It states that the associated L-series has value zero at a fixed point if an elliptic curve has an infinite number of solutions.
It has been partially proven by the Gross-Zagier formula, Kolyvagin's theorem, and the Coates-Wiles theorem. It states that the associated L-series has value zero at a fixed point if an elliptic curve has an infinite number of solutions. FTP, name this Millennium Prize problem, a conjecture named for two mathematicians and sometimes Birch.
It has been partially proven by the Gross-Zagier formula, Kolyvagin's theorem, and the Coates-Wiles theorem.
(Birch and) {Swinnerton-Dyer} conjecture