Modulus (algebraic number theory)

For the operation that gives a number's remainder, see Modulo operation.

In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle,[1] or extended ideal[2]) 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[3][4]

${\displaystyle \mathbf {m} =\prod _{\mathbf {p} }\mathbf {p} ^{\nu (\mathbf {p} )},\,\,\nu (\mathbf {p} )\geq 0}$

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] and in the number field case, a modulus can be considered as special form of 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 pν) depends on what type of prime p is:[7][8]

• if it is finite, then
${\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} ^{\nu })\Leftrightarrow \mathrm {ord} _{\mathbf {p} }\left({\frac {a}{b}}-1\right)\geq \nu }$
where ordp is the normalized valuation associated to p;
• if it is a real place (of a number field) and ν = 1, then
${\displaystyle a\equiv ^{\ast }\!b\,(\mathrm {mod} \,\mathbf {p} )\Leftrightarrow {\frac {a}{b}}>0}$
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 pν(p)) for all p such that ν(p) > 0.

Ray class group

Main article: Ray class group

The ray modulo m is[9][10][11]

${\displaystyle K_{\mathbf {m} ,1}=\left\{a\in K^{\times }:a\equiv ^{\ast }\!1\,(\mathrm {mod} \,\mathbf {m} )\right\}.}$

A modulus m can be split into two parts, mf and m, the product over the finite and infinite places, respectively. Let Im to be one of the following:

In both case, there is a group homomorphism i : Km,1Im 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).[14][15] 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.[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

1. ^ Lang 1994, §VI.1
2. ^ Cohn 1985, definition 7.2.1
3. ^ Janusz 1996, §IV.1
4. ^ Serre 1988, §III.1
5. ^ Serre 1988, §III.1
6. ^ Neukirch 1999, §III.1
7. ^ Janusz 1996, §IV.1
8. ^ Serre 1988, §III.1
9. ^ Milne 2008, §V.1
10. ^ Janusz 1996, §IV.1
11. ^ Serre 1988, §VI.6
12. ^ Janusz 1996, §IV.1
13. ^ Serre 1988, §V.1
14. ^ Janusz 1996, §IV.1
15. ^ Serre 1988, §VI.6
16. ^ Neukirch 1999, §VII.6
17. ^ Janusz & 1996 §4.1