Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 458

representation space W of ρ with

V ⊆ J [ M ] ⊆ J [ ] .

We now have the representation ρ living in J 0 ( N )[ ]andin J [ ]. The rep-
resentation ρ can be detected using the reduction of J 0 ( N )mod q and also
using the reduction of J mod p , and Ribet uses a calculation with quater-
nion algebras to establish a relationship between these two reductions. This
relationship allows him to show that p can be removed from the level N .

REMARK 15.9 A correspondence between modular forms for GL 2 and
modular forms for the multiplicative group of a quaternion algebra plays a
major role in work of Jacquet-Langlands. This indicates a relation between
J 0 ( N )and J . In fact, there is a surjection from J 0 ( N )to J . However, this
map is not being used in the present case since such a map would relate the
reduction of J 0 ( N )mod q to the reduction of J mod q . Instead, Ribet works
with the reduction of J 0 ( N )mod q and the reduction of J mod p .Thisswitch
between p and q is a major step in the proof of Ribet's theorem.

15.4 Sketch of Wiles's Proof

In this section, we outline the proof that all semistable elliptic curves over
Q are modular. For more details, see [29], [32], [118], [133]. Let E be a

semistable elliptic curve and let
f E =
n

a n q n

≥ 1

be the associated potential modular form. We want to prove that f E is a
modular form (for some Γ 0 ( N )).
Suppose we have two potential modular forms
f =
n

g =
n

c n q n , c n q n

≥ 1 ≥ 1

arising from Galois representations G O p is some ring

containing the p -adic integers. We assume that all of the coe cients c n ,c n
are embedded in

→ GL 2 ( O p )(where

O p is the ring of
p -adic integers, then p = p .) If c ≡ c (mod p ) for almost all primes (that
is, we allow finitely many exceptions), then we write

O p ). Let p be the prime above p in O p .(If

f ≡ g (mod p ) .