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 p^ν) 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 ord_p 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 p^ν(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, m_f and m_∞, the product over the finite and infinite places, respectively. Let I^m 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 m_f; ^[12]
• if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m. ^[13]

In both case, there is a group homomorphism i : K_m,1 → I^m obtained by sending a to the principal ideal (resp. divisor) (a).

The ray class group modulo m is the quotient C_m = I^m / i(K_m,1). ^{[14] [15]} A coset of i(K_m,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

References

• Cohn, Harvey (1985), Introduction to the construction of class fields, Cambridge studies in advanced mathematics, 6, Cambridge University Press, ISBN   978-0-521-24762-7
• Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics, 7, American Mathematical Society, ISBN   978-0-8218-0429-2
• Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics, 110 (2 ed.), New York: Springer-Verlag, ISBN   978-0-387-94225-4, MR   1282723
• Milne, James (2008), Class field theory (v4.0 ed.), retrieved 2010-02-22
• Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: 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, 117, New York: Springer-Verlag, ISBN   978-0-387-96648-9