Proof.
[Proof (sketch)]
First reduce to the case when
is
a prime greater than
(see Exercise
1.2). Suppose that
with
and
.
Permuting
, we may suppose that
is even and
that we have
.
Following Gerhard Frey, consider the elliptic curve
defined by
The discriminant of
is
(see Exercise
1.3 below).
Andrew Wiles and Richard Taylor [Annals of Math., May 1995] proved that
must be ``modular''. This means that there is a ``modular form''
of ``level
''
such that for all primes
,
Ken Ribet [Inventiones Math., 1991] used that the
discriminant of is a perfect th power (away from )
to prove that there is a cuspidal modular form
of ``level
'' such that
for all
This is a contradiction because
the space of ``cuspidal modular forms'' of level
has dimension 0 (see Section
3.1).