Introduction to Topics on Modular Curves
David Kohel

In the first talk I introduce the modular curves X_0(N) 
and their connections to the theory of elliptic curves.
For this purpose I will present the classical description 
of a modular curve as the Riemann surface covered by the 
upper half complex plane, and the correspondence between 
points on modular curves and "enhanced" elliptic curves.  
This perspective lends itself to the description of 
specialized points CM and supersingular points on X_0(N) 
in terms of the endomorphism rings of elliptic curves, 
and an explicit description of Hecke operators in terms 
of the families of coverings of X_0(N).

Topics on Modular Curves II

In the second talk I describe the geometry of special CM 
and supersingular points on X_0(N), in terms of the objects 
which they classify.  This allows one to understand the 
arithmetic of these points in terms of families of maps 
X_0(Np) -> X_0(N) and the Hecke operators which they induce.