Modulus (algebraic number theory)


In mathematics, in the field of algebraic number theory, a modulus is a formal product of places of a global 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
where p runs over all places of K, finite or infinite, the exponents ν are zero except for finitely many p. If K is a number field, ν = 0 or 1 for real places and ν = 0 for complex places. If K is a function field, ν = 0 for all infinite places.
In the function field case, a modulus is the same thing as an effective divisor, and in the number field case, a modulus can be considered as special form of Arakelov divisor.
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 depends on what type of prime p is:
Then, given a modulus m, a ≡∗b if a ≡∗b for all p such that ν >; 0.

Ray class group

The ray modulo m is
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,1 → Im obtained by sending a to the principal ideal .
The ray class group modulo m is the quotient Cm = Im / i. A coset of i 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.

Properties

When K is a number field, the following properties hold.