Bombieri–Lang conjecture
E571015
The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Bombieri–Lang conjecture canonical | 1 |
| Lang conjectures | 1 |
| Vojta conjectures | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6150020 — 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: Bombieri–Lang conjecture Context triple: [Enrico Bombieri, knownFor, Bombieri–Lang conjecture]
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A.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
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B.
Birch and Swinnerton-Dyer Conjecture
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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C.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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D.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
E.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
- 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: Bombieri–Lang conjecture Target entity description: The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
-
A.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
B.
Birch and Swinnerton-Dyer Conjecture
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.
-
C.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
D.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
E.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture in arithmetic geometry
ⓘ
conjecture in number theory ⓘ mathematical conjecture ⓘ |
| assumes |
base field is a number field
ⓘ
variety is of general type ⓘ |
| concerns |
distribution of rational points
ⓘ
varieties of general type over number fields ⓘ |
| consequence | strong restrictions on rational points on general type varieties ⓘ |
| domain | number fields ⓘ |
| expressedIn |
Diophantine approximation language
NERFINISHED
ⓘ
algebraic geometry ⓘ |
| field |
arithmetic geometry
ⓘ
number theory ⓘ |
| formulatedBy |
Enrico Bombieri
NERFINISHED
ⓘ
Serge Lang NERFINISHED ⓘ |
| generalizes | Mordell conjecture NERFINISHED ⓘ |
| implies |
Mordell conjecture over number fields in suitable settings
ⓘ
finiteness of rational points on curves of genus at least 2 over number fields ⓘ |
| importance |
central open problem about rational points
ⓘ
major conjecture in arithmetic geometry ⓘ |
| influencedBy |
Mordell conjecture
NERFINISHED
ⓘ
Weil conjectures on curves NERFINISHED ⓘ |
| isPartOf | Lang's program on Diophantine geometry NERFINISHED ⓘ |
| motivation | understanding arithmetic of higher-dimensional varieties ⓘ |
| namedAfter |
Enrico Bombieri
NERFINISHED
ⓘ
Serge Lang NERFINISHED ⓘ |
| predicts | varieties of general type over number fields have only finitely many rational points ⓘ |
| quantifier | finitely many rational points ⓘ |
| relatedTo |
Campana conjecture
NERFINISHED
ⓘ
Faltings's theorem NERFINISHED ⓘ Lang conjectures on rational points ⓘ Northcott property for heights ⓘ Vojta conjectures NERFINISHED ⓘ abc conjecture ⓘ height functions on varieties ⓘ hyperbolicity of varieties ⓘ |
| scope | smooth projective varieties of general type ⓘ |
| status | open problem ⓘ |
| timePeriod | late 20th century ⓘ |
| topic |
Diophantine geometry
NERFINISHED
ⓘ
rational points on algebraic varieties ⓘ varieties of general type ⓘ |
| typeOfStatement | finiteness conjecture ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Bombieri–Lang conjecture Description of subject: The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lang conjectures
this entity surface form:
Vojta conjectures