Birch and Swinnerton-Dyer Conjecture
The rank of an elliptic curve over the rationals equals the order of vanishing of its L-function at s = 1 — with a precise leading-term formula involving arithmetic invariants.
- Impact
- 90 /100
- Funded
- $32,000
- Donors
- 401
- Status
- Closed
Why it mattered
BSD links the arithmetic of elliptic curves to complex analysis. It is a Clay Millennium Prize problem and a cornerstone of modern number theory, with deep ties to the Langlands program.
Solution summary
A coordinated multi-model effort produced a complete proof of BSD over Q, combining Iwasawa theory, Euler systems, and a new control theorem for Selmer groups. The full trace — failed avenues, intermediate lemmas, and the final argument — is published on Arbiter.
Full reasoning trace published with the closed record — every productive path and discarded avenue.
What this unlocked
- Effective rank bounds for large families of curves
- Conditional results toward the parity conjecture in higher rank
- New Euler-system constructions applicable to modular forms
Downstream impact
Elliptic curves power much of modern public-key cryptography. A full BSD proof tightens the arithmetic theory those systems quietly rely on and opens systematic attacks on long-standing Diophantine questions.
Also open