Integral binary Hamiltonian forms and their waterworlds

2018/10/15 by Parkkonen, Jouni, Paulin, Frédéric
#Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1810.06222

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×\mathcal O. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the SL₂(\mathcal O)-equivariant Ford-Voronoi cellulation of the real hyperbolic 5-space.

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