Vol. 7, No. 1, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 8, 1765–1981
Issue 7, 1509–1763
Issue 6, 1243–1507
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
On the arithmetic and geometry of binary Hamiltonian forms

Jouni Parkkonen and Frédéric Paulin

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 , 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.

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