modulus

A modulus for a number field $K$ is a formal product $$ \prod_{\p} \p^{n_\p} $$ where

• The product is taken over all finite primes and infinite primes of $K$
• The exponents $n_\p$ are nonnegative integers
• All but finitely many of the $n_\p$ are zero
• For every real prime $\p$ , the exponent $n_\p$ is either 0 or 1
• For every complex prime $\p$ , the exponent $n_\p$ is 0

A modulus can be written as a product of its finite part $$ \prod_{\p \text{ finite}} \p^{n_\p} $$ and its infinite part $$ \prod_{\p \text{ real}} \p^{n_\p}, $$ with the finite part equal to some ideal in the ring of integers $\mathcal{O}_K$ of $K$ , and the infinite part equal to the product of some subcollection of the real primes of $K$ .