Bloch–Kato conjecture

E911354

mathematical conjecture statement in algebraic K-theory statement in arithmetic geometry

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.

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All labels observed (6)

Label	Occurrences
Bloch–Kato conjecture canonical	5
Bloch–Kato conjectures	2
Bloch–Kato Selmer groups (in the context of motives)	1
Bloch–Kato norm residue conjecture	1
Bloch–Kato norm residue isomorphism conjecture	1
norm residue isomorphism 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.
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
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 ⓘ linked to: Bloch–Kato conjecture norm residue isomorphism conjecture ⓘ linked to: Bloch–Kato conjecture
concerns	Milnor K-groups modulo l ⓘ fields of characteristic not equal to a fixed prime l ⓘ l-torsion in Galois cohomology ⓘ
field	Galois cohomology ⓘ algebraic K-theory ⓘ arithmetic geometry ⓘ motivic cohomology ⓘ number theory ⓘ
formulatedBy	Kazuya Kato ⓘ Spencer Bloch ⓘ
generalizes	Milnor conjecture ⓘ linked to: proof of the Milnor conjecture
implies	Milnor conjecture on quadratic forms (in suitable form) ⓘ linked to: proof of the Milnor conjecture
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 ⓘ Rost motives ⓘ motivic cohomology ⓘ norm varieties ⓘ
provedBy	Charles Weibel ⓘ Markus Rost ⓘ Vladimir Voevodsky ⓘ other collaborators in the Rost–Voevodsky program ⓘ
relatedTo	Beilinson–Lichtenbaum conjecture ⓘ Bloch–Kato Selmer groups (in the context of motives) ⓘ linked to: Bloch–Kato conjecture Quillen K-theory ⓘ
relatesConcept	Galois cohomology ⓘ Galois representations ⓘ Milnor K-theory ⓘ 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