Families of Riemann Surfaces, Uniformization and Arithmeticity

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2-dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2-dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read off from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk

On the number of coincidences of morphisms between closed Riemann surfaces

We give a bound for the number of coincidentes of two morphisms between given compact Riemann surfaces (complete complex algebraic curves). Our results generalize well known facts about the number of fixed points of an automorphism

Theme 2: Belyi theory & the absolute Galois group

Introduction (by Ernesto Girondo and Gabino González-Diez) to Belyi theory and the absolute Galois group, including some important open questions in this theme areaNon UBCUnreviewedAuthor affiliation: Universidad Autónoma de MadridFacult

BELYI’S THEOREM FOR COMPLEX SURFACES

Belyi’s theorem states that a compact Riemann surface C can be defined over a number field if and only if there is on it a meromorphic function f with three critical values. Such functions (resp. Riemann surfaces) are called Belyi functions (resp. Belyi surfaces). Alternatively Belyi surfaces can be characterized as those which contain a proper Zariski open subset uniformised by a torsion free subgroup of the classical modular group PSL2(Z). In this article we establish a result analogous to Belyi’s theorem in complex dimension two. It turns out that the role of Belyi functions is now played by (composed) Lefschetz pencils with three critical values while the analogous to torsion free subgroups of the modular group will be certain extensions of them acting on a Bergmann domain of C². These groups were first introduced by Bers and Griffiths

Introduction to Compact Riemann Surfaces and Dessins d'Enfants

An elementary account of the theory of compact Riemann surfaces and an introduction to the Belyi-Grothendieck theory of dessins d'enfants