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

Taniguchi, Takashi

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 625-670.

Soit $k$ un corps de nombres. Dans cet article, nous donnons une formule pour la valeur moyenne du carré du nombre de classe multiplié par le régulateur pour certaines familles d’extensions quadratiques de $k$ caractérisées par un nombre fini de conditions locales. Notre approche utilise la théorie de la fonction zêta associée à l’espace de paires d’algèbres de quaternions. Nous prouvons aussi une formule asymptotique pour le coefficient de corrélation du nombre de classe multiplié par le régulateur de certaines familles d’extensions quadratiques.

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.

Classification : 11M41
Keywords: Density theorem, prehomogeneous vector space, quaternion algebra, local zeta function
Mot clés : théorème densité, espace vectoriel préhomogène, fonction zêta local, algèbres de quaternions

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