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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