Integral binary Hamiltonian forms and their waterworlds

Parkkonen, Jouni; Paulin, Frédéric

doi:10.1090/ecgd/362

Integral binary Hamiltonian forms and their waterworlds
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by Jouni Parkkonen and Frédéric Paulin

Conform. Geom. Dyn. 25 (2021), 126-169

DOI: https://doi.org/10.1090/ecgd/362

Published electronically: October 20, 2021

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Abstract:

We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathscr {O}$ 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 $\mathscr {O}\times \mathscr {O}$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the $\operatorname {SL}_{2}(\mathscr {O})$-equivariant Ford-Voronoi cellulation of the real hyperbolic $5$-space, and the conformal action of $\operatorname {SL}_{2}(\mathscr {O})$ on the Hamilton quaternions.

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