Parkkonen, Jouni; Paulin, Frédéric [Emery, Vincent] On the arithmetic and geometry of binary Hamiltonian forms. With an appendix by Vincent Emery. (English) Zbl 1273.11065 Algebra Number Theory 7, No. 1, 75-115 (2013). Authors’ abstract: Given an indefinite binary quaternionic Hermitian form \(f\) with coefficients in a maximal order of a definite quaternion algebra over \(\mathbb Q\), 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 \( + \infty \). 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. Reviewer: Ranjeet Sehmi (Chandigarh) Cited in 4 Documents MSC: 11E39 Bilinear and Hermitian forms 11R52 Quaternion and other division algebras: arithmetic, zeta functions 53A35 Non-Euclidean differential geometry 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) Keywords:Binary Hamiltonian form; Hamiltonian Bianchi group; hyperbolic volume PDFBibTeX XMLCite \textit{J. Parkkonen} and \textit{F. Paulin}, Algebra Number Theory 7, No. 1, 75--115 (2013; Zbl 1273.11065) Full Text: DOI arXiv Link