Modulus (algebraic number theory)

In mathematics, in the field of algebraic number theory, a modulus (or an extended ideal) is a formal product of places of an algebraic number field. It is used to encode ramification data for abelian extensions of number field.

Definition

Let K be an algebraic number field with ring of integers R. A modulus is a formal product

$\mathbf {m} =\prod _{\mathbf {p} }\mathbf {p} ^{\nu (\mathbf {p} )}$

where p runs over all places of K, finite or infinite, the exponents ν are zero except for finitely many p, for real places r we have ν(r)=0 or 1 and for complex places ν=0.

We extend the notion of congruence to this setting. Let x and y be elements of K. For a finite place p, that is, a prime ideal of the ring of integers, we define x and y to be congruent modulo pⁿ if x/y is in the valuation ring R_p of p and congruent to 1 modulo pⁿ in R_p in the usual sense of ring theory. For a real place r we define x and y to be congruent modulo r if x/y is positive in the real embedding of K associated to r. Finally, we define x and y to be congruent modulo m if they are congruent modulo p^ν(p) whenever ν(p) > 0.

Ray class group

We split the modulus m into m_fin and m_inf, the product over the finite and infinite places respectively. Define

$K_{\mathbf {m} }=\left\lbrace a/b\in K\mid a,b\in R,~ab~{\mbox{coprime to}}~\mathbf {m} _{\mbox{fin}}\right\rbrace ,$

$K_{\mathbf {m} ,1}=\left\lbrace x\in K_{\mathbf {m} }\mid x\equiv 1{\pmod {\mathbf {m} }}\right\rbrace .$

We call the group K_m,1 the ray modulo m.

Further define the subgroup of the ideal group I^m to be the subgroup generated by ideals coprime to m_fin. The ray class group modulo m is the quotient I^m / i(K_m,1), where i is the map from K to principal ideals in the ideal group. A coset of i(K_m,1) is a ray class.

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.

Properties

• When m = 1, the ray class group is just the ideal class group.
• The ray class group is finite. Its order is the ray class number.
• The ray class number divides the class number of K.

References

• Harvey Cohn (1978). A classical invitation to algebraic numbers and class fields. Springer-Verlag, 163-187. ISBN 0-387-90345-3. 
• Harvey Cohn (1985). Introduction to the construction of class fields. Cambridge University Press, 99. ISBN 0-521-24762-4. 
• Gerald J. Janusz (1973). Algebraic Number Fields. Academic Press, 107-113. ISBN 0-12-380250.