107 research outputs found
On the arithmetic and geometry of binary Hamiltonian forms
Given an indefinite binary quaternionic Hermitian form with coefficients
in a maximal order of a definite quaternion algebra over , 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 by , as 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 Prasad's general formula.
We use hyperbolic geometry in dimension 5 to describe the reduction theory of
both definite and indefinite binary quaternionic Hermitian forms.Comment: With an appendix by Vincent Emery. Revised versio
Counting arcs in negative curvature
Let M be a complete Riemannian manifold with negative curvature, and let C_-,
C_+ be two properly immersed closed convex subsets of M. We survey the
asymptotic behaviour of the number of common perpendiculars of length at most s
from C_- to C_+, giving error terms and counting with weights, starting from
the work of Huber, Herrmann, Margulis and ending with the works of the authors.
We describe the relationship with counting problems in circle packings of
Kontorovich, Oh, Shah. We survey the tools used to obtain the precise
asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe
several arithmetic applications, in particular the ones by the authors on the
asymptotics of the number of representations of integers by binary quadratic,
Hermitian or Hamiltonian forms.Comment: Revised version, 44 page
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
- …