Mathematics Colloquium

According to a theorem of Belyi (and Weil) a complex curve is defined over a number field if and only if it admits of a nonconstant meromorphic function with at most three critical values (a Belyi function). Fascinated by this relation, Grothendieck suggested the development of a geometric and combinatorial structure on which the absolute Galois group G = Gal(ℚbar/ℚ) acts with the expectation that this new framework will lead to major new insights.

For a Belyi function Β on a smooth curve X with critical values {0,1, ∞}, Β^-1([0,1]) is a two colorable graph Γ = Γ_Β on the underlying topological surface X_μ whose complement consists of discs. Conversely, such a graph on an closed orientable topological surface endows it with a unique complex structure defined over a number field. Thus the set of such graphs on X_μ is a candidate for a geometric or combinatorial object on which G acts. The term dessin d'enfants (children's drawings) was coined by Grothendieck to refer to graphs Γ_Β. In spite of its simplicity, this idea has resisted development.

In this lecture I will explain two applications. The first application demonstrates how dessins (or children's drawings) are an efiective tool for making explicit calculations that may have been dificult or impossible otherwise. A second application is more theoretical in nature. The notion of n-flat refinement of a dessin is introduced. This functorial notion enables one to assign to a given dessin infinite families of canonically defined dessins. These towers are the geometric analogues of infinite towers of number fields and are expected to reect the inverse system structure of G. The remarkable property of these infinite towers of dessins is that they define infinite towers of number fields with controlled ramification in the sense that the set of ramified primes is stable.

There are a number of open basic questions about these towers of number fields that are currently under investigation and could have interesting number theoretic implications.

(This lecture represents joint work with Majid Hadian-Jazi and Ali Kamalinejad.)