Open Access
2013 On the arithmetic and geometry of binary Hamiltonian forms
Jouni Parkkonen, Frédéric Paulin
Algebra Number Theory 7(1): 75-115 (2013). DOI: 10.2140/ant.2013.7.75


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.


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Jouni Parkkonen. Frédéric Paulin. "On the arithmetic and geometry of binary Hamiltonian forms." Algebra Number Theory 7 (1) 75 - 115, 2013.


Received: 11 May 2011; Revised: 14 December 2011; Accepted: 30 January 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1273.11065
MathSciNet: MR3037891
Digital Object Identifier: 10.2140/ant.2013.7.75

Primary: 11E39 , 11R52 , 20G20
Secondary: 11F06 , 11N45 , 15A21 , 20H10 , 53A35

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

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 1 • 2013
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