FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 249

•

= v and ψ

f

is “ﬂat” at v.

7

Then there exists a cuspidal normalized newform g of level

N/ and weight 2 with ψ

g

= ψ

f

.

Note that the equation ψ

g

= ψ

f

tells you about the coefﬁcients in

the q-expansions of g and f after they are reduced (mod v). In other

words, if w is a prime that is not a factor of vN, then a

w

(g) ≡ a

w

(f )

(mod v).

Proof of FLT Given the Truth of the Modularity

Conjecture for Certain Elliptic Curves

In this section, set v = p, and suppose that FLT is false. Then by

Lemma 22.1, there exist nonzero integers a, b, and c such that

a ≡ 3 (mod 4), b is even, and a

p

+ b

p

= c

p

. We want to derive a

contradiction from this equation, and thereby prove FLT. (Remem-

ber we are assuming that p is a prime greater than 3.) Form the

Frey curve E = E

a

p

,b

p

,c

p

and consider the Galois representation ψ

E

obtained from the p-torsion points of E(C). It is known that ψ

E

obeys

the hypotheses of the Level Lowering Theorem.

8

The conductor N of E can be computed to be the product of

all primes dividing abc. By the Modularity Conjecture, there is a

cuspidal normalized newform f of level N and weight 2 such that

for all primes w that are not factors of N, a

w

(f ) = a

w

(E), and hence

these pairs of integers are also congruent modulo p.

Using a standard theorem of algebraic number theory, it follows

from the congruences a

w

(f ) ≡ a

w

(E) (mod p) that the Galois repre-

sentation ψ

f

is equivalent to ψ

E

. Using our detailed knowledge of

how we constructed the Frey curve E, it can be shown that ψ

E

,

7

This is a condition too hard to deﬁne in this book, but it more or less means “as well-

behaved as possible” if = v—it is too much to expect ψ

f

to be unramiﬁed at v.

8

In particular, the absolute irreducibility follows from a theorem proved by Barry Mazur

in the 1970s. Mazur’s theorem had to do with studying the rational solutions to a

different Z-variety, called a “modular curve.” Many of his ideas (including the concept

of “deformation” discussed later in this chapter) were used in the eventual proof of FLT.

Unfortunately, modular curves would require a very long digression to explain.

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