On definition fields of algebraic curves

Abstract:  A Belyi theorem says that a complex compact smooth algebraic curve X can be defined over the field of the algebraic numbers iff there exists a map from X to the projective line, ramified over three points only. This theorem has been very famous since it was used in Grothendieck's "dessin d'enfant" theory. In the talk we are going to present a conjectural generalization of the Belyi theorem, to prove it in certain cases, and to state some open questions.