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.

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## About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.