Modulus (algebraic number theory)

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]

$\mathbf{m} = \prod_{\mathbf{p}} \mathbf{p}^{\nu(\mathbf{p})},\,\,\nu(\mathbf{p})\geq0$

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 an 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

$a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p}^\nu)\Leftrightarrow \mathrm{ord}_\mathbf{p}\left(\frac{a}{b}-1\right)\geq\nu$

where ord_p is the normalized valuation associated to p;

• if it is a real place (of a number field), then

$a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p})\Leftrightarrow \frac{a}{b}>0$

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]

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

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 978-0-821-80429-2 
• Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics, 110 (2 ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR1282723 
• Milne, James (2008), Class field theory (v4.0 ed.), http://jmilne.org/math/CourseNotes/cft.html, retrieved 2010-02-22 
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR1697859 
• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9