## On the arithmetic and geometry of binary Hamiltonian forms. With an appendix by Vincent Emery.(English)Zbl 1273.11065

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.

### 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)
Full Text: