2013/08/26 by Jouni Parkkonen, Parkkonen, Jouni, Frédéric Paulin +1
Mathematics · #11D85 #11E39 #11F06 #11N45 #11R52 #20G20 #20H10 #30F40 #53A35 #53C22 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11D85 #msc:11E39 #msc:11F06 #msc:11N45 #msc:11R52 #msc:20G20 #msc:20H10 #msc:30F40 #msc:53A35 #msc:53C22
paper · pdf · doi:10.48550/arxiv.1308.5500
44 pages
arxiv created 2013/08/26 · openalex publication_date 2013/08/26 · arxiv updated 2013/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 up to 5. We prove generalisations of Mertens' formula for quadratic imaginary number fields and definite quaternion algebras over the rational numbers, 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.