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OpenNumber Theory · Arithmetic Geometry

The Birch and Swinnerton-Dyer Conjecture

Prize

$1,000,000 USD

Posed

1965

By

Bryan Birch · Peter Swinnerton-Dyer

Status

Open

TL;DR

An elliptic curve is a specific kind of cubic equation — for example, y2=x3+ax+by^2 = x^3 + ax + b. The rational points on it (solutions with xx and yy both rational) form a group, and this group has been studied for centuries.

The finite part of this group is easy to describe; the infinite part is measured by the rank, a non-negative integer. Some elliptic curves have rank 0 (only finitely many rational points), some have rank 1, and higher ranks become progressively rare. Nobody currently knows an algorithm guaranteed to compute the rank of a given curve.

Attached to every elliptic curve is another mathematical object — its L-function, L(E,s)L(E, s) — which encodes deep information about how the curve behaves modulo primes.

The Birch and Swinnerton-Dyer Conjecture asserts that the rank of the curve equals the order of vanishing of L(E,s)L(E, s) at s=1s = 1. If L(E,1)≠0L(E, 1) \ne 0, the rank is 0 (only finitely many points). If L(E,s)L(E, s) has a simple zero, the rank is 1. And so on. Numerical evidence overwhelmingly supports the conjecture. A proof would give us the first algorithm to compute the rank in general.

Birch–Swinnerton-Dyer — an elliptic curve with rational pointsPlot of an elliptic curve y-squared equals x-cubed minus x, with several rational points marked. The rank counts independent infinite-order points.xy(-1, 0)(0, 0)(1, 0)P2Py² = x³ − x · rank = order of vanishing of L(E, s) at s = 1

Formal statement

For an elliptic curve EE over Q\mathbb{Q}, the rank of the group of rational points E(Q)E(\mathbb{Q}) equals the order of vanishing of the L-function L(E,s)L(E, s) at s=1s = 1.

Why it matters

Elliptic curves underpin modern public-key cryptography (elliptic curve cryptography secures HTTPS, blockchain, and much more) and appear throughout number theory, including in Andrew Wiles's proof of Fermat's Last Theorem. The BSD Conjecture is the central open question about their arithmetic.

History

Bryan Birch and Peter Swinnerton-Dyer stumbled onto the conjecture in the early 1960s while running numerical experiments on the EDSAC-2 computer at Cambridge. They noticed a striking pattern in the behaviour of L-functions of elliptic curves near s=1s = 1 and formulated the conjecture in a 1965 paper.

Their work was one of the first significant conjectures in mathematics driven by computer experiment — a lineage that now includes many landmark results.

Current status of research

Major partial results have been established:

  • Coates–Wiles (1977): BSD holds for elliptic curves with complex multiplication, when the L-function is non-vanishing at s=1s = 1.
  • Gross–Zagier (1986) + Kolyvagin (1989): BSD holds for curves of analytic rank 0 or 1 over Q\mathbb{Q}.
  • Skinner–Urban (2014) and others have extended these results in various directions.

For rank 2 or higher, the conjecture remains completely open. Even the "full BSD" — including the precise formula for the leading coefficient of the L-function in terms of the regulator, Tamagawa numbers, and Sha (the Tate–Shafarevich group) — is unresolved except in low-rank cases.

Notable attempts

The BSD conjecture is one of the most intensely studied problems in number theory. Progress happens in specialised subcases. No dramatic false-alarm proofs, but continuous incremental progress from a large community.

What a resolution would mean

A proof would give the first general algorithm to compute the rank of an elliptic curve — currently a fundamental problem in computational number theory. It would settle the precise arithmetic structure of solutions to cubic equations in two variables. It would likely be a stepping stone to broader conjectures (the Bloch–Kato conjecture, generalized to higher-dimensional varieties).

All 7 Millennium Prize Problems