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
= 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
B = b
, and C = c
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
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
The Modularity Conjecture
In chapter 21, we brieﬂy mentioned modular forms. A cuspidal
normalized newform is an inﬁnite series
f = q + a
This inﬁnite series is called the “q-expansion of f .”