Summary. A partial result on a problem of Siegel is given: Hypergeometric functions with rational parameters have transcendental values in almost all algebraic points -up to some natural exceptions; these exceptions are the well-known algebraic functions and an (unexpected) second class of examples related to certain Shimura-curves.
According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and oo. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincare upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyi's theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wol], the conference proceedings article [Baultz] and the "Comptes Rendus" note [CoWo2].
Dessins d'enfants can be seen as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, even an algebraic structure as a projective algebraic curve defined over a number field. Combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve, such as the field of moduli or the field of definition. However, apart from a few series of examples, very few general results concerning such correspondences are known. As a step in this direction, we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields, based on Wilson's operations on maps: these leave invariant the graph but change the cyclic ordering of edges around the vertices, therefore they change the embeddings, and hence the dessins, and hence the conformal and algebraic structure of the underlying curves. Under suitable assumptions, satisfied by many series of regular dessins, these changes coincide with the effect of Galois conjugation. This coincidence allows one to draw conclusions about Galois orbits and fields of definition of dessins. The possibilities afforded by these techniques, and their limitations, are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs.
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