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Fearless Symmetry by Robert Gross, Avner Ash

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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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