Bloch–Kato conjecture
E911354
The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bloch–Kato conjecture canonical | 2 |
| Bloch–Kato norm residue conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219370 — 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: Bloch–Kato conjecture Context triple: [Milnor K-theory, relatedTo, Bloch–Kato conjecture]
-
A.
Fontaine–Mazur conjecture
The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
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B.
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.
-
C.
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.
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D.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
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E.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
- 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: Bloch–Kato conjecture Target entity description: The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
-
A.
Fontaine–Mazur conjecture
The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
-
B.
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.
-
C.
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.
-
D.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
-
E.
Serre’s conjecture on Galois representations
Serre’s conjecture on Galois representations is a landmark statement in number theory that predicts which two-dimensional mod p Galois representations of the absolute Galois group of the rationals arise from modular forms.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
statement in algebraic K-theory ⓘ statement in arithmetic geometry ⓘ |
| alsoKnownAs |
Bloch–Kato norm residue isomorphism conjecture
NERFINISHED
ⓘ
norm residue isomorphism conjecture NERFINISHED ⓘ |
| concerns |
Milnor K-groups modulo l
ⓘ
fields of characteristic not equal to a fixed prime l ⓘ l-torsion in Galois cohomology ⓘ |
| field |
Galois cohomology
NERFINISHED
ⓘ
algebraic K-theory ⓘ arithmetic geometry ⓘ motivic cohomology ⓘ number theory ⓘ |
| formulatedBy |
Kazuya Kato
NERFINISHED
ⓘ
Spencer Bloch NERFINISHED ⓘ |
| generalizes | Milnor conjecture NERFINISHED ⓘ |
| implies | Milnor conjecture on quadratic forms (in suitable form) NERFINISHED ⓘ |
| influenced |
development of motivic homotopy theory
ⓘ
research on special values of L-functions ⓘ study of cohomological invariants of algebraic groups ⓘ |
| involves |
continuous Galois cohomology of absolute Galois groups
ⓘ
graded pieces of Milnor K-theory ⓘ |
| proofUses |
Bloch–Ogus theory
NERFINISHED
ⓘ
Rost motives ⓘ motivic cohomology ⓘ norm varieties ⓘ |
| provedBy |
Charles Weibel
NERFINISHED
ⓘ
Markus Rost NERFINISHED ⓘ Vladimir Voevodsky NERFINISHED ⓘ other collaborators in the Rost–Voevodsky program ⓘ |
| relatedTo |
Beilinson–Lichtenbaum conjecture
NERFINISHED
ⓘ
Bloch–Kato Selmer groups (in the context of motives) NERFINISHED ⓘ Quillen K-theory NERFINISHED ⓘ |
| relatesConcept |
Galois cohomology
ⓘ
Galois representations ⓘ Milnor K-theory NERFINISHED ⓘ algebraic K-groups ⓘ field arithmetic ⓘ motivic complexes ⓘ norm residue homomorphism ⓘ étale cohomology ⓘ |
| states | norm residue homomorphism from Milnor K-theory modulo l to Galois cohomology is an isomorphism ⓘ |
| status | proved ⓘ |
| timePeriod | late 20th century mathematics ⓘ |
| topic |
cohomological invariants of fields
ⓘ
description of Galois cohomology in terms of K-theory ⓘ exact correspondence between Milnor K-theory and Galois cohomology ⓘ norm residue isomorphism in degree n ⓘ |
How these facts were elicited
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Input
Subject: Bloch–Kato conjecture Description of subject: The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Bloch–Kato norm residue conjecture