Siegel's theorem on integral points

E790515

result in Diophantine geometry result in number theory theorem

Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.

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Observed surface forms (2)

Surface form	Occurrences
Siegel’s theorem on integral points	1
Siegel’s theorem on the finiteness of integer points on curves of genus at least one	1

Statements (48)

Predicate	Object
instanceOf	result in Diophantine geometry ⓘ result in number theory ⓘ theorem ⓘ
appearsIn	theory of elliptic curves ⓘ theory of hyperelliptic curves ⓘ
appliesTo	affine curves of genus 0 with at least three points at infinity ⓘ curves of genus at least 1 ⓘ
asserts	finiteness of integral points on certain algebraic curves ⓘ
citedIn	advanced textbooks on Diophantine equations ⓘ monographs on Diophantine geometry ⓘ
concerns	Diophantine equations ⓘ affine algebraic curves over number fields ⓘ integral points on algebraic curves ⓘ
doesNotApplyTo	affine line with at most two points removed ⓘ projective line with at most two points at infinity ⓘ
field	Diophantine geometry NERFINISHED ⓘ number theory ⓘ
generalizedBy	Faltings's theorem NERFINISHED ⓘ Mordell–Lang conjecture NERFINISHED ⓘ
hasConsequence	integral points on elliptic curves are finite ⓘ integral points on hyperelliptic curves of genus at least 1 are finite ⓘ integral solutions of many polynomial equations in two variables are finite ⓘ
hasProperty	ineffective ⓘ non-constructive ⓘ
implies	only finitely many S-integral points on suitable curves ⓘ
ineffectivityReason	proof gives no explicit bound for the size of integral points ⓘ
influenced	development of modern Diophantine geometry ⓘ work on heights and Arakelov theory ⓘ
namedAfter	Carl Ludwig Siegel NERFINISHED ⓘ
proofTechnique	Diophantine approximation methods ⓘ Thue–Siegel method NERFINISHED ⓘ
provedBy	Carl Ludwig Siegel NERFINISHED ⓘ
relatedTo	Faltings's theorem NERFINISHED ⓘ Mordell's conjecture NERFINISHED ⓘ Mordell–Weil theorem NERFINISHED ⓘ Roth's theorem NERFINISHED ⓘ Thue–Siegel–Roth theorem NERFINISHED ⓘ
statedFor	S-integral points with respect to a finite set of places S ⓘ
statedOver	number fields ⓘ
strengthenedBy	Roth's theorem NERFINISHED ⓘ
usesConcept	Diophantine approximation NERFINISHED ⓘ S-integers ⓘ affine curves ⓘ genus of a curve ⓘ number fields ⓘ points at infinity ⓘ projective curves ⓘ
yearProved	1929 ⓘ

Referenced by (3)

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

Diophantine geometry → relatedTo → Siegel's theorem on integral points ⓘ

Carl Ludwig Siegel → notableWork → Siegel's theorem on integral points ⓘ

this entity surface form: Siegel’s theorem on integral points

Carl Ludwig Siegel → notableWork → Siegel's theorem on integral points ⓘ

this entity surface form: Siegel’s theorem on the finiteness of integer points on curves of genus at least one