The Modular Approach to Diophantine Equations

Abstract

One of the most powerful tools in the study of Diophantine equations, extensively developed in the past few years, has been the use of special types of elliptic curves associated with possible solutions of the Diophantine equation (but not considered as Diophantine equations in themselves), now called Hellegouarch-Frey curves, or simply Frey curves. The three very deep theorems that are necessary to use these tools are on the one hand the Taniyama- Shimura-Weil conjecture, now proved thanks to Wiles and successors (Theorems 8.1.4 and 8.1.5), Ribet’s level-lowering Theorem 15.2.5, and Mazur’s Theorem 15.2.6; see below. The aim of this chapter is to explain these tools so that they can be used by the reader as a black box, in particular with a minimal knowledge of the underlying (beautiful) mathematics. Since the first great success of this method was the complete proof of Fermat’s last theorem in 1995, it is not surprising that the method is difficult, and requires more prerequisites than assumed in the rest of this book. However, considering its importance, we have decided to include it as a chapter in the last part of this book. We will see for instance that FLT is the easiest case to which the method applies.