Birch and Swinnerton-Dyer
Does the number of rational points on an elliptic curve match the behaviour of its L-function at 1? The rank part is proved only when that behaviour is simple.
Open in the interactive tree →An elliptic curve is a cubic equation whose rational solutions form a group. The conjecture says the rank of that group equals the order of vanishing of the curve's L-function at s=1, and gives an exact formula for the leading coefficient. The rank statement is proved when that order is 0 or 1 (Gross-Zagier, Kolyvagin); for higher orders no proof exists. Clay prize: one million US dollars.
As of October 2026
October 2026: unsolved; Bhargava, Skinner and Zhang showed in 2014 that more than 66% of elliptic curves over Q, ordered by height, satisfy the rank part of the conjecture. In January 2026 Banwait and Huang (preprint) turned recent criteria of Burungale, Skinner, Tian and Wan into an algorithm and identified all curves of conductor up to 500,000 in the LMFDB with infinitely many quadratic twists satisfying the strong conjecture. A Clay workshop on recent results ran at Oxford on 21-25 September 2026; the record rank is at least 29 (Elkies and Klagsbrun, 2024; exactly 29 if the generalized Riemann hypothesis holds). In March 2026 the discoverers of murmurations, a pattern in elliptic-curve data first found with machine learning, described them in an arXiv preprint as a case study in AI-assisted mathematics linked to themes around the conjecture.
What is missing
- A proof for curves where the L-function vanishes to order 2 or more; no method links rational points to higher-order zeros
- Finiteness of the Tate-Shafarevich group in general; over Q it is known only when the analytic rank is at most 1
- The exact leading-coefficient formula for all curves, not only for positive proportions or special families
- The conjecture over general number fields, where even finiteness remains open
Becomes possible once solved
- A complete rule, computable from the L-function, for whether a curve has infinitely many rational points
- A complete answer to the congruent number problem, whose full criterion is currently conditional on this conjecture
- Exact control of the Tate-Shafarevich group in elliptic-curve arithmetic
Open steps
- Lean proof of the rank 0 and 1 cases High AI leverageFormalize Gross-Zagier and Kolyvagin (the conjecture for analytic rank at most 1) in Lean, to give a machine-checked base for new results.
- More curves with proven BSD Medium AI leverageExtend the families of curves, such as quadratic twists, for which the full formula is proved, and tabulate them in the LMFDB.
- Explain murmurations High AI leverageProve why average Frobenius traces oscillate with rank, and test whether the pattern gives a handle on ranks above 1.
- Rank statistics of elliptic curves Medium AI leverageSettle the conjectured distribution of ranks, improving the proportions of curves proved to have rank 0 or 1 (Bhargava and coauthors).
- A route beyond rank 1 Low AI leverageFind a method that connects rational points to L-function zeros of order two or more; no approach is known, although curves of rank at least 29 exist.
Where AI could help
Medium AI leverage. AI helped find murmurations in elliptic-curve data and computation is central, but a proof for rank 2 and above needs new arithmetic geometry.
- Mine the LMFDB's millions of curves for patterns in rank, L-values and Sha
- Formalize Gross-Zagier and Kolyvagin in Lean as a machine-checked base
- Automate checking of the hypotheses of known theorems across all curves
- Run large computations of ranks and L-functions
Shown so far
- In March 2026, the discoverers of murmurations (He, Lee, Oliver, Pozdnyakov) described them in an arXiv preprint as a case study in AI-assisted mathematics: a pattern in elliptic-curve data found computationally, analysable with machine-learning interpretability tools and connected to BSD themes. source
- On 4 September 2026, Anthropic reported that Claude agents produced a complete Lean proof of Fermat's Last Theorem in about 11 days (company report). source