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Fearless Symmetry by Robert Gross, Avner Ash

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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,wesayis
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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