Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 459

This means that the Galois representations mod p associated to f and g are
equivalent.
The following result of Langlands and Tunnell gives us a place to start.

THEOREM 15.10

Let E be an elliptic curve defined over Q and let f E = n≥ 1 a n q n be the

associated potentialm odular form.Thereexistsamodular form
g 0 =
n

b n q n

≥ 1

su ch that

(mod 3) a ≡ b

for almostallprimes (that is, w ith possibly finitelymanyexceptions), and
where 3 denotes a primeof O 3 .

Recall that O 3 denotes an unspecified ring containing the 3-adic integers.

If O 3 is su ciently large, the coe cients b , which are algebraic integers, can

be regarded as lying in O 3 .

The reason that 3 is used is that the group GL 2 ( F 3 ) has order 48, hence
is solvable. The representation ρ 3 of G on E [3] therefore has its image in
a solvable group. The techniques of base change developed in the Langlands
program apply to cyclic groups, hence to solvable groups, and these techniques
are the key to proving the result. The groups GL 2 ( F p )for p ≥ 5arenot
solvable, so the base change techniques do not apply. On the other hand, the
representation ρ 2 for the Galois action on E [2] is trivial for the Frey curves
since the 2-torsion is rational for these curves. Therefore, it is not expected
that ρ 2 should yield any information.
Note that the modular form g 0 does not necessarily have rational coe -
cients. Therefore, g 0 is not necessarily the modular form associated to an
elliptic curve. Throughout Wiles's proof, Galois representations associated to
arbitrary modular forms are used.
The result of Langlands and Tunnell leads us to consider the following.

GENERAL PROBLEM

Fixaprime p .Let g = n≥ 1 a n q n be a potentialm odular form (associated
toa2-dimensional G aloisrepresentation). Suppose there isamodu ar form

g 0 = b n q n g 0 (mod ) p . C an w e prove that g isamodu ar

su ch that g

≡

form ?

The work of Wiles shows that the answer to the general problem is often
yes. Let A be the set of all potential modular forms g with g ≡ g 0 (mod ) p
(subject to certain restrictions). Let M ⊆ A be the set of modular g 's in A .
We are assuming that g 0 ∈ M . The basic idea is the following. Let T A be the