Diophantine geometry
E223662
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Diophantine geometry canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T1994279 — 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: Diophantine geometry Context triple: [Hilbert’s irreducibility theorem, usedIn, Diophantine geometry]
-
A.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
B.
Hasse principle
The Hasse principle is a concept in number theory stating that a Diophantine equation has a rational solution if and only if it has solutions in all completions of the rationals, such as the real numbers and p-adic numbers.
-
C.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
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D.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
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E.
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.
- 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: Diophantine geometry Target entity description: Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
-
A.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
B.
Hasse principle
The Hasse principle is a concept in number theory stating that a Diophantine equation has a rational solution if and only if it has solutions in all completions of the rationals, such as the real numbers and p-adic numbers.
-
C.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
D.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
area of algebraic geometry
ⓘ
branch of mathematics ⓘ subfield of number theory ⓘ |
| appliesTo |
function fields
ⓘ
number fields ⓘ |
| concerns |
distribution of rational points
ⓘ
effectivity of Diophantine results ⓘ |
| developedFrom |
algebraic geometry
ⓘ
classical Diophantine analysis ⓘ |
| fieldOfStudy |
Diophantine equations
ⓘ
polynomial equations with integer coefficients ⓘ polynomial equations with rational coefficients ⓘ |
| hasKeyConcept |
Néron–Tate height
ⓘ
arithmetic variety ⓘ heights of points ⓘ integral point ⓘ moduli spaces ⓘ rational point ⓘ |
| hasKeyResult |
finiteness of rational points on curves of genus greater than 1
ⓘ
structure of rational points on elliptic curves as finitely generated abelian groups ⓘ |
| relatedTo |
Birch and Swinnerton-Dyer Conjecture
ⓘ
surface form:
Birch and Swinnerton-Dyer conjecture
Faltings' theorem ⓘ
surface form:
Faltings's theorem
Hasse principle ⓘ Bombieri–Lang conjecture ⓘ
surface form:
Lang conjectures
Manin obstruction ⓘ Faltings' theorem ⓘ
surface form:
Mordell conjecture
Mordell–Weil theorem ⓘ Siegel's theorem on integral points ⓘ Bombieri–Lang conjecture ⓘ
surface form:
Vojta conjectures
Weil conjectures ⓘ abc conjecture ⓘ local-global principles ⓘ |
| studies |
Diophantine approximation problems
ⓘ
abelian varieties ⓘ curves of higher genus ⓘ elliptic curves ⓘ integral points on algebraic varieties ⓘ rational points on algebraic varieties ⓘ rational points on curves ⓘ rational points on surfaces ⓘ solutions to polynomial equations over global fields ⓘ solutions to polynomial equations over number fields ⓘ |
| usesMethod |
Arakelov theory
ⓘ
Galois representations ⓘ algebraic geometry ⓘ arithmetic geometry ⓘ height theory ⓘ p-adic methods ⓘ scheme theory ⓘ |
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: Diophantine geometry Description of subject: Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.