Proved only in the rank zero and rank one cases, through work by Coates and Wiles in 1977, Gross and Zagier in 1986, and Kolyvagin in 1989. No general argument is known for higher rank, and no counterexample has ever been found either.
What would resolve this A general proof covering every rank, or a single explicit elliptic curve whose rank and L-function order of vanishing disagree, reaching beyond the rank zero and rank one cases current methods can prove.
OpenNumber theoryClay Mathematics Institute