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.

Let $(G,V)$ be a regular prehomogeneous vector space (abbreviated to $PV$), where $G$ is a reductive algebraic group over $ℂ$. If $V={\oplus }_{i=1}^n{V}_i$ is a decomposition of $V$ into irreducible representations, then, in general, the PV’s $(G,V_i)$ are no longer regular. In this paper we introduce the notion of quasi-irreducible $PV$ (abbreviated to $Q$-irreducible), and show first that for completely $Q$-reducible $PV$’s, the $Q$-isotypic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate...

On étudie la fonction zêta d’Igusa ζ(P,s) associée à une hypersurface projective complexe P = 0. On montre qu’elle est une intégrale d’Euler généralisée et on précise le système différentiel A-hypergéométrique qu’elle satisfait. On indique un algorithme pour la détermination explicite d’une équation aux différences satisfaite par ζ(P,s). On calcule explicitement cette fonction pour quelques cas particuliers. On prouve que la fonction zêta associée au résultant $R_{(1,2)}$ n’est pas une somme de produits de...

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...

For a number field $K$ with ring of integers ${𝒪}_K$, we prove an analogue over finite rings of the form ${𝒪}_K/{𝒫}^m$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $𝒫$ is a big enough prime ideal of ${𝒪}_K$ and $m\>1$. In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of $L$-functions...