Hecke’s integral formula for relative quadratic extensions of algebraic number fields. (English) Zbl 1138.11052

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


11R42 Zeta functions and \(L\)-functions of number fields
11R11 Quadratic extensions
Full Text: DOI arXiv Euclid


[1] T. Asai, On a certain function analogous to \(\log\lvert\eta(z)\rvert\) , Nagoya Math. J., 40 (1970), 193–211. · Zbl 0213.05701
[2] N. Bourbaki, Algèbre Commutative, Masson, 1985.
[3] E. Hecke, Über die Kroneckersche Grenzformel für reelle quadratische Körper und die Klassenzahl relative-Abelscher Körper , Mathematische Werke, 198–207. · JFM 47.0144.01
[4] K. Hiroe and T. Oda, Hecke-Siegel’s pull back formula for the Epstein zeta function with a harmonic polynomial , · Zbl 1174.11071
[5] J. Jorgenson and S. Lang, Hilbert-Asai Eisenstein series, regularized products, and heat kernels , Nagoya Math. J., 153 (1999), 155–188. · Zbl 0936.11033
[6] S. Konno, Eisenstein series in hyperbolic \(3\)-space and Kronecker limit formula for biquadratic field , Nagoya Math. J., 113 (1989), 129–146. · Zbl 0645.12005
[7] Y. I. Manin, Real multiplication and noncommutative geometry ( ein Alterstraum), The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 685–727. · Zbl 1091.11022
[8] C. Meyer, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957. · Zbl 0079.06001
[9] C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980. · Zbl 0478.10001
[10] H. Yoshida, Absolute CM-Periods, Mathematical Surveys and Monographs, Vol. 106, American Mathematical Society, 2003. · Zbl 1041.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.