On arithmetic properties of Fuchsian groups and Riemann surfaces

Abstract

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.<br /> Afterwards a rigidity theorem for semi-arithmetic Fuchsian groups with respect to the congruence topology is shown.<br /> 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.