Vol. 7, No. 1, 2013

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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

Given an indefinite binary quaternionic Hermitian form f with coefficients in a maximal order of a definite quaternion algebra over , 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 + . 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.

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
Received: 11 May 2011
Revised: 14 December 2011
Accepted: 30 January 2012
Published: 28 March 2013
Jouni Parkkonen
Department of Mathematics and Statistics
University of Jyväskylä
P.O. Box 35
FI-40014 University of Jyväskylä
Frédéric Paulin
Département de mathématique, UMR 8628 CNRS
Université Paris-Sud
Bât. 425
91405 ORSAY Cedex
Vincent Emery
Section de mathématiques
2-4 rue du Lièvre
Case postale 64
1211 Genève 4