Number Theory Seminar: Nuno Freitas

Fermat's Last Theorem states that the equation x^n + y^n = z^n for n > 2 has no integer solutions such that xyz \neq 0. It's proof was completed in 1995 by the groundbreaking work of Andrew Wiles on the modularity of semistable elliptic curves over Q. From its proof a new revolutionary method to attack Diophantine equations was born. This method, now known as the modular method, builds on the work of Frey, Serre, Ribet, Mazur and makes use of the Galois representations attached to modular forms and elliptic curves. In the first part of this talk, guided by the proof of FLT, we will introduce the tools and sketch the basic strategy behind the modular method. In the second part, we will discuss the main obstacles that arise when we try to apply the method to other type of equations or over number fields.