Birch and Swinnerton-Dyer Conjecture

E175567

mathematical conjecture unsolved problem in mathematics unsolved problem in number theory

The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.

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All labels observed (4)

Label	Occurrences
Birch and Swinnerton-Dyer conjecture	3
Birch and Swinnerton-Dyer Conjecture canonical	2
Birch–Swinnerton-Dyer conjecture	1
Birch–Swinnerton-Dyer formula	1

How this entity was disambiguated

This entity first appeared as the object of triple T1523323 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Birch and Swinnerton-Dyer Conjecture
Context triple: [Millennium Prize Problem, hasProblem, Birch and Swinnerton-Dyer Conjecture]

A. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
B. Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
C. Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
D. Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
E. Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Birch and Swinnerton-Dyer Conjecture
Target entity description: The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.

A. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
B. Fermat's Last Theorem
Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
C. Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
D. Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
E. Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	mathematical conjecture ⓘ unsolved problem in mathematics ⓘ unsolved problem in number theory ⓘ
basedOn	computational experiments on elliptic curves ⓘ numerical evidence for behavior of L-functions at s = 1 ⓘ
concerns	L-functions of elliptic curves ⓘ elliptic curves ⓘ rational points on elliptic curves ⓘ
difficulty	considered extremely difficult ⓘ
field	number theory ⓘ
hasPart	strong Birch and Swinnerton-Dyer conjecture ⓘ weak Birch and Swinnerton-Dyer conjecture ⓘ
hasPrize	US$1,000,000 prize ⓘ
holdsFor	many elliptic curves in special cases ⓘ
implies	deep relations between analytic and arithmetic invariants of elliptic curves ⓘ finiteness of the Tate–Shafarevich group for elliptic curves ⓘ
importance	central problem in modern number theory ⓘ
namedAfter	Bryan Birch ⓘ Peter Swinnerton-Dyer ⓘ
offeredBy	Clay Mathematics Institute ⓘ
partiallyProvedBy	Andrew Wiles ⓘ Bryan Birch ⓘ John Coates ⓘ Karl Rubin ⓘ Peter Swinnerton-Dyer ⓘ Victor Kolyvagin ⓘ
predicts	elliptic curve has finitely many rational points if and only if its L-function does not vanish at s = 1 ⓘ elliptic curve has infinitely many rational points if and only if its L-function vanishes at s = 1 ⓘ leading Taylor coefficient of L-function at s = 1 is given by arithmetic invariants of the elliptic curve ⓘ rank of an elliptic curve equals order of vanishing of its L-function at s = 1 ⓘ
recognizedAs	one of the Millennium Prize Problems ⓘ
relatedTo	Beilinson conjectures ⓘ Hasse–Weil conjecture ⓘ linked to: Tate Conjecture Iwasawa theory ⓘ Modularity theorem ⓘ linked to: Taniyama–Shimura–Weil conjecture Mordell conjecture ⓘ linked to: Faltings' theorem
relatesConcept	Birch–Swinnerton-Dyer formula ⓘ linked to: Birch and Swinnerton-Dyer Conjecture Hasse–Weil L-function ⓘ linked to: Hasse–Weil zeta function Mordell–Weil group ⓘ linked to: Mordell–Weil theorem Tamagawa numbers ⓘ Tate–Shafarevich group ⓘ linked to: Cassels–Tate pairing conductor of an elliptic curve ⓘ order of vanishing of an L-function ⓘ rank of an elliptic curve ⓘ regulator of an elliptic curve ⓘ
status	open ⓘ
subfield	arithmetic geometry ⓘ arithmetic of elliptic curves ⓘ
timeProposed	1960s ⓘ

How these facts were elicited

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Millennium Prize Problem → hasProblem → Birch and Swinnerton-Dyer Conjecture ⓘ

Millennium Prize Problem → includes → Birch and Swinnerton-Dyer Conjecture ⓘ

Birch and Swinnerton-Dyer Conjecture → relatesConcept → Birch–Swinnerton-Dyer formula ⓘ