On the arithmetic of cross-ratios and generalised Mertens’ formulas
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 5, pp. 967-1022.

We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension 5. We prove generalisations of Mertens’ formula for quadratic imaginary number fields and definite quaternion algebras over , counting results of quadratic irrationals with respect to two different natural complexities, and counting results of representations of (algebraic) integers by binary quadratic, Hermitian and Hamiltonian forms with error bounds. For each such statement, we prove an equidistribution result of the corresponding arithmetically defined points. Furthermore, we study the asymptotic properties of crossratios of such points, and expand Pollicott’s recent results on the Schottky-Klein prime functions.

Nous développons les liens entre la géométrie hyperbolique et les problèmes arithmétiques d’équidistribution, provenant de l’action de groupes arithmétiques sur des espaces hyperboliques réels, surtout en dimension au plus 5. Nous démontrons des généralisations de la formule de Mertens pour les corps de nombres quadratiques imaginaires et les algèbres de quaternion définies sur , des résultats de comptage d’irrationnels quadratiques en utilisant deux complexités naturelles, et des résultats de comptage avec termes d’erreur de représentations d’entiers (algébriques) par des formes binaires quadratiques, hermitiennes ou hamiltoniennes. Pour tout tel énoncé, nous démontrons un résultat d’équidistribution des points arithmétiquement définis correspondants. De plus, nous étudions les propriétés asymptotiques des birapports de tels points, et nous étendons les résultats récents de Pollicott sur les fonctions premières de Schottky-Klein.

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Parkkonen, Jouni; Paulin, Frédéric. On the arithmetic of cross-ratios and generalised Mertens’ formulas. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 5, pp. 967-1022. doi : 10.5802/afst.1433. http://archive.numdam.org/articles/10.5802/afst.1433/

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