Vol. 7, No. 1, 2013

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On the arithmetic and geometry of binary Hamiltonian forms

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 $ℚ$, 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.

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