On the arithmetic of cross-ratios and generalised Mertens’ formulas

Abstract

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 ℚ, 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.