Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 1 (2013), 75-115.
On the arithmetic and geometry of binary Hamiltonian forms
Given an indefinite binary quaternionic Hermitian form 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 by , as tends to . We compute the volumes of hyperbolic -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 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.
Algebra Number Theory, Volume 7, Number 1 (2013), 75-115.
Received: 11 May 2011
Revised: 14 December 2011
Accepted: 30 January 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11E39: Bilinear and Hermitian forms 11R52: Quaternion and other division algebras: arithmetic, zeta functions 20G20: Linear algebraic groups over the reals, the complexes, the quaternions
Secondary: 11N45: Asymptotic results on counting functions for algebraic and topological structures 15A21: Canonical forms, reductions, classification 53A35: Non-Euclidean differential geometry 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
Parkkonen, Jouni; Paulin, Frédéric. On the arithmetic and geometry of binary Hamiltonian forms. Algebra Number Theory 7 (2013), no. 1, 75--115. doi:10.2140/ant.2013.7.75. https://projecteuclid.org/euclid.ant/1513729930