## ABSTRACT

We continue sketching Wiles’s proof of Fermat’s Last Theorem. The previous chapter discusses the classical theory of modular functions. This theory provides the context required to state the Taniyama-Shimura-Weil Conjecture, which forms the centrepiece of Wiles’s approach to-and proof of-Fermat’s Last Theorem. The present chapter is about more recent discoveries, and develops the circle of new ideas that leads to the proof of a special case of the Taniyama-Shimura-Weil Conjecture. Wiles, building on the work of Frey and others, realised that this special case, the semistable Taniyama-Shimura-Weil Conjecture, immediately implies the truth of Fermat’s Last Theorem. A vital first step is the definition of the Frey elliptic curve, which links Fermat’s Last Theorem to elliptic curves.