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}
}
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… 

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References

SHOWING 1-10 OF 65 REFERENCES

On the representation of integers by indefinite binary Hermitian forms

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

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

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

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

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

Rational Quadratic Forms

...