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
m=\prodpp\nu(p),\nu(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, 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:
a\equiv\astb(modp\nu)\Leftrightarrow
| ord | ||||
|
-1\right)\geq\nu
where ordp is the normalized valuation associated to p;
a\equiv\astb(modp)\Leftrightarrow
| 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.
See main article: Ray class group. The ray modulo m is
Km,1=\left\{a\inK x :a\equiv\ast1(modm)\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,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.
When K is a number field, the following properties hold.