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]

\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 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]

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;
a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p})\Leftrightarrow \frac{a}{b}>0
under the real embedding associated to p.

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

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, 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]

Notes

References