Kable, Anthony C.; Yukie, Akihiko The mean value of the product of class numbers of paired quadratic fields. III. (English) Zbl 1039.11086 J. Number Theory 99, No. 1, 185-218 (2003). For Parts I and II, see Tôhoku Math. J. (2) 54, 513–565 (2002; Zbl 1020.11079) and J. Math. Soc. Japan 55, 739–764 (2003; Zbl 1039.11087).This is the third part of a series of papers on the explicit computations of the mean value of the product of class numbers and regulators of two quadratic extensions \(F\), \(F^*\neq\widetilde k\) contained in the biquadratic extensions of \(k\subseteq\widetilde k\). Let \(k\) be a number field, let \(\Delta_k\), \(h_k\) and \(R_k\) be the absolute discriminant, which is an integer, the class number and the regulator, respectively. We fix a number field \(k\) and a quadratic extension \(\widetilde k\) of \(k\). If \(F\neq\widetilde k\) is another quadratic extension of \(k\), let \(\widetilde F\) be the composite of \(F\) and \(\widetilde k\). Then \(\widetilde F\) is a biquadratic extension of k and so contains precisely three quadratic extensions, \(k\), \(F\) and the third one \(F^*\) of \(k\). \(F\) and \(F^*\) are said to be paired. The main theorem of this series of papers are the following two results: (1) With either choice of sign we have \[ \lim_{X\to\infty} X^{-2} \sum_{[F:\mathbb Q]= 2,\;0< \pm \Delta_F< X} h_F R_F h_{F^*} R_{F^*}= c_{\pm}(d_0)^{-1} M(d_0). \] (2) With either choice of sign we have \[ \lim_{X\to \infty} X^{-2} \sum_{[F: \mathbb{Q}]= 2,\;0< \pm\Delta_F< X}h_{F(\sqrt {d_0})} R_{F(\sqrt{d_0})}= c_{\pm}(d_0)^{-1} h_{\mathbb Q(d_0)} R_{\mathbb Q(d_0)} M(d_0). \] Here \(M(d_0)\) is a number-theoretical quantity like an Euler product. In the third part, the authors compute the local density that involve wild ramification at dyadic places, which is rather elaborate. For this purpose they introduce an invariant attached to a pair of ramified quadratic extensions of a dyadic local field. The evaluation of this invariant may be of interest independent of its application here. Reviewer: Masakazu Muro (Gifu) Cited in 2 ReviewsCited in 7 Documents MSC: 11R45 Density theorems 11R29 Class numbers, class groups, discriminants 11S90 Prehomogeneous vector spaces 11S40 Zeta functions and \(L\)-functions Keywords:density theorem; prehomogeneous vector space; binary Hermitian forms; local zeta functions; quadratic fields Citations:Zbl 1020.11079; Zbl 1039.11087 PDFBibTeX XMLCite \textit{A. C. Kable} and \textit{A. Yukie}, J. Number Theory 99, No. 1, 185--218 (2003; Zbl 1039.11086) Full Text: DOI References: [1] Kable, A. C.; Yukie, A., Prehomogeneous vector spaces and field extensions II, Invent. Math., 130, 315-344 (1997) · Zbl 0889.12004 [2] A.C. Kable, A. Yukie, The mean value of the product of class numbers of paired quadratic fields II, J. Math. Soc. Japan, to appear.; A.C. Kable, A. Yukie, The mean value of the product of class numbers of paired quadratic fields II, J. Math. Soc. Japan, to appear. · Zbl 1039.11087 [3] Kable, A. C.; Yukie, A., The mean value of the product of class numbers of paired quadratic fields I, Tohoku Math. J., 54, 513-565 (2002) · Zbl 1020.11079 [4] Weil, A., Basic Number Theory (1974), Springer: Springer Berlin, Heidelberg, New York · Zbl 0326.12001 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.