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

2

+ y = x

3

− 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

mind are:

•

f is determined by its q-expansion;

5

•

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

v

) whose

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

f

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.

5

In fact, it is enough to know the integers a

(f ) for all primes to determine the entire

q-expansion, and hence determine f .

248 CHAPTER 22

DEFINITION: A Galois representation R : G → GL(2, k)is

irreducible if there is no line L in the vector space k

2

with the

property that R(σ)L = L for all σ in G.

Now let k

be a ﬁeld containing k with the property that any

polynomial with coefﬁcients in k

has a root in k

. (For any ﬁeld k,

there always exists such a ﬁeld k

—in fact, there are many of them.)

Note that because every element of k is also in k

, we can consider a

Galois representation R : G → GL(2, k) with values in k-matrices as

also deﬁning a Galois representation R

: G → GL(2, k

) with values

in k

-matrices.

DEFINITION: A Galois representation R : G → GL(2, k)is

absolutely irreducible if the corresponding R

is irreducible.

6

For example, if R : G → GL(2, k), and if R(σ ) takes on all possible

values in GL(2, k)asσ runs through all the elements of the absolute

Galois group G, then R is absolutely irreducible. (This is not

obvious.)

By the way, for all the elliptic curves E that are used in this

chapter, it turns out that if v ≥ 3, then the representation on the

v-torsion points of E(C) is absolutely irreducible if and only if it

is irreducible. (So why did we introduce both concepts? For truth

in advertising. The “if and only if” we just stated is a difﬁcult

theorem.)

Ken Ribet (Ribet, 1990) proved the following:

THEOREM 22.3 (Level Lowering Theorem): Let f be a

cuspidal normalized newform of level N and weight 2.

Suppose that is a prime dividing N but that

2

does not

divide N, and suppose that ψ

f

: G → GL(2, F

v

) is absolutely

irreducible. Assume further that either

•

= v and ψ

f

is unramiﬁed at (i.e., is a good prime for

ψ

f

), or

6

It can be proven that this concept does not depend on the choice of the ﬁeld k

.

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