244 CHAPTER 22
The equation a
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
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
). It is essential to use Galois representa-
tions that take values in matrices with entries from the big ﬁeld Q
(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
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