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] Kenneth Hardy, Richard H. Hudson, David Richman, and Kenneth S. Williams, Determination of all imaginary cyclic quartic fields with class number 2, Trans. Amer. Math. Soc. 311 (1989), no. 1, 1–55. MR 929663, 10.1090/S0002-9947-1989-0929663-9
  • [2] Jeffrey Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math. 55 (1979), no. 1, 37–47. MR 553994, 10.1007/BF02139701
  • [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 J. B. Friedlander’s paper: “On the class numbers of certain quadratic extensions” [Acta Arith. 28 (1975/76), no. 4, 391–393; MR 52 #10683], Acta Arith. 35 (1979), no. 1, 54–55. MR 536880
  • [7] J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248–256. MR 0429824
  • [8] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN, Warsaw, 1976.
  • [9] J. Barkley Rosser, Real roots of real Dirichlet 𝐿-series, J. Research Nat. Bur. Standards 45 (1950), 505–514. MR 0041161
  • [10] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 0342472
  • [11] Kôji Uchida, Imaginary abelian number fields of degrees 2^{𝑚} with class number one, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986) Nagoya Univ., Nagoya, 1986, pp. 151–170. MR 891894
  • [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

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1169041-0
Keywords: Class number, CM-fields, zeta function
Article copyright: © Copyright 1994 American Mathematical Society