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Fearless Symmetry by Robert Gross, Avner Ash

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FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 247
and complex numbers. This approach is discussed in most of the
popular books about Wiles’s proof of FLT.
For example, the elliptic curve defined by y
2
+ y = x
3
x satisfies
the Modularity Conjecture where the corresponding modular form
is given by the power series in equation (21.2).
By the way, the Modularity Conjecture is not a conjecture any
more. Following the ideas of Wiles and Taylor–Wiles, and with a lot
of extra hard work, the proof was completed by Christophe Breuil,
Brian Conrad, Fred Diamond, and Richard Taylor (Breuil et al.,
2001). However, at the time of Wiles’s work on FLT, it was still a
conjecture, so we will continue to refer to it as such in this chapter.
Because we cannot go into the details of what a cuspidal normal-
ized newform f really is, the main things that you should keep in
mind are:
f is determined by its q-expansion;
5
compared with Galois representations, newforms are easy
to compute and work with.
Lowering the Level
Suppose we have a cuspidal normalized newform f of level N and
weight 2. By the corollary to Theorem 21.1, for any choice of a
prime v, there is a Galois representation r : G GL(2, F
v
) whose
traces at Frobenius elements give you the values of coefficients in
the q-expansion of f modulo v. We will change the name of the
representation from r to ψ
f
to emphasize its dependence on f .
To state the Level Lowering Theorem and also to explain Wiles’s
main contribution, we have to introduce the concept of “irreducibil-
ity” of a representation. This is a technical concept and you can skip
it on a first reading. But we want to include it so that we can state
the theorems in the chapter accurately.
You will need to review the definitions in the first section of
chapter 20. Suppose k is a field.
5
In fact, it is enough to know the integers a
(f ) for all primes to determine the entire
q-expansion, and hence determine f .
248 CHAPTER 22
DEFINITION: A Galois representation R : G GL(2, k)is
irreducible if there is no line L in the vector space k
2
with the
property that R(σ)L = L for all σ in G.
Now let k
be a field containing k with the property that any
polynomial with coefficients in k
has a root in k
. (For any field k,
there always exists such a field k
—in fact, there are many of them.)
Note that because every element of k is also in k
, we can consider a
Galois representation R : G GL(2, k) with values in k-matrices as
also defining a Galois representation R
: G GL(2, k
) with values
in k
-matrices.
DEFINITION: A Galois representation R : G GL(2, k)is
absolutely irreducible if the corresponding R
is irreducible.
6
For example, if R : G GL(2, k), and if R(σ ) takes on all possible
values in GL(2, k)asσ runs through all the elements of the absolute
Galois group G, then R is absolutely irreducible. (This is not
obvious.)
By the way, for all the elliptic curves E that are used in this
chapter, it turns out that if v 3, then the representation on the
v-torsion points of E(C) is absolutely irreducible if and only if it
is irreducible. (So why did we introduce both concepts? For truth
in advertising. The “if and only if we just stated is a difficult
theorem.)
Ken Ribet (Ribet, 1990) proved the following:
THEOREM 22.3 (Level Lowering Theorem): Let f be a
cuspidal normalized newform of level N and weight 2.
Suppose that is a prime dividing N but that
2
does not
divide N, and suppose that ψ
f
: G GL(2, F
v
) is absolutely
irreducible. Assume further that either
= v and ψ
f
is unramified at (i.e., is a good prime for
ψ
f
), or
6
It can be proven that this concept does not depend on the choice of the field k
.

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