Faltings' theorem
E518465
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Mordell conjecture | 3 |
| Faltings' theorem canonical | 1 |
| Faltings's theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425385 — 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: Faltings' theorem Context triple: [Gerd Faltings, knownFor, Faltings' theorem]
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A.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
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B.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
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C.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
-
D.
Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
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E.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
- 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: Faltings' theorem Target entity description: 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.
-
A.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
B.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
C.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
-
D.
Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
-
E.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Faltings' theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Mordell conjecture
this entity surface form:
Mordell conjecture
this entity surface form:
Mordell conjecture
this entity surface form:
Faltings's theorem