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RESEARCH PRODUCT
Integral binary Hamiltonian forms and their waterworlds
Jouni ParkkonenFrédéric Paulinsubject
Mathematics - Differential GeometryPure mathematicsBinary number01 natural sciences[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]waterworlddifferentiaaligeometriamaximal orderhyperbolic 5-space0103 physical sciences0101 mathematicsAlgebraic numberreduction theoryMathematicslukuteoriaMathematics - Number TheoryQuaternion algebra010102 general mathematicsHamilton-Bianchi groupryhmäteoriaOrder (ring theory)Mathematics::Geometric TopologyHermitian matrix[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT][MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]Binary quadratic form010307 mathematical physicsGeometry and Topologyrational quaternion algebraMathematics - Group Theorybinary Hamiltonian formHamiltonian (control theory)description
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-10-15 | Conformal Geometry and Dynamics of the American Mathematical Society |