Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Lower bounds for relative class numbers of CM-fields
HTML articles powered by AMS MathViewer

by Stéphane Louboutin PDF
Proc. Amer. Math. Soc. 120 (1994), 425-434 Request permission

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
  • 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, DOI 10.1090/S0002-9947-1989-0929663-9
  • Jeffrey Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math. 55 (1979), no. 1, 37–47. MR 553994, DOI 10.1007/BF02139701
    • 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. S. Lang, Functional equation of the zeta function, Hecke’s proof, Algebraic Number Theory, Graduate Texts in Math., vol. 110, Springer-Verlag, New York. 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.
  • 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, DOI 10.4064/aa-35-1-53-54
  • J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286(287) (1976), 248–256. MR 429824
    • W. Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN, Warsaw, 1976.
  • J. Barkley Rosser, Real roots of real Dirichlet $L$-series, J. Research Nat. Bur. Standards 45 (1950), 505–514. MR 0041161
  • H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
  • 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
    • 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
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 120 (1994), 425-434
  • MSC: Primary 11R42; Secondary 11R29
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1169041-0
  • MathSciNet review: 1169041