Filippo Nuccio (IUT and Institut Camille Jordan, Université de Saint-Étienne)
This will be a colloquium-style talk aimed at presenting to a non-specialist audience the main ideas beneath Wiles' proof of Fermat's Last Theorem (Ann. of Math, 1995). I will first introduce Frey's idea of attaching to a non-trivial solution to Fermat's equation an "elliptic curve over Q", and discuss what kind of objects, from an arithmetic point of view, such an elliptic curve is. I will then explain how an elliptic curve as above gives rise to 2-dimensional representations of a certain group and how these representations are classified (this classification is the core of Wiles' contribution). The proof is based upon the idea that the representation that would arise from Frey's curve would not fit into the classification, and thus cannot exist. As a result, there is no non-trivial solution to Fermat's equation.