Exceptional Belyi coverings

Abstract

Exceptional Belyi covering is a connected Belyi covering uniquely determined by its ramification scheme or the respective dessin d’enfant. Well known examples are cyclic, dihedral, and Chebyshev coverings. We add to this list a new infinite series of rational exceptional coverings together with the respective Belyi functions. We shortly discuss the field of definition of a rational exceptional covering and show that it is either Q or its quadratic extension. Existing theories give no upper bound on degree of the field of definition of an exceptional covering of genus 1. It is an open question whether the number of such coverings is finite or infinite. Maple search for an exceptional covering of genus g > 1 found none of degree 18 or less. Absence of exceptional hyperbolic coverings is a mystery we could not explain.