254 CHAPTER 22

we are trying to prove in the ﬁrst place, so it would seem unlikely

that we could verify it a priori.

This is where the miracles start happening.

10

By a theorem

of Robert Langlands and Jerry Tunnell (Tunnell, 1981) from the

theory of “automorphic representations,” we know, by completely

different methods from anything discussed in this book that if

v = 3, then ψ

E,3

is automatically modular! So if ψ

E,3

is irreducible,

we can apply Theorem 22.6 for v = 3, and so

E,3

is modular.

But what if ψ

E,3

is not irreducible? Wiles came up with a very

clever argument that if ψ

E,3

is not irreducible, there is another

elliptic curve E

for which ψ

E

,3

is irreducible, and for which ψ

E,5

=

ψ

E

,5

. It follows that

E

is modular and hence that ψ

E

,5

is modular

and hence that ψ

E,5

is modular. It also turns out that if ψ

E,3

is

not irreducible, ψ

E,5

must be irreducible! So now we can apply

Theorem 22.6 for v = 5.

Either way, the Frey curve turns out to satisfy the Modularity

Conjecture and FLT is proven true. Perhaps even more signiﬁ-

cant than FLT is the Modularity Conjecture itself, which as we

mentioned has now been proven for all elliptic curves.

Generalized Fermat Equations

Now that FLT has been proven, number theorists are moving on to

other equations. An equation of the form

x

p

+ y

q

= z

r

where x, y, and z are unknown integers, and where the three

exponents p, q, and r are not all the same, is called a generalized

Fermat equation. Because it looks like the equation in FLT, it might

be expected that similar methods could be used to solve it. This is

partially true. But the fact that the three exponents p, q, and r are

not all the same makes it even more difﬁcult than FLT.

To make progress on these generalized Fermat equations, math-

ematicians usually make some assumptions to narrow the scope of

10

We mean “start” from a logical, not historical, point of view; these miraculous

mathematical facts were known before Wiles proved his theorem.

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