On Fields of Definition of an Algebraic Curve

Abstract

The paper deals with geometric invariants of an algebraic curve such as the minimal number of crucial values of rational functions and the minimal transcendence degree of definition fields. The main question is if the difference of these two invariants is always equal to 3 for any curve with genus g > 0. For curves defined over an algebraic number field, a positive answer is given by Belyi’s theorem. In the paper, the main question is answered in the affirmative for some other cases.