Artin conductor

E753160

arithmetic invariant invariant in number theory

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.

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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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hasse–Arf theorem → relatedTo → Artin conductor ⓘ

Grothendieck–Ogg–Shafarevich formula → relatesTo → Artin conductor ⓘ