For the past couple of years, I’ve been thinking about properties of Dessins d’Enfants. As my very first post, I’d like to give a rigorous definition. This is meant to be formal for the math folks out there who are comfortable with algebraic geometry.
Let be a compact, connected Riemann surface. It is well-known that is an algebraic variety, that is, in terms of a collection of homogeneous polynomials over in variables . Denote as the ring of regular functions on , that is, “polynomials” ; and denote as its quotient field, that is, rational functions . For example, if , then consists of polynomials in one variable, while consists of rational functions in one variable. In particular, any rational map induces a map which sends . The degree of such a map is the size of the group .
For each , let be the localization of at the kernel of the evaluation map defined by . Let denote the maximal ideal of ; we view this as the collection of rational maps which vanish at . As shown by Weil and Belyĭ, the Riemann surface can be defined in terms of homogeneous polynomials over an algebraic closure if and only if there exists a rational function such that . The difference for any because the function vanishes at ; the condition means the derivative of the function vanishes as well. A rational function as above where these critical values are at most , , and is called a Belyĭ map.
Following Grothendieck, we associate a bipartite graph to a Belyĭ map by denoting the “black” vertices as , “white” vertices as , midpoints of the faces as , and edges as . This is a loopless, connected, bipartite graph, called a Dessin d’Enfant, which can be embedded on without crossings. The group permutes the solutions to , and hence acts on the dessin . The hope is that in studying graphs one can better understand quotients of the absolute Galois group .
We are motivated by the following question:
Given a loopless, connected, bipartite graph on a compact, connected Riemann surface , when is the Dessin d’Enfant of a Belyĭ map ?
Given such a loopless, connected, planar, bipartite graph , we wish to use properties of the symmetry group to construct a Belyĭ map such that arises as its Dessin d’Enfant.