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

Taniguchi, Takashi

Annales de l'Institut Fourier, Tome 57 (2007) no. 4, pp. 1331-1358.

Dans cet article, nous considérons l’espace vectoriel préhomogène associé à une paire d’algèbres simples qui sont des formes intérieures de types $D_4$ et $E_6$. Nous traitons principalement les cas non-déployées. Le but principal de cet article est de déterminer les parties principales de la fonction zêta globale de ces espaces quand les algèbres simples sont non-déployés. Nous donnons aussi une description des ensembles des orbites rationnelles de ces espaces, qui clarifie les théorèmes de densité provenant des propriétés de ces fonctions zêta.

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 zeta functions.

Classification : 11M41
Keywords: prehomogeneous vector space, zeta function, simple algebra
Mot clés : espace vectoriel préhomogène, fonction zêta, algèbre simple.

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