Artin conductor
E753160
The Artin conductor is an invariant in number theory that measures the ramification of Galois representations or characters of local and global fields, playing a key role in the study of L-functions and class field theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Artin conductor canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733560 — 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: Artin conductor Context triple: [Hasse–Arf theorem, relatedTo, Artin conductor]
-
A.
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.
-
B.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
C.
Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
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D.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
-
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: Artin conductor Target entity description: The Artin conductor is an invariant in number theory that measures the ramification of Galois representations or characters of local and global fields, playing a key role in the study of L-functions and class field theory.
-
A.
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.
-
B.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
C.
Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
-
D.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
-
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 (49)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic invariant
ⓘ
invariant in number theory ⓘ |
| appearsIn |
functional equation of Artin L-functions
ⓘ
functional equation of L-functions ⓘ |
| appliesTo |
Artin representations
NERFINISHED
ⓘ
characters of global fields ⓘ characters of local fields ⓘ finite-dimensional complex representations of Galois groups ⓘ |
| context |
finite Galois extensions of local fields
ⓘ
finite Galois extensions of number fields ⓘ local fields with discrete valuation ⓘ |
| definedFor |
Galois representations of global fields
ⓘ
Galois representations of local fields ⓘ representations of the Weil group ⓘ representations of the Weil–Deligne group ⓘ |
| dependsOn |
higher ramification filtration
ⓘ
inertia subgroup of the Galois group ⓘ |
| field | number theory ⓘ |
| generalizes | conductor of a Dirichlet character ⓘ |
| hasVariant |
Artin conductor exponent
ⓘ
conductor ideal ⓘ |
| measures |
depth of ramification
ⓘ
ramification ⓘ wild ramification ⓘ |
| namedAfter | Emil Artin NERFINISHED ⓘ |
| parameterOf |
Artin L-function
NERFINISHED
ⓘ
global L-factors ⓘ local L-factors ⓘ |
| refines | information given by the discriminant ⓘ |
| relatedTo |
Swan conductor
ⓘ
conductor of a Dirichlet character ⓘ conductor of an elliptic curve ⓘ discriminant of number fields ⓘ global epsilon factors ⓘ local epsilon factors ⓘ |
| satisfies |
additivity on direct sums of representations
ⓘ
multiplicativity under induction in many cases ⓘ |
| usedIn |
Galois representation theory
NERFINISHED
ⓘ
algebraic number theory ⓘ class field theory NERFINISHED ⓘ global Galois representations ⓘ global class field theory NERFINISHED ⓘ local Galois representations ⓘ local class field theory ⓘ representation theory of Galois groups ⓘ theory of L-functions ⓘ |
| usedToDefine |
conductor of a motive
ⓘ
conductor-discriminant formula ⓘ level of an automorphic representation ⓘ |
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: Artin conductor Description of subject: The Artin conductor is an invariant in number theory that measures the ramification of Galois representations or characters of local and global fields, playing a key role in the study of L-functions and class field theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.