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On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras

Takashi Taniguchi — 2007

Annales de l’institut Fourier

In this paper we consider the prehomogeneous vector space for a pair of simple algebras which are inner forms of the D 4 type and the E 6 type. We mainly study the non-split cases. The main purpose of this paper is to determine the principal parts of the global zeta functions associated with these spaces when the simple algebras are non-split. We also give a description of the sets of rational orbits of these spaces, which clarifies the expected density theorems arising from the properties of these...

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

Takashi Taniguchi — 2008

Annales de l’institut Fourier

Let k be a number field. In this paper, we give a formula for the mean value of the square of class number times regulator 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 pairs of quaternion algebras. We also prove an asymptotic formula of the correlation coefficient for class number times regulator of certain families of quadratic extensions.

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