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The Millennium Museum·Number theory

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The Birch and Swinnerton-Dyer Conjecture

Is the rank of an elliptic curve over the rationals equal to the order of vanishing of its L-function at s = 1?

Open. Known for curves of analytic rank zero or one, and for a majority of curves on average.

The exhibit

Rational points on y² + y = x³ − xThe real points of the elliptic curve of conductor 37, and its rational points P, 2P, 3P, and so on, generated by P = (0, 0) and listed exactly at left, their denominators growing without end. The chord through P and 2P meets the curve again at −3P, and reflecting in the line y = −½ gives 3P. The curve has rank one, a case of the conjecture proved by Gross, Zagier and Kolyvagin.P2P3P4P5P7P8P9P11P12P−3P P = (0, 0) 2P = (1, 0) 3P = (−1, −1) 4P = (2, −3) 5P = (1/4, −5/8) 6P = (6, 14) 7P = (−5/9, 8/27) 8P = (21/25, −69/125) 9P = (−20/49, −435/343)10P = (161/16, −2065/64)y² + y = x³ − x
Plate VIRational points on y² + y = x³ − x. The real points of the elliptic curve of conductor 37, and its rational points P, 2P, 3P, and so on, generated by P = (0, 0) and listed exactly at left, their denominators growing without end. The chord through P and 2P meets the curve again at −3P, and reflecting in the line y = −½ gives 3P. The curve has rank one, a case of the conjecture proved by Gross, Zagier and Kolyvagin.

The problem, three ways

Curious

An elliptic curve is an equation such as y² = x³ − x. Some have infinitely many rational solutions and some only finitely many. Bryan Birch and Peter Swinnerton-Dyer, computing on one of the first computers, guessed that a single analytic function predicts which, and how many.

Undergraduate

For an elliptic curve E over ℚ, Mordell's theorem gives E(ℚ) ≅ ℤ^r × T with T finite. The conjecture says r equals the order of vanishing of L(E, s) at s = 1, and gives the leading coefficient in terms of the regulator, the Tate–Shafarevich group, Tamagawa numbers and torsion.

Specialist

Known when the analytic rank is 0 or 1, by Gross–Zagier and Kolyvagin together with modularity (Wiles; Breuil, Conrad, Diamond and Taylor). Bhargava, Skinner and Zhang showed that more than 66% of elliptic curves over ℚ satisfy the rank part. Finiteness of the Tate–Shafarevich group is open in general.

What would count

  • A proof for all elliptic curves over the rationals, as in Andrew Wiles' Clay description, or a counterexample.
  • Clay's conditions: publication in a qualifying outlet, two years, general acceptance, then a decision by its board.

Fidelity traps

  • The rank statement and the full leading-coefficient formula are different claims.
  • Results for analytic rank at most one, or for a proportion of curves, are progress rather than the conjecture.
  • Formal Conjectures states the conjecture in Lean; a formal proof should be checked against that statement.

Who is attacking it

  • Formal Conjectures: States the conjecture in Lean, and is drafting related statements. Source
  • OpenAI: Says it began large-scale work on all the open Millennium problems on 1 September 2026. Source

The history

8 events
  1. 1960s
  2. 1965

    posed

    Computations on EDSAC

    Bryan Birch and Peter Swinnerton-Dyer publish the conjecture, drawn from computations on the EDSAC computer at Cambridge.

    Source

  3. 1970s
  4. 1977

    progress

    Coates and Wiles

    John Coates and Andrew Wiles prove that a curve with complex multiplication and non-vanishing L-value at 1 has only finitely many rational points.

    Source

  5. 1980s
  6. 1986

    progress

    The Gross–Zagier formula

    Benedict Gross and Don Zagier relate the derivative of the L-function to the height of a Heegner point.

    Source

  7. 1989

    progress

    Kolyvagin

    Victor Kolyvagin's Euler systems, with Gross–Zagier, prove the rank part for curves of analytic rank zero or one.

    Source

  8. 2000s
  9. 24 May 2000

    prize

    A Millennium Prize Problem

    The Clay Mathematics Institute names the conjecture one of seven carrying a $1,000,000 prize, with an official description by Andrew Wiles.

    Source

  10. 2001

    progress

    Every curve is modular

    Breuil, Conrad, Diamond and Taylor complete the modularity theorem, so the L-function of every elliptic curve over ℚ extends to the whole plane.

    Source

  11. 2010s
  12. 2014

    progress

    Most curves satisfy it

    Bhargava, Skinner and Zhang show that more than 66% of elliptic curves over ℚ satisfy the rank part of the conjecture.

    Source

  13. 2026
  14. 1 Sep 2026

    AI

    OpenAI turns to the Millennium problems

    OpenAI says it began large-scale work on all the open Millennium problems.

    Source

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