A mean value theorem for the square of class number times regulator of quadratic extensions

Let $k$ be a number field. In this paper, we give a formula for the mean value of the square of class numbers times regulators for certain families of quadratic extensions of $k$ characterized by finitely many local conditions. We approach this by using the theory of the zeta function associated with the space of pair of quaternion algebras.Comment: 35 pages, AMS LaTeX; added a reference and rewrote Section

Dusty Tori of Seyfert Nuclei Probed by the Water Vapor Maser Emission: How Large are the Dusty Tori ?

Geometrical and physical properties of dusty tori of Seyfert nuclei probed by the water vapor maser emission at 22 GHz are discussed. We assume that the dusty torus has a simple cylindrical form and the maser emission can be detected only when we observe the torus from almost edge-on views. The observed low frequency of occurrence of the water vapor maser emission (less than 10 percent) suggests that the torus is a vertically thin cylinder whose outer radius between a few pc and ~ 10 pc. However, the observed masing regions are concentrated in the inner 1 pc regions of the torus. This property can be explained by that only the inner a few pc regions have physical conditions enough to cause the maser emission; the temperature is as high as several hundred K and the density is as high as ~ 10^(10) cm^(-3).Comment: To appear in the Astrophysical Journal (Letters

Why Does NGC 1068 Have a More Powerful Active Galactic Nucleus than NGC 4258 ?

The nuclear gas kinematics probed by water vapor maser emission has shown that two nearby active galaxies, NGC 1068 and NGC 4258, have a supermassive object in their nuclei and their masses are nearly comparable; a few 10^7 solar masses. Nevertheless, the activity of the central engine of NGC 1068 is more powerful by two orders of magnitude than that of NGC 4258. Since it is generally considered that the huge luminosities of active galactic nuclei are attributed to the mass accretion onto a supermassive black hole, the above observational results suggest that the accretion rate in NGC 1068 is much higher than that in NGC 4258. Comparing the kinematical properties of the accreting molecular clouds between NGC 1068 and NGC 4258, we find possible evidence for dynamical gas accretion in NGC 1068, which may be responsible for the more powerful central engine in this galaxy.Comment: 9 Latex (AASTeX) pages, 2 PostScript figures. To appear in Publications of the Astronomical Society of Japa