turkmath.org

Exceptional covering is a connected Belyi covering uniquely determined by its ramification scheme. 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 minimal 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 g>1 found none of degree 18 or less. Absence of exceptional hyperbolic coverings is a mystery we couldn’t explain. This is a joint project with Cemile Kürkoğlu