244 CHAPTER 22

The equation a

4

+ b

4

= c

4

was shown to have no solutions by

Fermat, using a method called “inﬁnite descent.” So, throughout

this chapter we will ﬁx an odd prime p. In fact, for technical reasons,

we will assume that p ≥ 5. This is not a problem because FLT had

been proven for exponents n = 3 as well as n = 4 since at least 1770.

Because we have ﬁxed the prime p, we have to use other letters

for other prime numbers that we will need to use. We will use the

letters v, w, and for these.

There are three basic components to the winning strategy:

1. Frey curves;

2. The Modularity Conjecture;

3. The Level Lowering Theorem.

The ﬁrst two of these components can be explained without men-

tioning Galois representations. But explaining the third component

requires a reciprocity law (Theorem 21.1). This reciprocity law will

then continue to be essential to Wiles’s successful execution of the

strategy.

Another feature of Wiles’s work is that it requires us to enlarge

our menagerie of linear Galois representations beyond those that

take values in GL(2, F

p

). It is essential to use Galois representa-

tions that take values in matrices with entries from the big ﬁeld Q

p

(and even more complicated algebraic systems that we will not be

able to discuss in any detail). We have already introduced examples

of these Galois representations in Theorem 21.1.

From now on, new ideas and concepts are going to come thick

and fast. To explain them in detail would take a whole additional

book.

Frey Curves

Start with the innocuous looking equation

A + B = C

where A, B, and C are integers. It is not difﬁcult to solve this

equation! But if you put some additional requirements on A, B,and

C, it can become difﬁcult. For example, if we require A, B, and C

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