Abstract:

Modualar Curves are algebraic curves whose points parametrize families of 
elliptic curves. For this reason they are useful for studying properties of 
elliptic curves defined over a fixed number field $K$.
For example, the non existence of "many" automorphisms, provides evidence for Serre's uniformity conjecture, an important statement about the action of the absolute Galois group of $K$ on torsion points of elliptic curves defined over $K$.
In this talk, we will give a brief introduction to the theory of Modular Curves, as well as a summary of the results and techniques used to determine their automorphism groups.