinnerton-Dyer Conjecture

Relates rational points on elliptic curves to the behavior of their L-functions at s = 1.

Formula

BSD conjecture
rank⁡ E(Q)=ord⁡s=1L(E,s)\operatorname{rank}\,E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)

The algebraic rank of the group of rational points on E equals the analytic rank (order of vanishing of the L-function at s = 1).

L-function
L(E,s)=∏p good11−app−s+p1−2s⋅∏p bad(⋯ )L(E,s)=\prod_{p\text{ good}}\frac{1}{1-a_p p^{-s}+p^{1-2s}}\cdot\prod_{p\text{ bad}}(\cdots)

The Hasse–Weil L-function of E encodes local point counts a_p = p + 1 − #E(𝔽_p) at each prime.

Leading coefficient (refined BSD)
lim⁡s→1L(E,s)(s−1)r=∣Ш∣⋅ΩE⋅RE⋅∏cp∣E(Q)tors∣2\lim_{s\to 1}\frac{L(E,s)}{(s-1)^r}=\frac{|\text{Ш}|\cdot\Omega_E\cdot R_E\cdot\prod c_p}{|E(\mathbb{Q})_{\text{tors}}|^2}

The refined conjecture predicts the leading Taylor coefficient in terms of the Sha group, regulator, period, and Tamagawa numbers.

Summary

The Birch and Swinnerton-Dyer conjecture predicts that the algebraic rank of an elliptic curve E over the rationals -- the number of independent rational points of infinite order -- equals the analytic rank, the order of vanishing of its Hasse-Weil L-function L(E,s) at s = 1. A refined version gives a precise formula for the leading Taylor coefficient involving the Tate-Shafarevich group, the regulator, real periods, and Tamagawa numbers. The conjecture emerged from extensive numerical computation on the Cambridge EDSAC-2 in the early 1960s. Major partial results include the Coates-Wiles theorem (1977) for CM curves, the Gross-Zagier formula (1986) connecting Heegner points to L-function derivatives, and Kolyvagin's Euler system method (1989) settling the rank 0 and 1 cases for modular elliptic curves.

Sources

On the conjecture of Birch and Swinnerton-Dyer

1977

Heegner points and derivatives of L-series

1986

Finiteness of E(Q) and Sha(E/Q) for a subclass of Weil curves

1989

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