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Fearless Symmetry by Robert Gross, Avner Ash

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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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