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)

 
 

 

On proportional constants of the mean value of class numbers of quadratic extensions


Author: Takashi Taniguchi
Journal: Trans. Amer. Math. Soc. 359 (2007), 5517-5524
MSC (2000): Primary 11R45, 11S90
DOI: https://doi.org/10.1090/S0002-9947-07-04221-3
Published electronically: April 17, 2007
MathSciNet review: 2327040
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we give a refinement of the mean value theorem for the class number of quadratic extensions obtained by Goldfeld-Hoffstein and Datskovsky. More specifically, we determine the proportional constants of the mean value for fields that satisfy any local conditions including wild ramification at places dividing $ 2$.


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

  • [D] B. Datskovsky.
    A mean value theorem for class numbers of quadratic extensions.
    Contemporary Mathematics, 143:179-242, 1993. MR 1210518 (94m:11137)
  • [G] C.F. Gauss.
    Disquisitiones arithmeticae.
    Yale University Press, New Haven, London, 1966. MR 0197380 (33:5545)
  • [GH] D. Goldfeld and J. Hoffstein.
    Eisenstein series of $ 1/2$-integral weight and the mean value of real Dirichlet series.
    Invent. Math., 80:185-208, 1985. MR 788407 (86m:11029)
  • [KY] A.C. Kable and A. Yukie.
    The mean value of the product of class numbers of paired quadratic fields, II.
    J. Math. Soc. Japan, 55:739-764, 2003. MR 1978221 (2004g:11104a)
  • [GIT] D. Mumford and J. Fogarty.
    Geometric invariant theory.
    Springer-Verlag, Berlin, Heidelberg, New York, 2nd edition, 1982. MR 719371 (86a:14006)
  • [Sh] T. Shintani.
    On zeta-functions associated with vector spaces of quadratic forms.
    J. Fac. Sci. Univ. Tokyo, Sect IA, 22:25-66, 1975. MR 0384717 (52:5590)
  • [Si] C.L. Siegel.
    The average measure of quadratic forms with given discriminant and signature.
    Ann. of Math., 45:667-685, 1944. MR 0012642 (7:51a)
  • [T] T. Taniguchi.
    A mean value theorem for the square of class number times regulator of quadratic extensions.
    Preprint 2004. MR 2106480 (2006c:11137)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11R45, 11S90

Retrieve articles in all journals with MSC (2000): 11R45, 11S90


Additional Information

Takashi Taniguchi
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3–8–1 Komaba Megoro-ku, Tokyo 153-0041, Japan
Email: tani@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-07-04221-3
Keywords: Density theorems, prehomogeneous vector spaces
Received by editor(s): November 15, 2004
Received by editor(s) in revised form: October 31, 2005
Published electronically: April 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society