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Modular Elliptic Curves and Fermat's Last Theorem

Andrew Wiles

The Annals of Mathematics, 2nd Ser., Vol. 141, No. 3. (May, 1995), pp. 443-551.

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Fri Nov 2 21:41:06 2007

444

ANDREW WILES

Our approach to the study of elliptic curves is via their associated Galois

representations.

Suppose that

is the representation of Gal(Q/Q) on the

pdivision points of an elliptic curve over Q, and suppose for the moment that

is irreducible. The choice of 3 is critical because a crucial theorem of Lang-

lands and Tunnell shows that if

is irreducible then it is also modular. We

then proceed by showing that under the hypothesis that

is semistable at 3,

together with some milder restrictions on the ramification of

at the other

primes, every suitable lifting of

is modular. To do this we link the problem,

via some novel arguments from commutative algebra, to a class number prob-

lem of a well-known type. This we then solve with the help of the paper [TW].

This suffices to prove the modularity of

as it is known that

is modular if

and only if the associated 3-adic representation is modular.

The key development in the proof is a new and surprising link between two

strong but distinct traditions in number theory, the relationship between Galois

representations and modular forms on the one hand and the interpretation of

special values of L-functions on the other. The former tradition is of course

more recent.

Following the original results of Eichler and Shimura in the

1950's and 1960's the other main theorems were proved by Deligne, Serre and

Langlands in the period up to 1980. This included the construction of Galois

representations associated to modular forms, the refinements of Langlands and

Deligne (later completed by Carayol), and the crucial application by Langlands

of base change methods to give converse results in weight one. However with

the exception of the rather special weight one case, including the extension by

Tunnell of Langlands' original theorem, there was no progress in the direction

of associating modular forms to Galois representations. From the mid

1980's

the main impetus to the field was given by the conjectures of Serre which

elaborated on the &-conjecture alluded to before. Besides the work of Ribet and

others on this problem we draw on some of the more specialized developments

of the 1980's, notably those of Hida and Mazur.

The second tradition goes back to the famous analytic class number for-

mula of Dirichlet, but owes its modern revival to the conjecture of Birch and

Swinnerton-Dyer. In practice however, it is the ideas of Iwasawa in this field on

which we attempt to draw, and which to a large extent we have to replace. The

principles of Galois cohomology, and in particular the fundamental theorems

of Poitou and Tate, also play an important role here.

The restriction that

be irreducible at 3 is bypassed by means of an

intriguing argument with families of elliptic curves which share a common

p5.

Using this, we complete the proof that all semistable elliptic curves are

modular. In particular, this finally yields a proof of Fermat's Last Theorem. In

addition, this method seems well suited to establishing that all elliptic curves

over Q are modular and to generalization to other totally real number fields.

Now we present our methods and results in more detail.

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM

445

Let

be an eigenform associated to the congruence subgroup Fl(N) of

SL2(Z) of weight

2 and character

Thus if

is the Hecke operator

associated to an integer n there is an algebraic integer c(n,

such that

c(n,

for each n. We let Kf be the number field generated over Q by the

{c (n,

f))

together with the values of

and let Of be its ring of integers.

For any prime

of Of let Of,x be the completion of Of at A. The following

theorem is due to Eichler and Shimura (for

2) and Deligne (for

2).

The analogous result when

is a celebrated theorem of Serre and Deligne

but is more naturally stated in terms of complex representations. The image

in that case is finite and a converse is known in many cases.

THEOREM

For each prime

and each prime

of Of there

0.1.

continuous representation

which is

unramijied outside the primes dividing Np and such that for all primes

NP,

We will be concerned with trying to prove results in the opposite direction,

that is to say, with establishing criteria under which a A-adic representation

arises in this way from a modular form.

We have not found any advantage

in assuming that the representation is part of a compatible system of

A-adic

representations except that the proof may be easier for some

than for others.

Assume

Gal(Q/Q)

GL2(Fp)

is a continuous representation with values in the algebraic closure of a finite

field of characteristic

and that det po is odd. We say that po is modular

if po and pf,x mod

are isomorphic over

for some

and

and some

embedding of Of/X in F,. Serre has conjectured that every irreducible po of

odd determinant is modular. Very little is known about this conjecture except

when the image of po in PGL~(F,) is dihedral,

S4.

In the dihedral case

it is true and due (essentially) to Hecke, and in the

and

cases it is again

true and due primarily to Langlands, with one important case due to Tunnel1

(see Theorem

5.1

for a statement). More precisely these theorems actually

associate a form of weight one to the corresponding complex representation

but the versions we need are straightforward deductions from the complex

case. Even in the reducible case not much is known about the problem in

the form we have described it, and in that case it should be observed that

one must also choose the lattice carefully as only the semisimplification of

pf,~mod

is independent of the choice of lattice in Kf2,x.

446

ANDREW

WILES

If O is the ring of integers of a local field (containing Q,) we will say that

Gal(Q/Q)

GL2(0) is a lifting of

if, for a specified embedding of the

residue field of O in

F,,

and

are isomorphic over F,. Our point of view

will be to assume that

is modular and then to attempt to give conditions

under which a representation

lifting

comes from a modular form in the

sense that

pf,~

over

Kf,x

for some

We will restrict our attention to

two cases:

(I)

is ordinary (at p) by which we mean that there is a one-dimensional

subspace of

F;,

stable under a decomposition group at p and such that

the action on the quotient space is unramified and distinct from the

action on the subspace.

(11)

is flat (at

p),

meaning that

a representation of a decomposition

group at

is equivalent to one that arises from a finite flat group

scheme over

Z,,

and det

restricted to an inertia group at

is the

cyclotomic character.

We say similarly that

is ordinary (at

if, viewed

a representation to

QE,

there is a one-dimensional subspace of Q; stable under a decomposition group

and such that the action on the quotient space is unramified.

Let

Gal(Q/Q)

denote the cyclotomic character. Conjectural

converses to Theorem

0.1

have been part of the folklore for many years but

have hitherto lacked any evidence. The critical idea that one might dispense

with compatible systems was already observed by Drinfeld in the function field

case [Dr]. The idea that one only needs to make a geometric condition on the

restriction to the decomposition group at

was first suggested by Fontaine and

Mazur. The following version is a natural extension of Serre's conjecture which

is convenient for stating our results and is, in a slightly modified form, the one

proposed by Fontaine and Mazur. (In the form stated this incorporates Serre's

conjecture. We could instead have made the hypothesis that

is modular.)

CONJECTURE.

Suppose that

Gal(Q/Q)

GL2(C3)

is an irreducible

lifting of

and that

is unramified outside of a finite set of primes. There

are two cases:

(i)

Assume that

is ordinary. Then if

is ordinary and

det

&"IX

for

some integer

and some

of finite order,

comes from a modular

form.

(ii)

Assume that

is flat and that p is odd. Then if

restricted to a de-

composition group at

is equivalent to a representation on a p-divisible

group, again

comes from a modular form.

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