FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 249
= v and ψ
is “ﬂat” at v.
Then there exists a cuspidal normalized newform g of level
N/ and weight 2 with ψ
Note that the equation ψ
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
(g) ≡ a
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
. 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
and consider the Galois representation ψ
obtained from the p-torsion points of E(C). It is known that ψ
the hypotheses of the Level Lowering Theorem.
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
(f ) = a
(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
(f ) ≡ a
(E) (mod p) that the Galois repre-
is equivalent to ψ
. Using our detailed knowledge of
how we constructed the Frey curve E, it can be shown that ψ
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
to be unramiﬁed at v.
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.