On the arithmetic and geometry of binary Hamiltonian forms

Parkkonen, Jouni; Paulin, Frédéric

doi:10.2140/ant.2013.7.75

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On the arithmetic and geometry of binary Hamiltonian forms

Jouni Parkkonen and Frédéric Paulin

Appendix: Vincent Emery

Vol. 7 (2013), No. 1, 75–115

DOI: 10.2140/ant.2013.7.75

Abstract

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $ℚ$ , we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$ , as $s$ tends to $+ \infty$ . We compute the volumes of hyperbolic $5$ -manifolds constructed by quaternions using Eisenstein series. In the appendix, V. Emery computes these volumes using Prasad’s general formula. We use hyperbolic geometry in dimension $5$ to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Keywords

binary Hamiltonian form, representation of integers, group of automorphs, Hamilton–Bianchi group, hyperbolic volume, reduction theory

Mathematical Subject Classification 2010

Primary: 11E39, 11R52, 20G20

Secondary: 11N45, 15A21, 53A35, 11F06, 20H10

Milestones

Received: 11 May 2011

Revised: 14 December 2011

Accepted: 30 January 2012

Published: 28 March 2013

Authors

Jouni Parkkonen
Department of Mathematics and Statistics University of Jyväskylä P.O. Box 35 FI-40014 University of Jyväskylä Finland

Frédéric Paulin
Département de mathématique, UMR 8628 CNRS Université Paris-Sud Bât. 425 91405 ORSAY Cedex France

Vincent Emery
Section de mathématiques 2-4 rue du Lièvre Case postale 64 1211 Genève 4 Switzerland

Vol. 7, No. 1, 2013