Bombieri–Lang conjecture

E571015

conjecture in arithmetic geometry conjecture in number theory mathematical conjecture

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.

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All labels observed (3)

Label	Occurrences
Bombieri–Lang conjecture canonical	1
Lang conjectures	1
Vojta conjectures	1

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 ⓘ

Referenced by (3)

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

Enrico Bombieri → knownFor → Bombieri–Lang conjecture ⓘ

Diophantine geometry → relatedTo → Bombieri–Lang conjecture ⓘ

this entity surface form: Lang conjectures

Diophantine geometry → relatedTo → Bombieri–Lang conjecture ⓘ

this entity surface form: Vojta conjectures