The structure of Galois groups of 𝐶𝑀-fields

Dodson, B.

doi:10.1090/S0002-9947-1984-0735406-X

The structure of Galois groups of $\textrm {CM}$-fields
HTML articles powered by AMS MathViewer

by B. Dodson PDF

Trans. Amer. Math. Soc. 283 (1984), 1-32 Request permission

Abstract:

A $CM$-field $K$ defines a triple $(G,H,\rho )$, where $G$ is the Galois group of the Galois closure of $K$, $H$ is the subgroup of $G$ fixing $K$, and $\rho \in G$ is induced by complex conjugation. A "$\rho$-structure" identifies $CM$-fields when their triples are identified under the action of the group of automorphisms of $G$. A classification of the $\rho$-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of $\rho$-structues and reflex fields are provided for $[K:\mathbb {Q}] = 2n$, with $n = 3,4,5$ and $7$. In addition, simple degenerate Abelian varieties of $CM$-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group $G = {D_{2n}}$, with $n$ odd and $H$ of order $2$, and a relative class number formula is found.

References

D. Bertrand and M. Waldschmidt (eds.), Fonctions abéliennes et nombres transcendants, Bordas, Paris, 1981 (French). Papers from the Colloquium held at the École Polytechnique, Palaiseau, May 23–26, 1979; Mém. Soc. Math. France (N.S.) 1980/81, no. 2. MR 608633

List of the transitive substitutions groups of ten and of eleven letters

27

H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913, DOI 10.1007/978-3-662-21943-0
Walter Feit, Some consequences of the classification of finite simple groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 175–181. MR 604576
Marshall Hall Jr. and James K. Senior, The groups of order $2^{n}\,(n\leq 6)$, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1964. MR 0168631
P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. MR 3389, DOI 10.1515/crll.1940.182.130

Bestimmung der Klassenzahl einer Neuen Reihe von A Igebraischen Zahlkorpern

Tomio Kubota, On the field extension by complex multiplication, Trans. Amer. Math. Soc. 118 (1965), 113–122. MR 190144, DOI 10.1090/S0002-9947-1965-0190144-8
F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, North-Holland Mathematical Library, Vol. 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0465509
G. A. Miller, Memoir on the Substitution-Groups Whose Degree Does not Exceed Eight, Amer. J. Math. 21 (1899), no. 4, 287–338. MR 1505804, DOI 10.2307/2369624

Collected works

Henry Pohlmann, Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. (2) 88 (1968), 161–180. MR 228500, DOI 10.2307/1970570
B. M. Puttaswamaiah and John D. Dixon, Modular representations of finite groups, Pure and Applied Mathematics, No. 73, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 0442071
K. A. Ribet, Division fields of abelian varieties with complex multiplication, Mém. Soc. Math. France (N.S.) 2 (1980/81), 75–94. MR 608640
K. A. Ribet, Generalization of a theorem of Tankeev, Seminar on Number Theory, 1981/1982, Univ. Bordeaux I, Talence, 1982, pp. Exp. No. 17, 4. MR 695334
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
Goro Shimura, On the class-fields obtained by complex multiplication of abelian varieties, Osaka Math. J. 14 (1962), 33–44. MR 170893
Goro Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144–222. MR 257031, DOI 10.2307/1970604
Goro Shimura, On the zeta-function of an abelian variety with complex multiplication, Ann. of Math. (2) 94 (1971), 504–533. MR 288089, DOI 10.2307/1970768
Goro Shimura, On abelian varieties with complex multiplication, Proc. London Math. Soc. (3) 34 (1977), no. 1, 65–86. MR 572987, DOI 10.1112/plms/s3-34.1.65
Goro Shimura, Automorphic forms and the periods of abelian varieties, J. Math. Soc. Japan 31 (1979), no. 3, 561–592. MR 535097, DOI 10.2969/jmsj/03130561
Goro Shimura, The arithmetic of certain zeta functions and automorphic forms on orthogonal groups, Ann. of Math. (2) 111 (1980), no. 2, 313–375. MR 569074, DOI 10.2307/1971202
Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Mathematical Society of Japan, Tokyo, 1961. MR 0125113

Abelian varieties and the Hodge ring

Garrett Birkhoff and Marshall Hall Jr. (eds.), Computers in algebra and number theory, SIAM-AMS Proceedings, Vol. IV, American Mathematical Society, Providence, R.I., 1971. MR 0327414