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The Millennium Museum·Number theory

Open

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

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. 2000-05-24

    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. 2026-09-01

    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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