Modulus of a number field

A modulus is a formal product $\mathfrak{m}={\mathfrak{m}}_0{\mathfrak{m}}_{\infty }$, where ${\mathfrak{m}}_0$ is a nonzero integral ideal and ${\mathfrak{m}}_{\infty }$ is a product of distinct real places.

For a modulus of a number field $\mathfrak{c}={\mathfrak{c}}_0{\mathfrak{c}}_{\infty }$, the multiplicity at a finite place is the exponent of its prime ideal in ${\mathfrak{c}}_0$. At a real Archimedean place it is one if that place occurs in ${\mathfrak{c}}_{\infty }$ and zero otherwise. Complex places have multiplicity zero. These effective modulus multiplicities encode congruences and positivity requirements in the ray class group.

The ray class group modulo $\mathfrak{m}$ is $I_K(\mathfrak{m})/P_K(\mathfrak{m})$, where the numerator consists of fractional ideals prime to ${\mathfrak{m}}_0$ and the denominator consists of $(\alpha )$ with $\alpha \equiv 1$ at the finite modulus and positive at every real place in ${\mathfrak{m}}_{\infty }$.

A subgroup of the group of fractional ideals prime to a modulus of a number field, containing the principal ray subgroup. It is the inverse image of a subgroup of the ray class group. The existence theorem of global class field theory realizes it as the kernel of an Artin map, and identifies its quotient with the corresponding Galois group. This is the ideal-theoretic class-field meaning of congruence subgroup.

A ray unit for a modulus of a number field is a unit congruent to one modulo every finite prime power in the modulus and positive at every real place in it. The group of ray units is the kernel of the unit group map to the residue and signature group of a modulus.

For a finite modulus $\mathfrak{m}$, there is an exact sequenceWith real places in the modulus, the residue-unit term and the unit map also record the required signs.

For a modulus with no infinite component, suppose the ordinary ideal class group is trivial. Then the ray class group is trivial exactly when the global unit group maps surjectively onto the product of local unit residue groups at the finite modulus. Negative real units are allowed: positivity restrictions occur only when the modulus includes the corresponding real places.

This group records finite unit residues and signs at the selected real places of a modulus of a number field. Omit the residue factor for trivial finite modulus. Its quotient by the image of the unit group is naturally the kernel of the map from the ray class group to the ordinary ideal class group. The weak approximation for number fields theorem supplies elements realizing the specified data, and units account for changing a generator of a principal fractional ideal.

If $U_{\mathfrak{m},1}$ consists of units congruent to one modulo ${\mathfrak{m}}_0$ and positive at the selected real places, then