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Fearless Symmetry by Robert Gross, Avner Ash

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FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 245
all to be nonzero integral perfect pth powers, then a solution to
A + B = C would be a counterexample to FLT.
Given any solution to A + B = C, we can form an elliptic curve,
called a “Frey curve” (named after Gerhard Frey)
E : y
2
= x(x A)(x + B).
For any prime v, we can consider the two-dimensional Galois
representation ψ made with the v-torsion points on E(C)asin
chapter 18. Then we can try to find out where ψ is ramified, that
is, what are the bad primes for ψ (see chapter 16). It can be proven
that any bad prime for ψ either divides the product ABC or equals
v or equals 2.
When A, B, and C satisfy certain extra properties, the set of bad
primes for ψ can be reduced drastically. For example, if A = a
p
,
B = b
p
, and C = c
p
are all nonzero integral pth powers, and if we
assume that a 3 (mod 4) and b is even, and that a, b, and c have
no common prime factor, and if we take v = p, it turns out that the
only bad primes for ψ are 2 and p!
This is a great result, because in that case, we do not have to
know what a, b, and c are in order to know exactly the set of bad
primes for ψ. Because we are trying to prove there are no such a,
b, and c, it is nice not to need to know what they are, as we try to
prove a contradiction from their hypothetical existence.
By the way, it is not hard to prove the following:
LEMMA 22.1: If the Fermat equation x
p
+ y
p
= z
p
has a
solution with x, y, and z all nonzero integers, then it has such
a solution with x 3 (mod 4) and y even.
If we want to make explicit the dependence of ψ on the parame-
ters A, B, and C, we will denote it by ψ
A,B,C
.
The Modularity Conjecture
In chapter 21, we briefly mentioned modular forms. A cuspidal
normalized newform is an infinite series
3
f = q + a
2
q
2
+ a
3
q
3
+···
3
This infinite series is called the q-expansion of f .”
246 CHAPTER 22
that satisfies some complicated symmetry conditions we did not
specify and which are beyond the scope of this book. Remember
that f comes with a positive integer N called its level. We did not
mention it in chapter 21, but it also comes with another positive
integer k called the weight of the modular form. Both N and k
appear as parameters in the complicated symmetry conditions we
did not specify.
Given two integers N and k, there are only finitely many cuspidal
normalized newforms with level N and weight k. Moreover, there is
a relatively simple algorithm, suitable for computers, with which
you can find them all, as long as N and k are not too big.
4
So a cuspidal normalized newform gives you a series of integers
a
for all primes : Just read them off its q-expansion. We call these
a
(f ) to remind us that they come from the modular form f .
Now remember that an elliptic curve E also gives you a series of
integers a
for all primes via the equation #E(F
) = 1 + a
.We
will call these a
(E) to remind us that they come from the elliptic
curve E.
Although we never mentioned it, an elliptic curve has an extra
important property: It comes with a positive integer N, called its
“conductor. There is an algorithm for finding the conductor, given
E. There is a theorem that says if is a prime not dividing vN,
then is a good prime for the Galois representation G GL(2, F
v
)
obtained from the v-torsion points of E(C). (As usual, G is the
absolute Galois group of Q.)
We can now state the Modularity Conjecture.
CONJECTURE 22.2 (The Modularity Conjecture): Given
any elliptic curve E with conductor N there exists a cuspidal
normalized newform f with level N and weight 2 such that
a
(E) = a
(f ) for all but finitely many primes .
In this form, the conjecture is rather mysterious. It can be ex-
plained in a purely geometric way by using non-Euclidean geometry
4
On the other hand, if you only fix N, there will be infinitely many cuspidal normalized
newforms with level N and various weights. If you only fix k, again there will be infinitely
many cuspidal normalized newforms with weight k and various levels.

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