Number Theory Seminar
Mehrdad M. Shahshahani
IPM , Tehran
On the geometrization of the absolute Galois group
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.
Wednesday September 22, 2010 at 2:00 PM in SEO 427