FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 247
and complex numbers. This approach is discussed in most of the
popular books about Wiles’s proof of FLT.
For example, the elliptic curve deﬁned by y
+ y = x
− x satisﬁes
the Modularity Conjecture where the corresponding modular form
is given by the power series in equation (21.2).
By the way, the Modularity Conjecture is not a conjecture any
more. Following the ideas of Wiles and Taylor–Wiles, and with a lot
of extra hard work, the proof was completed by Christophe Breuil,
Brian Conrad, Fred Diamond, and Richard Taylor (Breuil et al.,
2001). However, at the time of Wiles’s work on FLT, it was still a
conjecture, so we will continue to refer to it as such in this chapter.
Because we cannot go into the details of what a cuspidal normal-
ized newform f really is, the main things that you should keep in
f is determined by its q-expansion;
compared with Galois representations, newforms are easy
to compute and work with.
Lowering the Level
Suppose we have a cuspidal normalized newform f of level N and
weight 2. By the corollary to Theorem 21.1, for any choice of a
prime v, there is a Galois representation r : G → GL(2, F
traces at Frobenius elements give you the values of coefﬁcients in
the q-expansion of f modulo v. We will change the name of the
representation from r to ψ
to emphasize its dependence on f .
To state the Level Lowering Theorem and also to explain Wiles’s
main contribution, we have to introduce the concept of “irreducibil-
ity” of a representation. This is a technical concept and you can skip
it on a ﬁrst reading. But we want to include it so that we can state
the theorems in the chapter accurately.
You will need to review the deﬁnitions in the ﬁrst section of
chapter 20. Suppose k is a ﬁeld.
In fact, it is enough to know the integers a
(f ) for all primes to determine the entire
q-expansion, and hence determine f .