Number TheoryUnsolved
Birch and Swinnerton-Dyer Conjecture
Is the rank of an elliptic curve determined by the behavior of its L-function at s = 1?
Overview
The conjecture relates the number of rational points on an elliptic curve to the behavior of its L-function at s = 1.
Why It Matters
It links arithmetic geometry, elliptic curves, and L-functions. Impacts cryptography and number theory.
Current Status
Known in many special cases, deep partial results exist, but the full statement remains unproven.
What a Solution Would Require
A general proof connecting rank and L-function behavior as stated.