Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 451

If p = , the definition of finite is much more technical (it involves finite flat
group schemes) and we omit it. However, for the representation ρ coming
from an elliptic curve, there is the following:

PROPOSITION 15.5
Let E be an elliptic curve defined over Q and let Δ be the m inimaldiscrimi-
nantof E .Let and p be primes(the case p = isallow ed) and let ρ be the
representation of G on E [ ] .Then ρ is finiteat p ifand onlyif v p (Δ) ≡ 0
(mod ) ,where v p denotes the p -adicva uation (see A ppendixA).

For a proof, see [105].
Consider the Frey curve. The minimal discriminant is

Δ=2 − 8 ( abc ) 2 .

Therefore, v p (Δ) ≡ 0(mod ) for all p =2,so ρ is finite at all odd primes.

Moreover, ρ is not finite at 2.

15.3 Sketch of Ribet's Proof

The key theorem that Ribet proved is the following.

THEOREM 15.6
Let

≥ 3 and let

ρ : G → GL 2 ( F )

be an irredu ciblerepresentation . A ssu m e that ρ ismodular of squarefree level
N and that there existsaprime q | N , q = ,atwhich ρ is notfinite. Suppose

p | N isaprimeatwhich ρ is finite. T hen ρ ismodular of level N/p .

In other words, if ρ comes from a modular form of level N , then, under
suitable hypotheses, it also comes from a modular form of level N/p .

COROLLARY 15.7
E Frey cannotbe m odular.

PROOF Since there are no solutions to the Fermat equation, and hence
no Frey curves, when = 3, we may assume ≥ 5. If E Frey is modular, then
the associated representation ρ is modular of some level N .Since E Frey is