Home›Open Questions›Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?Open QuestionsDoes the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?Citation FormatsGeneral ReferenceGeneral Reference Citation TextMathematics Atlas. "Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?." Accessed August 30, 2026. https://mathematics.interactiveatlas.org/open-questions-index/birch-and-swinnerton-dyer-conjecture-open-question.Copy General ReferenceAPA StyleAPA Style Citation TextMathematics Atlas. (n.d.). Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?. Retrieved August 30, 2026, from https://mathematics.interactiveatlas.org/open-questions-index/birch-and-swinnerton-dyer-conjecture-open-questionCopy APA StyleBibTeXBibTeX Citation Text@misc{mathematicsatlas-does-the-rank-of-an-elliptic-curve-s-gro, author = {Mathematics Atlas}, title = {Does the rank of an elliptic curve's group of rational points really equal the order of vanishing of its L-function at s equals one, for every elliptic curve?}, year = {2026}, url = {https://mathematics.interactiveatlas.org/open-questions-index/birch-and-swinnerton-dyer-conjecture-open-question}, note = {Accessed August 30, 2026} }Copy BibTeXOpen QuestionCross-Tradition ConnectionsSourcesComments (0)Reader Challenges (0 open reader challenges)Open QuestionProved 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 InstituteCross-Tradition ConnectionsQuestion OnBirch and Swinnerton-Dyer Conjecture, Conjectures Well-attested Source Clay Mathematics Institutetier 1SourcesClay Mathematics Institutetier 1Clay Mathematics Institutehttps://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdfView the SourceComments (0)No comments yet. Be the first to share a thought.Sign in to join the discussion.Reader Challenges (0 open reader challenges)No disputes yet. Spotted an error or a better source? Open the first one.Sign in to dispute this or suggest a correction.View At A Past YearThe atlas records no dated fact of its own for this entry, so there is no other year to choose.Show This Year