Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Lower bounds for relative class numbers of CM-fields

Author: Stéphane Louboutin
Journal: Proc. Amer. Math. Soc. 120 (1994), 425-434
MSC: Primary 11R42; Secondary 11R29
MathSciNet review: 1169041
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathbf{K}}$ be a CM-field that is a quadratic extension of a totally real number field $ {\mathbf{k}}$. Under a technical assumption, we show that the relative class number of $ {\mathbf{K}}$ is large compared with the absolute value of the discriminant of $ {\mathbf{K}}$, provided that the Dedekind zeta function of $ {\mathbf{k}}$ has a real zero $ s$ such that $ 0 < s < 1$. This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

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

  • [1] K. Hardy, R. H. Hudson, D. Richman, and K. S. Williams, Determination of all imaginary cyclic quartic fields with class number $ 2$, Trans. Amer. Math. Soc. 311 (1989), 1-55. MR 929663 (89f:11148)
  • [2] J. Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math. 55 (1979), 37-47. MR 553994 (80k:12019)
  • [3] K. Horie and M. Horie, On the exponent of the class group of $ CM$-fields, Lecture Notes in Math., vol. 1434, Springer-Verlag, Berlin and New York, 1990, pp. 143-148.
  • [4] S. Lang, Functional equation of the zeta function, Hecke's proof, Algebraic Number Theory, Graduate Texts in Math., vol. 110, Springer-Verlag, New York.
  • [5] S. Louboutin, Détermination des corps quartiques cycliques totalement imaginaires à groupes des classes d'idéaux d'exposant $ \leqslant 2$, C. R. Acad. Sci. Paris. Sér. I Math. 315 (1992), 251-254; Manuscripta Math. 77 (1992), 385-404.
  • [6] A. Mallik, A note on Friedlander's paper "On the class numbers of certain quadratic extensions", Acta Arith. 35 (1976), 53-54. MR 536880 (80h:12010)
  • [7] M. Masley and H. L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248-256. MR 0429824 (55:2834)
  • [8] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN, Warsaw, 1976.
  • [9] B. Rosser, Real roots of real Dirichlet series, J. Res. Nat. Bur. Standards 45 (1950), 505-514. MR 0041161 (12:804i)
  • [10] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135-152. MR 0342472 (49:7218)
  • [11] K. Uchida, Imaginary abelian number fields of degree $ {2^m}$ with class number one, Proc. Internat. Conf. on Class Numbers and Fundamental Units of Algebraic Number Fields, Kataka, Japan, 1986, pp. 151-170. MR 891894 (88j:11075)
  • [12] L. C. Washington, Cyclotomic fields of class number one, Introduction to cyclotomic fields, Graduate Texts in Math., vol. 83, Springer-Verlag, New York, 1982.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11R42, 11R29

Retrieve articles in all journals with MSC: 11R42, 11R29

Additional Information

Keywords: Class number, CM-fields, zeta function
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society