# Birch and Swinnerton-Dyer conjecture

- kind: problem
- status: open
- year: 1965
- authors: Bryan Birch and Peter Swinnerton-Dyer, from early computer experiments

**Statement.** The rank of an elliptic curve's rational points equals the order of vanishing of its L-function at s = 1.

It predicts how many independent rational solutions a cubic equation has from an analytic function built out of counting solutions modulo primes. It is proved when the analytic rank is 0 or 1, by Gross–Zagier and Kolyvagin. It is one of the rare Millennium problems whose statement came out of machine computation.

Posed 1965, Bryan Birch and Peter Swinnerton-Dyer, from early computer experiments.

## What would settle it

Lean statement in formal-conjectures

## Prize

$1,000,000 Clay Millennium Prize

## Related problems

- The original curation links this problem to [Modularity theorem](/wiki/Problem:modularity-theorem.md); the argument for the link has not been written yet.

## Sources

- [Wikipedia: Birch and Swinnerton-Dyer conjecture](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture)

[Number theory](/wiki/Space:number-theory.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- related to [modularity-theorem](/wiki/Problem:modularity-theorem.md) — The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.

---
Revision 12, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:birch-swinnerton-dyer?action=history. Edit: POST /api/wiki/page/Problem:birch-swinnerton-dyer {source, summary, base_rev: 12}.