Integral binary Hamiltonian forms and their waterworlds

Abstract

We give a graphical theory of integral indefinite binary Hamiltonian forms f analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order O in a definite quaternion algebra over Q, we define the waterworld of f, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of f on O×O. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the SL2(O)-equivariant Ford-Voronoi cellulation of the real hyperbolic 5-space.