Herbrand function

E753159

function in number theory mathematical concept tool in local class field theory

The Herbrand function is a numerical tool in local class field theory that measures the ramification filtration of Galois groups, playing a key role in understanding how ramification behaves in extensions of local fields.

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Statements (47)

Predicate	Object
instanceOf	function in number theory ⓘ mathematical concept ⓘ tool in local class field theory ⓘ
appearsIn	Serre Local Fields NERFINISHED ⓘ classical local class field theory ⓘ
appliesTo	Galois group of a finite Galois extension of local fields ⓘ
associatedWith	ramification groups in the lower numbering ⓘ ramification groups in the upper numbering ⓘ
characterizes	ramification filtration of Galois groups ⓘ
codomain	real numbers ⓘ
context	finite Galois extensions of non-archimedean local fields ⓘ ramified extensions of p-adic fields ⓘ
dependsOn	sizes of ramification subgroups ⓘ
domain	Galois extensions of local fields ⓘ non-archimedean local fields ⓘ
ensures	functorial behavior of upper ramification filtration under quotients ⓘ
field	local class field theory NERFINISHED ⓘ number theory ⓘ
formalVariable	real parameter u ≥ -1 ⓘ
historicalPeriod	20th century mathematics ⓘ
input	real parameter describing lower-numbered ramification groups ⓘ
measures	ramification behavior in extensions of local fields ⓘ
namedAfter	Jacques Herbrand NERFINISHED ⓘ
output	real parameter describing upper-numbered ramification groups ⓘ
property	breakpoints determined by jumps in ramification filtration ⓘ continuous from the right ⓘ increasing function of a real variable ⓘ piecewise linear ⓘ
relatedConcept	Herbrand quotient NERFINISHED ⓘ lower numbering of ramification groups ⓘ ramification filtration ⓘ upper numbering of ramification groups ⓘ
relatedTo	different of an extension of local fields ⓘ discriminant of an extension of local fields ⓘ
relates	lower numbering of ramification groups ⓘ upper numbering of ramification groups ⓘ
role	normalizes ramification filtration to behave well under quotients ⓘ transfers the filtration index from lower to upper numbering ⓘ
usedBy	number theorists studying local Galois representations ⓘ researchers in arithmetic geometry ⓘ
usedFor	comparing ramification in different extensions ⓘ defining upper ramification breaks ⓘ describing wild ramification ⓘ reindexing ramification groups from lower to upper numbering ⓘ
usedIn	Galois theory of local fields NERFINISHED ⓘ local class field theory NERFINISHED ⓘ ramification theory ⓘ

Referenced by (1)

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

Hasse–Arf theorem → relatedTo → Herbrand function ⓘ