Open problem, Arithmetic and number theory, posed 1965

Birch and Swinnerton-Dyer conjecture

OpenMillennium

For an elliptic curve $E$ over $\mathbb{Q}$ with Hasse–Weil $L$ -function $L(E,s)$ , the rank of the Mordell–Weil group $E(\mathbb{Q})$ equals the order of vanishing of $L(E,s)$ at $s=1$ : $\operatorname{rank} E(\mathbb{Q}) = \operatorname{ord}_{s=1} L(E,s).$ The refined conjecture further predicts the leading Taylor coefficient of $L(E,s)$ at $s=1$ in terms of the order of the Tate–Shafarevich group $Ш(E)$ , the regulator, the real period, the Tamagawa numbers, and the order of the torsion subgroup.

Research frontier as of 2026

As of 2026 the rank part of BSD is proved only for elliptic curves of analytic rank 0 or 1 (Gross–Zagier, Kolyvagin, combined with modularity); for analytic rank 2 or higher, essentially nothing is known unconditionally. Statistical results (Bhargava–Skinner–Zhang) show that a majority of all elliptic curves over $\mathbb{Q}$ , ordered by height, fall into the known rank-0/1 cases, but this says nothing about any individual higher-rank curve. The refined formula for the leading L-function coefficient, including finiteness of the Tate–Shafarevich group in general, remains conjectural.

Best known results

Gross–Zagier (1986) + Kolyvagin (1989), combined with modularity (Wiles et al., completed by Breuil–Conrad–Diamond–Taylor 2001): the rank part of BSD holds whenever the analytic rank is 0 or 1.
Bhargava–Skinner–Zhang (2014): more than 66% of elliptic curves over $\mathbb{Q}$ , ordered by height, satisfy the rank part of BSD and have finite Tate–Shafarevich group.
For curves of analytic rank ≥ 2, essentially nothing is known unconditionally, since Heegner-point Euler systems have no known analogue there.

Tools and where they stop

Tool	Achieved	Where it stops
Heegner points and Euler systems (Gross–Zagier, Kolyvagin)	Proves the rank part of BSD for analytic rank 0 and 1	The construction has no known analogue for analytic rank ≥ 2
Iwasawa main conjectures (Skinner–Urban) and geometry-of-numbers counting (Bhargava–Shankar)	Shows a majority of all elliptic curves, ordered by height, satisfy BSD	Says nothing about any individual high-rank curve

Open questions

Does the rank part of BSD hold for every elliptic curve of analytic rank ≥ 2?
Is the Tate–Shafarevich group finite for every elliptic curve over $\mathbb{Q}$ , and does the full refined BSD formula hold in general?

References

Bryan J. Birch, H.P.F. Swinnerton-Dyer (1965). Notes on elliptic curves. II.
Benedict H. Gross, Don B. Zagier (1986). Heegner points and derivatives of L-series
Victor A. Kolyvagin (1990). Euler systems
Manjul Bhargava, Christopher Skinner, Wei Zhang (2014). A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture · arXiv:1407.1826 [preprint, not peer-reviewed]