GetWiki

{{For|the operation that gives a number’s remainder|Modulo operation}}In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle,{{harvnb|Lang|1994|loc=§VI.1}} or extended ideal{{harvnb|Cohn|1985|loc=definition 7.2.1}}) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field.

Definition

Let K be a global field with ring of integers R. A modulus is a formal product{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§III.1}} where p runs over all places of K, finite or infinite, the exponents ν(p) are zero except for finitely many p. If K is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If K is a function field, ν(p) = 0 for all infinite places.In the function field case, a modulus is the same thing as an effective divisor,{{harvnb|Serre|1988|loc=§III.1}} and in the number field case, a modulus can be considered as special form of Arakelov divisor.{{harvnb|Neukirch|1999|loc=§III.1}}The notion of congruence can be extended to the setting of moduli. If a and b are elements of K×, the definition of a â‰¡âˆ—b (mod pν) depends on what type of prime p is:{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§III.1}}

• if it is finite, then

• if it is a real place (of a number field) and ν = 1, then

• if it is any other infinite place, there is no condition.

Then, given a modulus m, a â‰¡âˆ—b (mod m) if a â‰¡âˆ—b (mod pν(p)) for all p such that ν(p) > 0.

Ray class group

The ray modulo m is{{harvnb|Milne|2008|loc=§V.1}}{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§VI.6}} A modulus m can be split into two parts, mf and m∞, the product over the finite and infinite places, respectively. Let Im to be one of the following:

• if K is a number field, the subgroup of the group of fractional ideals generated by ideals coprime to mf;{{harvnb|Janusz|1996|loc=§IV.1}}
• if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m.{{harvnb|Serre|1988|loc=§V.1}}

In both case, there is a group homomorphism i : Km,1 → Im obtained by sending a to the principal ideal (resp. divisor) (a).The ray class group modulo m is the quotient Cm = Im / i(Km,1).{{harvnb|Janusz|1996|loc=§IV.1}}{{harvnb|Serre|1988|loc=§VI.6}} A coset of i(Km,1) is called a ray class modulo m.Erich Hecke’s original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m.{{harvnb|Neukirch|1999|loc=§VII.6}}

Properties

When K is a number field, the following properties hold.{{harvnb|Janusz|1996|loc=§4.1}}

• When m = 1, the ray class group is just the ideal class group.
• The ray class group is finite. Its order is the ray class number.
• The ray class number is divisible by the class number of K.

Notes

{{reflist|2}}

References

• {hide}Citation| last=Cohn| first=Harvey| title=Introduction to the construction of class fields| series=Cambridge studies in advanced mathematics| volume=6| publisher=Cambridge University Press| year=1985| isbn=978-0-521-24762-7

{edih}

• {{Citation| last=Janusz| first=Gerald J.| title=Algebraic number fields| publisher=American Mathematical Society| series=Graduate Studies in Mathematics| volume=7| year=1996| isbn=978-0-8218-0429-2

}}

• {hide}Citation| last=Lang| first=Serge| author-link=Serge Lang| title=Algebraic number theory| edition=2| publisher=Springer-Verlag| year=1994| series=Graduate Texts in Mathematics| volume=110| place=New York| isbn=978-0-387-94225-4| mr=1282723

{edih}

• {{Citation| last=Milne| first=James| title=Class field theory| url=http://jmilne.org/math/CourseNotes/cft.html| edition=v4.0| year=2008| accessdate=2010-02-22

}}

• {{Neukirch ANT}}
• {{Citation| last=Serre| first=Jean-Pierre| author-link=Jean-Pierre Serre| title=Algebraic groups and class fields| year=1988| isbn=978-0-387-96648-9| publisher=Springer-Verlag| location=New York| series=Graduate Texts in Mathematics| volume=117| url-access=registration| url=https://archive.org/details/algebraicgroupsc0000serr

}}