# CM body

CM body (CM-field) was algebra body K of the special type and, in mathematics, was accompanied by this name from the close relations with the imaginary number multiplication (complex multiplication) idea. I may be called J-body (J-field).

Abbreviated form "CM" was introduced by (Shimura & Taniyama 1961).

## Definition

When several K is CM bodies, by the second expansion of basic body F which is a total fruit, I say that K is empty all-out. In other words, the implantation to C of F is completely included all in R, but the implantation from K to R does not exist.

In other words, subfield F of K exists; and a certain one original square root following in K in F ${\displaystyle \beta ={\sqrt {\alpha}}}$  It is generated によって. It is all complex number that is not a real number at bottom of the smallest multinomial expression on Yuri several Q of β. Therefore, α must be chosen as "total minus number"; is clogged up; for each implantation σ to a real number field of K σ (α) It is <0.

## Property

One property of the CM body is that complex conjugate on C causes the self same model on the body which does not depend on the implantation body to C. This self-same model must change a mark of β in an upper sign.

It is the equivalent that several K is CM bodies and that K has "units defect" namely that it is true subfield F of K, and the single thing that several have a single Z-rank of K same as several exists (Remak 1954). In fact, F is total real part fission of K which I spoke at the top. This obeys it from a singular theorem of ディリクレ.

## Example

• The example of the CM body which is the easiest and becomes the incentive is an emptiness second body, and the total real part fission is Yuri number field.
• The Japanese yen fission that most important one in question of CM bodies is 1 primitive n root, and is generated ${\displaystyle \mathbb {Q} (\zeta _{n})}$  である. This body is total substance ${\displaystyle \mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1})}$  It is の total emptiness second expansion.${\displaystyle \mathbb {Q} (\zeta _{n}+\zeta _{n}^{-1})}$  It is a fixed body of the は complex conjugate representation,${\displaystyle \mathbb {Q} (\zeta _{n})}$  はそれに ${\displaystyle \zeta _{n}^{2}+\zeta _{n}^{-2}-2=(\zeta _{n}-\zeta _{n}^{-1})^{2}}$  I am provided by adding the の square root.
• Merger QCM of all CM bodies resembles CM body except that it is the infinite next expansion. It is the second expansion of merger QR of all total substance.) Gal(Q/QR) is generated (as shut subgroup) by all causes of order of magnitude 2 of Gal(Q/Q), and Gal(Q/QCM) is absolutely subgroup of index 2. Galois group Gal(QCM/Q) has the center generated by one cause (complex conjugate) of order of magnitude 2, and the quotient by the center is group Gal(QR/Q).
• If V is n dimension double bare Abel manifold, in any commutation algebra F of the self-associate same model of V, a rank is 2n at most in Z. If a rank is 2n, and V is simplicity, F is order of the CM body. On the contrary, any CM body arises from single purely double bare Abel manifold in this way except similar (isogeny).

## References

• Remak, Robert (1954), "Über algebraische Zahlkörper mit schwachem Einheitsdefekt" (German), Compositio Math. 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.