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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All labels observed (5)

Label	Occurrences
Siegel's theorem	1
Siegel's theorem on integral points canonical	1
Siegel’s theorem on integral points	1
Siegel’s theorem on the finiteness of integer points on curves of genus at least one	1
Thue–Siegel theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T9297006 — 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: Siegel's theorem on integral points
Context triple: [Diophantine geometry, relatedTo, Siegel's theorem on integral 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. Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
C. Baker theorem on linear forms in logarithms
The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
D. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
E. Mordell–Weil theorem
The Mordell–Weil theorem is a fundamental result in number theory stating that the group of rational points on an abelian variety (in particular, an elliptic curve) over a number field is finitely generated.
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: Siegel's theorem on integral points
Target entity description: 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.

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. Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
C. Baker theorem on linear forms in logarithms
The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
D. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
E. Mordell–Weil theorem
The Mordell–Weil theorem is a fundamental result in number theory stating that the group of rational points on an abelian variety (in particular, an elliptic curve) over a number field is finitely generated.
F. None of above. chosen

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 ⓘ number theory ⓘ
generalizedBy	Faltings's theorem ⓘ linked to: Faltings' theorem Mordell–Lang conjecture ⓘ
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 ⓘ
proofTechnique	Diophantine approximation methods ⓘ Thue–Siegel method ⓘ linked to: Thue–Siegel–Roth theorem
provedBy	Carl Ludwig Siegel ⓘ
relatedTo	Faltings's theorem ⓘ linked to: Faltings' theorem Mordell's conjecture ⓘ linked to: Faltings' theorem Mordell–Weil theorem ⓘ Roth's theorem ⓘ linked to: Roth theorem Thue–Siegel–Roth theorem ⓘ
statedFor	S-integral points with respect to a finite set of places S ⓘ
statedOver	number fields ⓘ
strengthenedBy	Roth's theorem ⓘ linked to: Roth theorem
usesConcept	Diophantine approximation ⓘ S-integers ⓘ affine curves ⓘ genus of a curve ⓘ number fields ⓘ points at infinity ⓘ projective curves ⓘ
yearProved	1929 ⓘ

How these facts were elicited

Referenced by (5)

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 ⓘ

linked to: Siegel's theorem on integral points

Carl Ludwig Siegel → notableWork → Siegel’s theorem on the finiteness of integer points on curves of genus at least one ⓘ

linked to: Siegel's theorem on integral points

Diophantine equations → relatedTo → Siegel's theorem ⓘ

linked to: Siegel's theorem on integral points

Roth's theorem → improvesOn → Thue–Siegel theorem ⓘ

subject linked to: Roth theorem

linked to: Siegel's theorem on integral points

All labels observed (5)

How this entity was disambiguated Show

Statements (48)

How these facts were elicited Show

Referenced by (5)

How this entity was disambiguated

How these facts were elicited