Faltings' theorem

E518465

mathematical theorem theorem in arithmetic geometry

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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (2)

Surface form	Occurrences
Mordell conjecture	3
Faltings's theorem	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in arithmetic geometry ⓘ
alsoKnownAs	Mordell conjecture NERFINISHED ⓘ
appliesTo	curves of genus at least two ⓘ smooth projective curves over number fields ⓘ
assumes	curve defined over a number field ⓘ genus greater than one ⓘ
category	theorems about rational points ⓘ theorems in algebraic geometry ⓘ
concerns	algebraic curves over number fields ⓘ curves of genus greater than one ⓘ rational points on algebraic curves ⓘ
conclusion	set of rational points is finite ⓘ
doesNotApplyTo	elliptic curves of genus one ⓘ genus zero curves ⓘ
field	Diophantine geometry NERFINISHED ⓘ arithmetic geometry ⓘ number theory ⓘ
generalizes	Mordell's theorem for curves over number fields ⓘ
hasConsequence	only finitely many rational points on any curve of genus greater than one over a fixed number field ⓘ
implies	Shafarevich conjecture for abelian varieties over number fields NERFINISHED ⓘ finiteness of rational points on curves of genus greater than one over number fields ⓘ
importance	landmark result in arithmetic geometry ⓘ
influenced	modern Diophantine geometry ⓘ research on rational points ⓘ
isAbout	finiteness of rational solutions ⓘ
namedAfter	Gerd Faltings NERFINISHED ⓘ
originallyConjecturedBy	Louis Mordell NERFINISHED ⓘ
over	number fields ⓘ
provedBy	Gerd Faltings NERFINISHED ⓘ
proves	Mordell conjecture NERFINISHED ⓘ
publishedIn	1983 ⓘ
relatedTo	Diophantine equations NERFINISHED ⓘ Faltings height NERFINISHED ⓘ Jacobians of curves ⓘ Shafarevich conjecture for abelian varieties NERFINISHED ⓘ abelian varieties over number fields ⓘ rational points on curves ⓘ
statement	Every algebraic curve of genus greater than one over a number field has only finitely many rational points ⓘ
status	proved ⓘ
type	finiteness theorem NERFINISHED ⓘ
uses	Arakelov theory NERFINISHED ⓘ Néron models NERFINISHED ⓘ Tate conjecture for abelian varieties over number fields ⓘ heights on abelian varieties ⓘ reduction theory of abelian varieties ⓘ
yearProved	1983 ⓘ

Referenced by (5)

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

Gerd Faltings → knownFor → Faltings' theorem ⓘ

Louis Mordell → knownFor → Faltings' theorem ⓘ

this entity surface form: Mordell conjecture

Birch and Swinnerton-Dyer Conjecture → relatedTo → Faltings' theorem ⓘ

this entity surface form: Mordell conjecture

Diophantine geometry → relatedTo → Faltings' theorem ⓘ

this entity surface form: Mordell conjecture

Diophantine geometry → relatedTo → Faltings' theorem ⓘ

this entity surface form: Faltings's theorem