Modulus (algebraic number theory)

In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) 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. 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(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, 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 (mod pν) depends on what type of prime p is: if it is finite, then where ordp is the normalized valuation associated to p; if it is a real place (of a number field) and ν = 1, then 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 pν(p)) for all p such that ν(p) > 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: if K is a number field, the subgroup of the group of fractional ideals generated by ideals coprime to mf; if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m. In both case, there is a group homomorphism i : Km,1 → Im 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). 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.