On the arithmetic of crossratios and generalised Mertens' formulas

TL;DR

This paper explores the connection between hyperbolic geometry and arithmetic distribution problems, generalizing Mertens' formulas and analyzing crossratios of algebraic points in low-dimensional hyperbolic spaces.

Contribution

It introduces new generalizations of Mertens' formulas for number fields and quaternion algebras, with counting and equidistribution results for algebraic points and forms.

Findings

Generalized Mertens' formulas for quadratic imaginary fields and quaternion algebras.

Proved equidistribution of arithmetically defined points in hyperbolic spaces.

Analyzed asymptotic behavior of crossratios and expanded on Schottky-Klein prime functions.

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 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.