250 CHAPTER 22

and hence ψ

f

, is unramiﬁed at all primes other than 2 and p and

ﬂat at p. Using the Level Lowering Theorem, there is a cuspidal

normalized newform g of level 2 and weight 2.

The punch line: We compute the set of cuspidal normalized

newforms g of level 2 and weight 2, and ﬁnd out that there

aren’t any! They do not exist! This is a logical contradiction, which

stemmed from our assuming the existence of a, b,andc such that

a

p

+ b

p

= c

p

. Therefore, no such a, b, and c can exist and FLT is

proved.

The only weak link in this proof (before Wiles) was that the

Modularity Conjecture had not been proven yet. Wiles and Taylor–

Wiles proved a large part of the Modularity Conjecture—enough to

make this strategy work.

Bring on the Reciprocity Laws

As we have mentioned, to go farther and explain what Wiles did,

we have to review the reciprocity laws connected with elliptic

curves and modular forms. However, we will have to up the ante

and discuss Galois representations not just to GL(2, F

v

) but to

GL(2, Q

v

) where Q

v

is the ﬁeld of v-adic numbers, brieﬂy introduced

in chapter 20.

Given an elliptic curve E and a prime v, in chapter 18 we ob-

tained a two-dimensional Galois representation ψ

E

: G → GL(2, F

v

)

on the v-torsion points of E(C). However, using the v

2

-torsion, the

v

3

-torsion, and so on, we can get a two-dimensional Galois repre-

sentation

E

: G → GL(2, Q

v

), where the matrices now have entries

in the big ﬁeld Q

v

. This Galois representation obeys the following

reciprocity law: If is a good prime for

E

, then χ

E

(Frob

) = 1 +

− #E(F

) exactly (not just modulo v—remember that Z is a subset

of Q

v

for every v.)

There is a way to reduce

E

modulo v. If you write elements in Q

v

as inﬁnite v-adic expansions, as in the footnote on page 229, then

you can deﬁne Z

v

as the set of all elements in Q

v

with no digits to the

right of the “decimal point.” You can reduce an element ···a

3

a

2

a

1

a

0

of Z

v

by sending it to a

0

modulo v. Now it turns out that for any

FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 251

element σ in the absolute Galois group G,

E

(σ ) is a 2-by-2 matrix

with entries in Z

v

. So you can reduce each of these entries and get

a 2-by-2 matrix with entries in F

v

. This deﬁnes the reduction of

E

modulo v. We will denote the reduction by

E

. It is itself a Galois

representation

E

: G → GL(2, F

v

). It turns out that

E

is actually

equal to ψ

E

.

Notice the system behind this notation: Galois representations

to GL(2, F

v

) are denoted by small Greek letters. Galois representa-

tions to GL(2, Q

v

) are denoted by large Greek letters. The reduction

of a Galois representation is denoted by putting a bar over the large

Greek letter. And if the uppercase and lowercase Greek letters are

the same letter, as, for example, and ψ, we mean to imply that

= ψ.

On the other hand, if f = q + a

2

q

2

+ a

3

q

3

+··· is a cuspidal

normalized newform and if all the coefﬁcients a

i

are integers, and if

v is any prime, then by Theorem 21.1 f has a level N and a two-

dimensional Galois representation G → GL(2, Q

v

) which we will

call

f

.If is a prime not dividing vN, then the trace of

f

(Frob

)

equals the integer a

.

As with

E

, we can reduce

f

modulo v to obtain

f

. It is also a

Galois representation

f

: G → GL(2, F

v

).

We can now state the Modularity Conjecture in a different

form that can be proven to be logically equivalent to the original

Modularity Conjecture:

CONJECTURE 22.4 : Given an elliptic curve E of conductor

N, then there exists a cuspidal normalized newform f with

level N and weight 2 such that

E

=

f

.

9

If an elliptic curve E satisﬁes Conjecture 22.4 we say it is

modular. Similarly, if a Galois representation : G → GL(2, Q

v

)

is equal to

f

for some cuspidal normalized newform f,wesay is

9

Remember the discussion of “equivalent” linear representations from chapter 15? A

more accurate way of stating the conclusion of this version of the Modularity Conjecture

is to say that

E

and

f

are equivalent. If that is the case, however, then we can make

certain choices so that in fact they are equal.

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