Algebra & Number Theory

On the arithmetic and geometry of binary Hamiltonian forms

Jouni Parkkonen and Frédéric Paulin

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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.

Article information

Source
Algebra Number Theory, Volume 7, Number 1 (2013), 75-115.

Dates
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
https://projecteuclid.org/euclid.ant/1513729930

Digital Object Identifier
doi:10.2140/ant.2013.7.75

Mathematical Reviews number (MathSciNet)
MR3037891

Zentralblatt MATH identifier
1273.11065

Subjects
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]

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

Citation

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


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