Mehrdad M. Shahshahani : On the geometrization of the absolute Galois group

Posted by Ramin Takloo-Bighash , part of the Number Theory Seminar.

At

Sept. 22, 2010, 2 p.m.

In

SEO 427

Abstract

Let $M$ be a compact orientable topological surface and $M^a$ a Riemann surface whose underlying surface is $M$. Then as an algebraic curve $M^a$ can be defined over a number field if and only if it admits of a non-constant meromorphic function $f : M^a \to {\mathbb C}{\mathbb P}(1)$ with at most three critical values according to theorems of Belyi and Weil. The critical values may be normalized to be $0$, $1$ and $\infty$ and such a normalized meromorphic function is referred to as a Belyi function and the set $f^{-1}[0; 1]$ is a graph (called \emph{dessin}) on $M$. Fixing the the equations defining the algebraic curve $M^a$ and the Belyi function $f$, the absolute Galois group $G = Gal(\bar{{\mathbb Q}}/{\mathbb Q})$ acts on the coefficients of the equations defining $M^a$ and the Belyi function $f$. Inspired by this remarkable theorem, Grothendieck suggested that studying graphs, satisfying certain conditions, on orientable topological surfaces and the action of the absolute Galois group accordingly provides a geometric and combinatorial framework for gaining a deeper understanding of $G$ which is known essentially only through its action on algebraic numbers. The seminal works of Drinfeld and Ihara on this subject embeds the absolute Galois group into the Grothendieck-Teichmuller group and was the starting point for research by a number of other mathematicians. After a general introduction, in this lecture I will discuss the work of my student, Ali Kamalinejad, on this program and specifically (1) The construction of dessins on curves of arbitrary genus. (2) The construction of families of dessins, the associated cartographic groups, and the corresponding Galois groups. (3) Relation with Jenkins-Strebel differentials.