Numerical calculation of three-point branched covers of the projective line

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate

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

Arithmetic Aspects of Bianchi Groups

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data.Comment: 35 pages, 171 references, 3 tables, 2 figure

On arithmetic properties of Fuchsian groups and Riemann surfaces

In this thesis, Riemann surfaces and their fundamental groups are studied from an arithmetic point of view. First the image of the absolute Galois group of a number field under Grothendieck's representation with values in the étale fundamental group of a curve is considered. Both for a projective line with three punctures and for an elliptic curve with one puncture this fundamental group is a free profinite group on two generators, and it is shown to which extent the image of the Galois group in it determines the original curve. With similar methods then Galois representations on étale fundamental group of hyperbolic Deligne-Mumford curves over number fields are approached. It is proved that the absolute Galois group acts faithfully on the finite Galois covers of a given such stack. As a corollary we obtain the faithfulness of the absolute Galois action on Hurwitz curves and origamis. Afterwards a rigidity theorem for semi-arithmetic Fuchsian groups with respect to the congruence topology is shown. Finally, a moduli interpretation for principal congruence subgroups of certain Fuchsian triangle groups in terms of hypergeometric curves is presented, and consequences for the Galois actions on the associated curves and dessins d'enfants are deduced

Schwarzian differential equations and Hecke eigenforms on Shimura curves

Let $X$ be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on $X$ in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}}$$ and other similar identities.Comment: 31 pages, 2 figure