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 ý(p) are zero except for finitely many p. If K is a number field, ý(p) = 0 or 1 for real places and ý(p) = 0 for complex places. If K is a function field, ý(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<sup>ÃÂ</sup>, the definition of a â¡<sup>âÂÂ</sup>b (mod p<sup>ý</sup>) depends on what type of prime p is:
Then, given a modulus m, a â¡<sup>âÂÂ</sup>b (mod m) if a â¡<sup>âÂÂ</sup>b (mod p<sup>ý(p)</sup>) for all p such that ý(p) > 0.
The ray modulo m is
A modulus m can be split into two parts, m<sub>f</sub> and m<sub>âÂÂ</sub>, the product over the finite and infinite places, respectively. Let I<sup>m</sup> to be one of the following:
In both case, there is a group homomorphism i : K<sub>m,1</sub> â I<sup>m</sup> obtained by sending a to the principal ideal (resp. divisor) (a).
The ray class group modulo m is the quotient C<sub>m</sub> = I<sup>m</sup> / i(K<sub>m,1</sub>). A coset of i(K<sub>m,1</sub>) 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.