Diophantine geometry

E223662

area of algebraic geometry branch of mathematics subfield of number theory

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.

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All labels observed (4)

Label	Occurrences
Diophantine geometry canonical	15
arithmetic geometry	2
Diophantine analysis	1
Lang's program on Diophantine geometry	1

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.
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
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 ⓘ linked to: Birch and Swinnerton-Dyer Conjecture Faltings's theorem ⓘ linked to: Faltings' theorem Hasse principle ⓘ Lang conjectures ⓘ linked to: Bombieri–Lang conjecture Manin obstruction ⓘ Mordell conjecture ⓘ linked to: Faltings' theorem Mordell–Weil theorem ⓘ Siegel's theorem on integral points ⓘ Vojta conjectures ⓘ linked to: Bombieri–Lang conjecture 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

Referenced by (19)

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

Hilbert’s irreducibility theorem → usedIn → Diophantine geometry ⓘ

Diophantine approximation → relatedTo → Diophantine geometry ⓘ

Neal Koblitz → hasResearchInterest → Diophantine geometry ⓘ

model theory → hasApplication → Diophantine geometry ⓘ

Faltings' theorem → field → Diophantine geometry ⓘ

abc conjecture → subfield → Diophantine analysis ⓘ

linked to: Diophantine geometry

Serge Lang → fieldOfWork → Diophantine geometry ⓘ

Bombieri–Pila determinant method → field → Diophantine geometry ⓘ

Bombieri–Lang conjecture → topic → Diophantine geometry ⓘ

Bombieri–Lang conjecture → isPartOf → Lang's program on Diophantine geometry ⓘ

linked to: Diophantine geometry

Diophantus of Alexandria → topicOf → Diophantine geometry ⓘ

Subspace theorem → relatedTo → Diophantine geometry ⓘ

Mordell curve → studiedIn → Diophantine geometry ⓘ

Mordell curve → studiedIn → arithmetic geometry ⓘ

linked to: Diophantine geometry

Mordell–Weil theorem → subfield → Diophantine geometry ⓘ

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

Davis–Putnam–Robinson–Matiyasevich theorem → field → Diophantine geometry ⓘ

proof of the Milnor conjecture → influenced → arithmetic geometry ⓘ

linked to: Diophantine geometry

Siegel’s lemma → context → Diophantine geometry ⓘ

All labels observed (4)

How this entity was disambiguated Show

Statements (49)

How these facts were elicited Show

Referenced by (19)

How this entity was disambiguated

How these facts were elicited