Arakelov theory
E790514
Arakelov theory is a framework in arithmetic geometry that extends intersection theory to arithmetic surfaces by incorporating both finite and infinite places, enabling analytic tools to study Diophantine problems.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Arakelov theory canonical | 3 |
| higher-dimensional Arakelov theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9296992 — 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: Arakelov theory Context triple: [Diophantine geometry, usesMethod, Arakelov theory]
-
A.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
B.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
E.
p-adic Hodge theory
p-adic Hodge theory is a branch of arithmetic geometry that studies p-adic Galois representations and their relationship to the cohomology of algebraic varieties over p-adic fields, using analogues of classical Hodge-theoretic structures.
- 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: Arakelov theory Target entity description: Arakelov theory is a framework in arithmetic geometry that extends intersection theory to arithmetic surfaces by incorporating both finite and infinite places, enabling analytic tools to study Diophantine problems.
-
A.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
B.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
E.
p-adic Hodge theory
p-adic Hodge theory is a branch of arithmetic geometry that studies p-adic Galois representations and their relationship to the cohomology of algebraic varieties over p-adic fields, using analogues of classical Hodge-theoretic structures.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theory
ⓘ
theory in arithmetic geometry ⓘ |
| aimsToSolve |
Diophantine problems
ⓘ
problems in Diophantine geometry ⓘ |
| appliesTo |
arithmetic surfaces
ⓘ
schemes over the spectrum of the ring of integers of a number field ⓘ |
| developedBy | Suren Arakelov ⓘ |
| field | arithmetic geometry ⓘ |
| furtherDevelopedBy |
Christophe Soulé
ⓘ
Gerd Faltings ⓘ Henri Gillet ⓘ Jean-Benoît Bost ⓘ Shou-Wu Zhang ⓘ |
| generalizationOf | classical intersection theory on algebraic surfaces ⓘ |
| hasVariant |
adelic Arakelov theory
ⓘ
higher-dimensional Arakelov theory ⓘ
linked to:
Arakelov theory
|
| mainConcept |
Arakelov Chow group
ⓘ
linked to:
Chow groups
Arakelov class group ⓘ
linked to:
Arakelov divisor
Arakelov divisor ⓘ Green function ⓘ adelic metrized line bundle ⓘ arithmetic intersection number ⓘ arithmetic surface ⓘ height function ⓘ hermitian line bundle ⓘ intersection theory ⓘ |
| namedAfter | Suren Arakelov ⓘ |
| provides |
arithmetic Riemann–Roch theorems
ⓘ
arithmetic analogues of classical geometric formulas ⓘ framework for heights of algebraic points ⓘ intersection theory including archimedean contributions ⓘ |
| relatedTo |
Beilinson–Bloch conjectures
ⓘ
linked to:
Beilinson conjectures
Diophantine approximation ⓘ Faltings’s theorem ⓘ
linked to:
Faltings' theorem
Mordell conjecture ⓘ
linked to:
Faltings' theorem
Néron–Tate height ⓘ equidistribution of small points ⓘ height theory ⓘ |
| timePeriod | 1970s ⓘ |
| usesConcept |
Dirichlet energy
ⓘ
Green’s function on a Riemann surface ⓘ Riemann surface ⓘ
linked to:
Riemann surfaces
archimedean places ⓘ complex analytic geometry ⓘ finite places of a number field ⓘ harmonic analysis ⓘ infinite places of a number field ⓘ non-archimedean places ⓘ potential 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: Arakelov theory Description of subject: Arakelov theory is a framework in arithmetic geometry that extends intersection theory to arithmetic surfaces by incorporating both finite and infinite places, enabling analytic tools to study Diophantine problems.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.