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

Jouni Parkkonen; Frédéric Paulin

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 5, page 967-1022
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Parkkonen, Jouni, and Paulin, Frédéric. "On the arithmetic of cross-ratios and generalised Mertens’ formulas." Annales de la faculté des sciences de Toulouse Mathématiques 23.5 (2014): 967-1022. <http://eudml.org/doc/275303>.

@article{Parkkonen2014,
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 $\le 5$. We prove generalisations of Mertens’ formula for quadratic imaginary number fields and definite quaternion algebras over $\mathbb\{Q\}$, 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.},
author = {Parkkonen, Jouni, Paulin, Frédéric},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
number = {5},
pages = {967-1022},
publisher = {Université Paul Sabatier, Toulouse},
title = {On the arithmetic of cross-ratios and generalised Mertens’ formulas},
url = {http://eudml.org/doc/275303},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Parkkonen, Jouni
AU - Paulin, Frédéric
TI - On the arithmetic of cross-ratios and generalised Mertens’ formulas
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 5
SP - 967
EP - 1022
AB - 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 $\le 5$. We prove generalisations of Mertens’ formula for quadratic imaginary number fields and definite quaternion algebras over $\mathbb{Q}$, 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.
LA - eng
UR - http://eudml.org/doc/275303
ER -

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