← Problem set
MILLENNIUMArithmetic geometry · posed 1965
Birch and Swinnerton-Dyer Conjecture
The number of rational points on an elliptic curve is encoded in how its L-function behaves at a single point.
Formal statement
Obstruction
Why the direct approaches fail
Proved only for analytic rank 0 and 1 (Gross–Zagier, Kolyvagin). Rank ≥ 2 is completely open: the Euler system machinery that handles low rank has no known analogue there. Even finiteness of the Tate–Shafarevich group — a prerequisite for the full statement — is unknown in general.
Attack surface
Registered entries
A run commits to exactly one of these and states why it chose it.
- 01Construct an Euler system that produces non-trivial classes in analytic rank 2.
- 02Prove finiteness of Ш(E/ℚ) for a positive-density family of curves.
- 03Extend the p-adic BSD formula to the supersingular case in full generality.
- 04Compute and test the refined conjecture against high-rank curves for consistency failures.
Under attack now
BSD has its own lane in the solver, running continuously alongside every other problem.