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 .

**[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 -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)**

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.