Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The structure of Galois groups of $ {\rm CM}$-fields


Author: B. Dodson
Journal: Trans. Amer. Math. Soc. 283 (1984), 1-32
MSC: Primary 11R32; Secondary 11G10, 20B25
MathSciNet review: 735406
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] F. N. Cole, List of the transitive substitutions groups of ten and of eleven letters, Quart. J. Pure Appl. Math. 27 (1895), 39-50.
  • [3] 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
  • [4] 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
  • [5] Marshall Hall Jr. and James K. Senior, The groups of order 2ⁿ(𝑛≤6), The Macmillan Co., New York; Collier-Macmillan, Ltd., London, 1964. MR 0168631
  • [6] P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. MR 0003389
  • [7] E. Hecke, Bestimmung der Klassenzahl einer Neuen Reihe von A Igebraischen Zahlkorpern, Nachr. K. Ges. Wiss. Gottingen, 1921, pp. 1-23 (= Mathematische Werke #15).
  • [8] Tomio Kubota, On the field extension by complex multiplication, Trans. Amer. Math. Soc. 118 (1965), 113–122. MR 0190144, 10.1090/S0002-9947-1965-0190144-8
  • [9] F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Mathematical Library, Vol. 16. MR 0465509
  • [10] 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, 10.2307/2369624
  • [11] -, Collected works, Vol. I, Univ. of Illinois Press, Urbana, Ill., 1935.
  • [12] Henry Pohlmann, Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. (2) 88 (1968), 161–180. MR 0228500
  • [13] B. M. Puttaswamaiah and John D. Dixon, Modular representations of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Pure and Applied Mathematics, No. 73. MR 0442071
  • [14] K. A. Ribet, Division fields of abelian varieties with complex multiplication, Mém. Soc. Math. France (N.S.) 2 (1980/81), 75–94. Abelian functions and transcendental numbers (Colloq., Étole Polytech., Palaiseau, 1979). MR 608640
  • [15] 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
  • [16] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380
  • [17] Goro Shimura, On the class-fields obtained by complex multiplication of abelian varieties, Osaka Math. J. 14 (1962), 33–44. MR 0170893
  • [18] Goro Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144–222. MR 0257031
  • [19] Goro Shimura, On the zeta-function of an abelian variety with complex multiplication., Ann. of Math. (2) 94 (1971), 504–533. MR 0288089
  • [20] Goro Shimura, On abelian varieties with complex multiplication, Proc. London Math. Soc. (3) 34 (1977), no. 1, 65–86. MR 0572987
  • [21] Goro Shimura, Automorphic forms and the periods of abelian varieties, J. Math. Soc. Japan 31 (1979), no. 3, 561–592. MR 535097, 10.2969/jmsj/03130561
  • [22] 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, 10.2307/1971202
  • [23] 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, The Mathematical Society of Japan, Tokyo, 1961. MR 0125113
  • [24] A. Weil, Abelian varieties and the Hodge ring, [1977c], Collected Papers, Vol. Ill, Springer-Verlag, Berlin and New York, pp. 421-429.
  • [25] Garrett Birkhoff and Marshall Hall Jr. (eds.), Computers in algebra and number theory, American Mathematical Society, Providence, R.I., 1971. MR 0327414

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11R32, 11G10, 20B25

Retrieve articles in all journals with MSC: 11R32, 11G10, 20B25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0735406-X
Keywords: Complex multiplication of Abelian varieties, reflex field, nondegenerate Abelian variety of $ CM$-type, imprimitive permutation groups, relative class number
Article copyright: © Copyright 1984 American Mathematical Society