# 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).

## Table of contents

## 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* 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 である. This body is total substance It is の total emptiness second expansion. It is a fixed body of the は complex conjugate representation, はそれに I am provided by adding the の square root. - Merger
**Q**^{CM}of all CM bodies resembles CM body except that it is the infinite next expansion. It is the second expansion of merger**Q**^{R}of all total substance. Galois group () Gal(**Q**/**Q**^{R}) is generated (as shut subgroup) by all causes of order of magnitude 2 of Gal(**Q**/**Q**), and Gal(**Q**/**Q**^{CM}) is absolutely subgroup of index 2. Galois group Gal(**Q**^{CM}/**Q**) has the center generated by one cause (complex conjugate) of order of magnitude 2, and the quotient by the center is group Gal(**Q**^{R}/**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 2*n*at most in**Z**. If a rank is 2*n*, 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 function*s, 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 theor*y, 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.

This article is taken from the Japanese Wikipedia **CM body**

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