# On the arithmetic and geometry of binary Hamiltonian forms

@article{Parkkonen2013OnTA,
title={On the arithmetic and geometry of binary Hamiltonian forms},
author={Jouni Parkkonen and Fr'ed'eric Paulin},
journal={Algebra \& Number Theory},
year={2013},
volume={7},
pages={75-115}
}
• Published 11 May 2011
• Mathematics
• Algebra & Number Theory
Given an indefinite binary quaternionic Hermitian form f with coefficients in a maximal order of a definite quaternion algebra over Q, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f , as s tends to +∞. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using…
9 Citations

### On the arithmetic of crossratios and generalised Mertens' formulas

• Mathematics
• 2013
We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension up to

### Counting and equidistribution in quaternionic Heisenberg groups

• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2021
Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on

### Integral binary Hamiltonian forms and their waterworlds

• Mathematics
• 2018
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.

### Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation

• Mathematics
• 2012
Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong

### Counting arcs in negative curvature

• Mathematics
• 2012
Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed locally convex subsets of M. We survey the asymptotic behaviour of the number of common

### Quaternionic hyperbolic lattices of minimal covolume

• Mathematics
Forum of Mathematics, Sigma
• 2022
Abstract For any $n>1$ we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group $\operatorname {\mathrm {Sp}}(n,1)$ . We explicitly describe them in terms of the

### Sur la classification des hexagones hyperboliques à angles droits en dimension 5

• Mathematics
• 2014
— The aim of this paper is to give a classification of the rightangled hyperbolic hexagons in the real hyperbolic space HR, by using a quaternionic distance between geodesics in HR. RÉSUMÉ. — Le but

### Sur les minima des formes hamiltoniennes binaires définies positives

• Publications Mathématiques de Besançon
• 2021
Etant donne un ordre maximal $O$ d'une algebre de quaternions rationnelle definie $A$ de discriminant $D_A$, nous montrons que le minimum des formes hamiltoniennes binaires sur $O$, definies

### On the minima of positive definite binary hamiltonian forms

• Mathematics
• 2019
Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$

## References

SHOWING 1-10 OF 65 REFERENCES

### On the representation of integers by indefinite binary Hermitian forms

• Mathematics
• 2010
Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some

### The covolume of quaternion groups on the four-dimensional hyperbolic space

• Mathematics
• 1996
where D denotes the discriminant of the corresponding rational quaternion algebra, is based on the Maas–Selberg relations using Stokes’ theorem as described in [EGM] and the Fourier expansion of the

### Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory

• Mathematics
• 1997
1. Three-Dimensional Hyperbolic Space.- 2. Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space.- 3. Automorphic Functions.- 4. Spectral Theory of the Laplace Operator.- 5. Spectral

### Quaternionic determinants

The classical matrix groups are of fundamental importance in many parts of geometry and algebra. Some of them, like Sp.n/, are most conceptually defined as groups of quaternionic matrices. But, the

### The Geometry of Discrete Groups

Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean

### Prescribing the behaviour of geodesics in negative curvature

• Mathematics
• 2010
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds,

### \'Equidistribution, comptage et approximation par irrationnels quadratiques

• Mathematics
• 2010
Let $M$ be a finite volume hyperbolic manifold, we show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic