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Integral binary Hamiltonian forms and their waterworlds

Jouni Parkkonen, Frédéric Paulin

TL;DR

This work extends Conway's graphical theory of binary quadratic forms to integral indefinite binary Hamiltonian forms over maximal orders in definite quaternion algebras by using the Ford-Voronoi spine in real hyperbolic 5-space and a normalized Busemann framework. It defines waterworlds as the combinatorial separator between positive and negative values of the associated cusp-normalized function F, and proves that the coned-off waterworld is a 4-dimensional piecewise hyperbolic plane whose closest-point projection to the zero-set hyperplane is equivariant under the automorphism group. The paper develops a Hermite-type reduction theory for Hamilton-Bianchi lattices SL_2(O), describes the spine X_O in detail, and analyzes how flooded and non-flooded FV-cells interact within this spine. Concrete instances, notably when the quaternion algebra has small discriminant (e.g., D_A = 2,3,5) and for the Hurwitz order, illustrate the resulting polyhedral decompositions and symmetry structures. An algebraic appendix provides a cusp-distance formula in terms of O-flags, tying the geometric picture to arithmetic invariants.

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.

Integral binary Hamiltonian forms and their waterworlds

TL;DR

This work extends Conway's graphical theory of binary quadratic forms to integral indefinite binary Hamiltonian forms over maximal orders in definite quaternion algebras by using the Ford-Voronoi spine in real hyperbolic 5-space and a normalized Busemann framework. It defines waterworlds as the combinatorial separator between positive and negative values of the associated cusp-normalized function F, and proves that the coned-off waterworld is a 4-dimensional piecewise hyperbolic plane whose closest-point projection to the zero-set hyperplane is equivariant under the automorphism group. The paper develops a Hermite-type reduction theory for Hamilton-Bianchi lattices SL_2(O), describes the spine X_O in detail, and analyzes how flooded and non-flooded FV-cells interact within this spine. Concrete instances, notably when the quaternion algebra has small discriminant (e.g., D_A = 2,3,5) and for the Hurwitz order, illustrate the resulting polyhedral decompositions and symmetry structures. An algebraic appendix provides a cusp-distance formula in terms of O-flags, tying the geometric picture to arithmetic invariants.

Abstract

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

Paper Structure

This paper contains 9 sections, 26 theorems, 112 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Background on definite quaternion algebras over ${\mathbb Q}$
  4. Background on Hamilton-Bianchi groups
  5. Background on binary Hamiltonian forms
  6. On the reduction theory of binary Hamiltonian forms and Hamilton-Bianchi lattices
  7. The spine of $\operatorname{SL}_2({\cal O})$
  8. Waterworlds
  9. An algebraic description of the distance to the cusps

Key Result

Proposition 1.1

If $h_A=1$, given a flooded Ford-Voronoi cell $C$, there exists a finite set of nonconstant affine maps $\{\varphi_i:{\mathbb H}\rightarrow {\mathbb R}\;:\;i\in F\}$ defined over ${\mathbb Q}$ such that the set of values of $F$ on the Ford-Voronoi cells meeting $C$ is $\bigcup_{i\in F}\varphi_i({\ca

Figures (13)

  • Figure 1: Disjointness of normalized horoballs $B_{\alpha'}(1)$ for $\alpha'\in{\mathbb P}^1_r(A)$.
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5: Boundary of equidistant hemispheres and halfplanes in ${\mathbb C}\subset {\mathbb H}$.
  • ...and 8 more figures

Theorems & Definitions (31)

  • Proposition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Lemma 3.6
  • ...and 21 more