#### Vol. 7, No. 1, 2013

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On the arithmetic and geometry of binary Hamiltonian forms

### Appendix: Vincent Emery

Vol. 7 (2013), No. 1, 75–115
##### 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