Cryptography Reference
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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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