Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T02:21:45.058Z Has data issue: false hasContentIssue false
  • English
  • Français

Hecke’s Integral Formula for Relative Quadratic Extensions of Algebraic Number Fields

Published online by Cambridge University Press:  11 January 2016

Shuji Yamamoto
Shuji Yamamoto*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker’s type which relates the 0-th Laurent coefficients at s = 1 of zeta functions of K and F.

Keywords

11R4211R11
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 189 , 2008 , pp. 139 - 154
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[1] Asai, T., On a certain function analogous to log|η(z)|, Nagoya Math. J., 40 (1970), 193–211.Google Scholar
[2] Bourbaki, N., Algèbre Commutative, Masson, 1985.Google Scholar
[3] Hecke, E., Über die Kroneckersche Grenzformel für reelle quadratische Körper und die Klassenzahl relative-Abelscher Körper, Mathematische Werke, 198–207.Google Scholar
[4] Hiroe, K. and Oda, T., Hecke-Siegel’s pull back formula for the Epstein zeta function with a harmonic polynomial, preprint.Google Scholar
[5] Jorgenson, J. and Lang, S., Hilbert-Asai Eisenstein series, regularized products, and heat kernels, Nagoya Math. J., 153 (1999), 155–188.Google Scholar
[6] Konno, S., Eisenstein series in hyperbolic 3-space and Kronecker limit formula for biquadratic field, Nagoya Math. J., 113 (1989), 129–146.Google Scholar
[7] Manin, Y. I., Real multiplication and noncommutative geometry (ein Alterstraum), The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 685–727.Google Scholar
[8] Meyer, C., Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957.Google Scholar
[9] Siegel, C. L., Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980.Google Scholar
[10] Yoshida, H., Absolute CM-Periods, Mathematical Surveys and Monographs, Vol. 106, American Mathematical Society, 2003.Google Scholar