CM-field

In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.

The abbreviation "CM" was introduced by (Shimura & Taniyama 1961).

Formal definition

A number field K is a CM-field if it is a quadratic extension K/F where the base field F is totally real but K is totally imaginary. I.e., every embedding of F into $\mathbb C$ lies entirely within $\mathbb R$ , but there is no embedding of K into $\mathbb R$ .

In other words, there is a subfield F of K such that K is generated over F by a single square root of an element, say β = $\sqrt{\alpha}$ , in such a way that the minimal polynomial of β over the rational number field $\mathbb Q$ has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of $K'$ into the real number field, σ(α) < 0.

Properties

One feature of a CM-field is that complex conjugation on $\mathbb C$ induces an automorphism on the field which is independent of its embedding into $\mathbb C$ . In the notation given, it must change the sign of β.

A number field K is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield F whose unit group has the same $\mathbb Z$ -rank as that of K (Remak 1954). In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.

Examples

The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals
One of the most important examples of a CM-field is the cyclotomic field $\mathbb Q (\zeta_n)$ , which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field $\mathbb Q (\zeta_n +\zeta_n^{-1}).$ The latter is the fixed field of complex conjugation, and $\mathbb Q (\zeta_n)$ is obtained from it by adjoining a square root of $\zeta_n^2+\zeta_n^{-2}-2 = (\zeta_n - \zeta_n^{-1})^2.$
The union Q^CM of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields Q^R. The absolute Galois group Gal(Q/Q^R) is generated (as a closed subgroup) by all elements of order 2 in Gal(Q/Q), and Gal(Q/Q^CM) is a subgroup of index 2. The Galois group Gal(Q^CM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(Q^R/Q).
If V is a complex abelian variety of dimension n, then any abelian algebra F of endomorphisms of V has rank at most 2n over Z. If it has rank 2n and V is simple then F is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.

References

Remak, Robert (1954), "Über algebraische Zahlkörper mit schwachem Einheitsdefekt", Compositio Mathematica (in German) 12: 35–80, Zbl 0055.26805
Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan 11, Princeton, N.J.: Princeton University Press
Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan 6, Tokyo: The Mathematical Society of Japan, MR 0125113
Washington, Lawrence C. (1996). "Introduction to Cyclotomic fields" (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.