Ray class field
The Ray class field (sometimes written ray class field) of a number field $K$ with respect to a modulus $\mathfrak{m}$ is the maximal abelian extension of $K$ whose Artin reciprocity map factors through the ray class group modulo $\mathfrak{m}$. It generalises the Hilbert class field, which corresponds to the trivial modulus.
Contents
- Definition
- Moduli and ray class groups
- Existence and uniqueness (Class field theory)
- Properties
- Examples
- Connections with other objects
- References
Definition
Let $K$ be a number field and let
$$ \mathfrak{m}= \mathfrak{m}0 ,\mathfrak{m}\infty $$
be a modulus, where $\mathfrak{m}0$ is an integral ideal of $K$ (the finite part) and $\mathfrak{m}\infty$ is a formal product of real places of $K$ (the infinite part).
The ray class group modulo $\mathfrak{m}$ is
$$ \operatorname{Cl}\mathfrak{m}(K)= I^{\mathfrak{m}}/P{K,1}^{\mathfrak{m}}, $$
where
- $I^{\mathfrak{m}}$ is the group of fractional ideals of $K$ coprime to $\mathfrak{m}_0$;
- $P_{K,1}^{\mathfrak{m}}$ consists of principal ideals $(\alpha)$ with $\alpha\equiv 1 \pmod{\mathfrak{m}}$ (i.e., $\alpha\equiv 1 \pmod{\mathfrak{m}0}$ and $\alpha>0$ at each real place occurring in $\mathfrak{m}\infty$).
The Ray class field $K_{\mathfrak{m}}$ is the (unique) finite abelian extension of $K$ characterised by the Artin reciprocity map
$$ \theta_{\mathfrak{m}} \colon \operatorname{Cl}\mathfrak{m}(K) ;\xrightarrow{;\sim;}; \operatorname{Gal}(K{\mathfrak{m}}/K), $$
which is an isomorphism of groups. In other words, $K_{\mathfrak{m}}$ is the maximal abelian extension of $K$ unramified outside the primes dividing $\mathfrak{m}0$ and with prescribed splitting behaviour at the real places in $\mathfrak{m}\infty$.
Moduli and ray class groups
A modulus $\mathfrak{m}$ encodes both finite and infinite ramification data:
| Part | Symbol | Description |
|---|---|---|
| Finite part | $\mathfrak{m}_0$ | Product $\prod_{\mathfrak{p}} \mathfrak{p}^{e_{\mathfrak{p}}}$ of prime ideals with exponents $e_{\mathfrak{p}}\ge 0$. |
| Infinite part | $\mathfrak{m}_\infty$ | Subset of the set of real embeddings of $K$; each selected real place contributes a factor representing the condition “positive at that place”. |
The resulting ray class group refines the ordinary ideal class group by imposing congruence conditions modulo $\mathfrak{m}$.
Existence and uniqueness (Class field theory)
Global class field theory guarantees:
- Existence: For every modulus $\mathfrak{m}$ there exists a unique (up to $K$-isomorphism) finite abelian extension $K_{\mathfrak{m}}$ satisfying the Artin reciprocity isomorphism above.
- Maximality: Any finite abelian extension $L/K$ whose conductor divides $\mathfrak{m}$ (i.e. is unramified outside $\mathfrak{m}$ and satisfies the same infinite‑place conditions) is contained in $K_{\mathfrak{m}}$.
When $\mathfrak{m}= (1)$ (the trivial modulus), the ray class field coincides with the Hilbert class field, the maximal unramified abelian extension of $K$.
Properties
| Property | Description |
|---|---|
| Galois group | $\operatorname{Gal}(K_{\mathfrak{m}}/K) \cong \operatorname{Cl}_\mathfrak{m}(K)$. |
| Conductor | The conductor $\mathfrak{f}(K_{\mathfrak{m}}/K)$ equals the smallest modulus $\mathfrak{n}$ such that $K_{\mathfrak{m}} \subseteq K_{\mathfrak{n}}$; it divides $\mathfrak{m}$. |
| Ramification | Only primes dividing $\mathfrak{m}0$ may ramify. Real places in $\mathfrak{m}\infty$ become complex (i.e., the extension is totally imaginary at those places). |
| Compatibility | If $\mathfrak{m}\mid\mathfrak{n}$ then $K_{\mathfrak{m}}\subseteq K_{\mathfrak{n}}$ and the natural projection $\operatorname{Cl}{\mathfrak{n}}(K)\twoheadrightarrow\operatorname{Cl}\mathfrak{m}(K)$ corresponds to restriction $\operatorname{Gal}(K_{\mathfrak{n}}/K)\to\operatorname{Gal}(K_{\mathfrak{m}}/K)$. |
| Explicit class field theory | In certain cases (e.g., $K=\mathbb{Q}$ or imaginary quadratic fields) $K_{\mathfrak{m}}$ can be described concretely: for $\mathbb{Q}$ it is the cyclotomic field $\mathbb{Q}(\zeta_m)$ where $m$ is the norm of $\mathfrak{m}$; for imaginary quadratic fields it is generated by values of the Weber, Siegel, or Weber‑function at CM‑points. |
Examples
-
Rational field $\mathbb{Q}$.
Let $\mathfrak{m}=m\mathbb{Z}$ (no infinite part). The ray class group is $(\mathbb{Z}/m\mathbb{Z})^\times$, and the corresponding ray class field is the cyclotomic field $\mathbb{Q}(\zeta_m)$. -
Imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$.
For a modulus $\mathfrak{m}= \mathfrak{f}$ (integral ideal) the ray class field $K_{\mathfrak{f}}$ is generated by the values of the j‑invariant (or more refined Weber functions) at elliptic curves with complex multiplication by the order of conductor $\mathfrak{f}$. In particular, the Hilbert class field corresponds to $\mathfrak{f}= (1)$. -
Real quadratic fields.
When the modulus contains the infinite part consisting of all real places, the ray class field is totally complex (i.e., a CM‑extension). An explicit description is usually more intricate, but for small conductors it can be computed via class field theory software (e.g., PARI/GP, Magma).
Connections with other objects
- Hilbert class field – special case with trivial modulus.
- Conductor – the smallest modulus giving a particular abelian extension; central in the study of Artin conductors.
- Abelian extensions of $\mathbb{Q}$ – all finite abelian extensions are subfields of cyclotomic fields, i.e., ray class fields of $\mathbb{Q}$.
- Kronecker–Weber theorem – the statement that every finite abelian extension of $\mathbb{Q}$ is a subfield of a ray class field $\mathbb{Q}(\zeta_m)$.
- Complex multiplication (CM) – the theory that provides explicit generators for ray class fields of imaginary quadratic fields via values of modular functions.
References
- J. Neukirch, Algebraic Number Theory, Springer GTM 322, 1999 – Chap. VIII (ray class fields, conductors).
- H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000 – §11.4 (explicit ray class fields).
- S. Lang, Algebraic Number Theory, Springer GTM 110, 1994 – Chap. III, §5 (moduli and ray class groups).
- J. Milne, Class Field Theory (online notes), 2022 – §5 (global class field theory, ray class fields).
- G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton Univ. Press, 1998 – application to imaginary quadratic ray class fields.