Cryptography Reference
In-Depth Information
semistable, (15.7) says that
N
=
p
p.
|
abc
It can be shown that
ρ
is irreducible when
5 (see [105], where it is
obtained as a corollary of Mazur's theorem (Theorem 8.11)). Let
q
=2in
Ribet's theorem. As we showed at the end of Section 13.2,
ρ
is not finite at
2 and is finite at all other primes. Therefore, Ribet's theorem allows us to
remove the odd primes from
N
one at a time. We eventually find that
ρ
is
modular of level 2. This means that there is a normalized cusp form of weight
2forΓ
0
(2) such that
ρ
is the associated mod
representation. But there
are no nonzero cusp forms of weight 2 for Γ
0
(2), so we have a contradiction.
Therefore,
E
Frey
≥
cannot be modular.
COROLLARY 15.8
TheTaniyam a-Shimura-Weilconjecture (for sem istableelliptic curves) im-
plies Ferm at's Last T heorem .
PROOF
5. If there is a nontrivial
solution to the Fermat equation for
, then the Frey curve exists. However,
Corollary 15.7 and the Taniyama-Shimura-Weil conjecture imply that the Frey
curve cannot exist. Therefore, there are no nontrivial solutions to the Fermat
equation.
We may restrict to prime exponents
≥
We now give a brief sketch of the proof of Ribet's theorem. The proof uses
the full power of Grothendieck's algebraic geometry and is not elementary.
Therefore, we give only a sampling of some of the ideas that go into the proof.
For more details, see [90], [89], [85], [29].
We assume that
ρ
is as in Theorem 15.6 and that
N
is chosen so that
1.
ρ
is modular of squarefree level
N
,
2. both
p
and
q
divide
N
,
3.
ρ
is finite at
p
but is not finite at
q
.
The goal is to show that
p
can be removed from
N
. The main ingredient
of the proof is a relation between Jacobians of modular curves and Shimura
curves. In the following, we describe modular curves and Shimura curves and
give a brief indication of how they occur in Ribet's proof.
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