Integral binary Hamiltonian forms and their waterworlds

TL;DR

This paper develops a graphical and combinatorial theory for integral indefinite binary Hamiltonian forms over quaternion algebras, introducing the concept of waterworlds to analyze their values using hyperbolic geometry and group actions.

Contribution

It introduces the waterworld concept for binary Hamiltonian forms, extending Conway and Bestvina-Savin's theories to quaternionic settings with hyperbolic geometry tools.

Findings

Defined the waterworld of a binary Hamiltonian form.

Provided a combinatorial description of form values.

Connected the theory to hyperbolic geometry and group actions.

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 $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, 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 $\mathcal O\times\mathcal O$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the $\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the real hyperbolic $5$-space.