The mean value of the product of class numbers of paired quadratic fields. III. (Q1869818)

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.