STNB2020(34th edition)

Introduction to the modular method


Nicolas Billerey


The modular method is a powerful approach to Diophantine equations which was first introduced by Frey and Serre for the case of Fermat's Last Theorem. In its original version, it is based on the modularity of semistable elliptic curves over Q proved by Wiles and relies on deep results on Galois representations due to Mazur and Ribet. In this first lecture we will explain the main steps of the modular method following first the classical FLT proof. Then we will discuss some of the issues that appear when applying extensions of this method to other Fermat type equations.


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