252 CHAPTER 22

modular. And if a Galois representation φ : G → GL(2, F

v

) is equal

to

f

for some cuspidal normalized newform f ,wesayφ is modular.

We will have to consider the possibility of using different primes

v, so if we want to emphasize which v we are using to deﬁne the

various Galois representations, we will put them into a subscript

on the Greek letter. Then we have the following:

THEOREM 22.5: If

E,v

is modular for some prime v, then

E,w

is modular for any prime w.

PROOF: If

E,v

is modular, then there exists a modular form f

such that

E,v

=

f ,v

. But it is a general fact (proven in

algebraic number theory) that

E,v

=

f ,v

if and only if

a

(f ) = a

(E)inQ

v

for all but a ﬁnite number of primes .But

a

(f ) and a

(E) are ordinary integers, so if they are equal in

Q

v

, then they are also equal in Q

w

for any prime w. Therefore,

E,w

=

f ,w

for any prime w.

What Wiles and Taylor–Wiles Did

Wiles and Taylor–Wiles proved the modularity conjecture for a

certain kind of elliptic curve that includes all of the Frey curves.

Starting with one of these elliptic curves E and an odd prime v,

consider

E,v

. We want to show that it is modular. So we consider

“deformations” of ψ

E,v

, that is, Galois representations : G →

GL(2, Q

v

) such that = ψ

E,v

. (The use of the odd term “deformation”

for this concept comes from an analogy with a previous use of the

term in algebraic geometry.)

We say that a deformation is a modular deformation if =

f ,v

for some cuspidal normalized newform f of weight 2. Now, it follows

immediately from our deﬁnitions that

E,v

is a deformation of ψ

E,v

.If

we can prove that every deformation of ψ

E,v

is modular, we are done,

because then

E,v

=

f ,v

for some f , which is what Conjecture 22.4 is

asserting. (We have to keep track of the level of f , of course, but we

omit mentioning it from now on to streamline the discussion.)

FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 253

Now let v be any odd prime. There is a way to package all the

deformations of a given Galois representation ψ : G → GL(2, F

v

)

into a single highly complicated Galois representation R

univ

: G →

GL(2, A

univ

). Here the subscript refers to the word “universal,”

because this Galois representation controls all the deformations

of ψ. The symbol A

univ

stands for the algebraic object that contains

the entries in the matrices R

univ

(σ ) for σ any element in G. This

algebraic object is something rather more complicated than a ﬁeld,

called a “complete noetherian local ring” (the deﬁnition of which is

beyond the scope of this book).

Actually, we lied: R

univ

does not control all the deformations of ψ,

but only those satisfying some highly technical conditions. One of

the many difﬁcult tasks that Wiles solved was ﬁguring out which

conditions to use.

There is also a way to package all the modular deformations of a

given Galois representation ψ : G → GL(2, F

v

) into another single

highly complicated Galois representation R

mod

: G → GL(2, A

mod

).

Here the subscript refers to the word “modular” for obvious reasons.

Again, A

mod

is a complete noetherian local ring.

The main theorem of Wiles and Taylor–Wiles can thus be stated:

THEOREM 22.6: If ψ : G → GL(2, F

v

) satisﬁes certain

hypotheses (H), which include being irreducible and being

modular, then A

univ

= A

mod

. Hence every deformation of ψ that

satisﬁes the highly technical conditions above is modular.

It follows from Theorem 22.6 that if E is a Frey curve and is

a deformation of ψ

E,v

satisfying those highly technical conditions,

then is modular. It so happens that

E,v

does satisfy those highly

technical conditions, so it is modular and therefore the Modularity

Conjecture holds for E. And that implies FLT, as we have seen

already.

But wait! Not so fast! To apply the main theorem, we need to

know that ψ

E,v

satisﬁes all of the hypotheses (H ). The hypotheses

we did not state hold all right, but being irreducible and being

modular are not obvious for ψ

E,v

. In fact, ψ

E,v

can sometimes fail to

be irreducible. And its being modular is the mod v version of what

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