Modulus (algebraic Number Theory) - Definition

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(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 ord_p 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.