Siegel's theorem on integral points
E790515
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.
All labels observed (3)
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]
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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.
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.
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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.
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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.
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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
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 ⓘ |
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Input
Subject: Siegel's theorem on integral points Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
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