It is used to encode ramification data for abelian extensions of a global field.
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.
The ray modulo m is[9][10][11] A modulus m can be split into two parts, mf and m∞, the product over the finite and infinite places, respectively.
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] When K is a number field, the following properties hold.