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 (1)
| Label | Occurrences |
|---|---|
| Arakelov theory canonical | 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 NERFINISHED ⓘ |
| field | arithmetic geometry ⓘ |
| furtherDevelopedBy |
Christophe Soulé
NERFINISHED
ⓘ
Gerd Faltings NERFINISHED ⓘ Henri Gillet NERFINISHED ⓘ Jean-Benoît Bost NERFINISHED ⓘ Shou-Wu Zhang NERFINISHED ⓘ |
| generalizationOf | classical intersection theory on algebraic surfaces ⓘ |
| hasVariant |
adelic Arakelov theory
ⓘ
higher-dimensional Arakelov theory NERFINISHED ⓘ |
| mainConcept |
Arakelov Chow group
NERFINISHED
ⓘ
Arakelov class group NERFINISHED ⓘ Arakelov divisor NERFINISHED ⓘ Green function ⓘ adelic metrized line bundle ⓘ arithmetic intersection number ⓘ arithmetic surface ⓘ height function ⓘ hermitian line bundle ⓘ intersection theory ⓘ |
| namedAfter | Suren Arakelov NERFINISHED ⓘ |
| provides |
arithmetic Riemann–Roch theorems
NERFINISHED
ⓘ
arithmetic analogues of classical geometric formulas ⓘ framework for heights of algebraic points ⓘ intersection theory including archimedean contributions ⓘ |
| relatedTo |
Beilinson–Bloch conjectures
NERFINISHED
ⓘ
Diophantine approximation NERFINISHED ⓘ Faltings’s theorem NERFINISHED ⓘ Mordell conjecture NERFINISHED ⓘ Néron–Tate height NERFINISHED ⓘ equidistribution of small points ⓘ height theory ⓘ |
| timePeriod | 1970s ⓘ |
| usesConcept |
Dirichlet energy
NERFINISHED
ⓘ
Green’s function on a Riemann surface ⓘ Riemann surface NERFINISHED ⓘ 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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.