In mathematics , in the field of algebraic number theory , a modulus (plural moduli ) (or cycle ,[ 1] extended ideal [ 2] 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[ 3] [ 4]
m
=
∏
p
p
ν
(
p
)
,
ν
(
p
)
≥
0
{\displaystyle \mathbf {m} =\prod _{\mathbf {p} }\mathbf {p} ^{\nu (\mathbf {p} )},\,\,\nu (\mathbf {p} )\geq 0}
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 ,[ 5] Arakelov divisor .[ 6]
The notion of congruence can be extended to the setting of moduli. If a and b are elements of K × , the definition of a ≡∗ b (mod p ν ) depends on what type of prime p is:[ 7] [ 8]
a
≡
∗
b
(
m
o
d
p
ν
)
⇔
o
r
d
p
(
a
b
−
1
)
≥
ν
{\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} ^{\nu })\Leftrightarrow \mathrm {ord} _{\mathbf {p} }\left({\frac {a}{b}}-1\right)\geq \nu }
where ordp normalized valuation associated to p ; if it is a real place (of a number field) and ν = 1, then
a
≡
∗
b
(
m
o
d
p
)
⇔
a
b
>
0
{\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} )\Leftrightarrow {\frac {a}{b}}>0}
under the real embedding associated to p . if it is any other infinite place, there is no condition. Then, given a modulus m , a ≡∗ b (mod m ) if a ≡∗ b (mod p ν(p ) ) for all p such that ν(p ) > 0.
Ray class group
The ray modulo m is[ 9] [ 10] [ 11]
K
m
,
1
=
{
a
∈
K
×
:
a
≡
∗
1
(
m
o
d
m
)
}
.
{\displaystyle K_{\mathbf {m} ,1}=\left\{a\in K^{\times }:a\equiv ^{\ast }\!1\,(\mathrm {mod} \,\mathbf {m} )\right\}.}
A modulus m can be split into two parts, m f and m ∞ , the product over the finite and infinite places, respectively. Let I m
In both case, there is a group homomorphism i : K m ,1I m a to the principal ideal (resp. divisor ) (a ).
The ray class group modulo m is the quotient C m I m K m ,1[ 14] [ 15] K m ,1ray 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 .[ 16]
Properties
When K is a number field, the following properties hold.[ 17]
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
^ Lang 1994 , §VI.1^ Cohn 1985 , definition 7.2.1^ Janusz 1996 , §IV.1^ Serre 1988 , §III.1^ Serre 1988 , §III.1^ Neukirch 1999 , §III.1^ Janusz 1996 , §IV.1^ Serre 1988 , §III.1^ Milne 2008 , §V.1^ Janusz 1996 , §IV.1^ Serre 1988 , §VI.6^ Janusz 1996 , §IV.1^ Serre 1988 , §V.1^ Janusz 1996 , §IV.1^ Serre 1988 , §VI.6^ Neukirch 1999 , §VII.6^ Janusz 1996 , §4.1
References
Cohn, Harvey (1985), Introduction to the construction of class fields , Cambridge studies in advanced mathematics, vol. 6, Cambridge University Press , ISBN 978-0-521-24762-7 Janusz, Gerald J. (1996), Algebraic number fields , Graduate Studies in Mathematics , vol. 7, American Mathematical Society , ISBN 978-0-8218-0429-2 Lang, Serge (1994), Algebraic number theory , Graduate Texts in Mathematics , vol. 110 (2 ed.), New York: Springer-Verlag , ISBN 978-0-387-94225-4 MR 1282723 Milne, James (2008), Class field theory , retrieved 2010-02-22 Neukirch, Jürgen (1999). Algebraische Zahlentheorie . Grundlehren der mathematischen Wissenschaften Springer-Verlag . ISBN 978-3-540-65399-8 MR 1697859 . Zbl 0956.11021 .Serre, Jean-Pierre (1988), Algebraic groups and class fields Graduate Texts in Mathematics , vol. 117, New York: Springer-Verlag , ISBN 978-0-387-96648-9 
