MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 447
In case (ii) it is not hard to see that if the form exists it has to be of
weight 2; in (i) of course it would have weight k. One can of course enlarge
this conjecture in several ways, by weakening the conditions in (i) and (ii), by
considering other number fields of Q and by considering groups other
than GL
2
.
We prove two results concerning this conjecture. The first includes the
hypothesis that ρ
0
is modular. Here and for the rest of this paper we will
assume that p is an odd prime.
Theorem 0.2. Suppose that ρ
0
is irreducible and satisfies either (I) or
(II) above. Suppose also that ρ
0
is modular and that
(i) ρ
0
is absolutely irreducible when restricted to Q
(−1)
p−1
2
p
.
(ii) If q ≡ −1 mod p is ramified in ρ
0
then either ρ
0
|
D
q
is reducible over
the algebraic closure where D
q
is a decomposition group at q or ρ
0
|
I
q
is
absolutely irreducible where I
q
is an inertia group at q.
Then any representation ρ as in the conjecture does indeed come from a mod-
ular form.
The only condition which really seems essential to our method is the re-
quirement that ρ
0
be modular.
The most interesting case at the moment is when p = 3 and ρ
0
can be de-
fined over F
3
. Then since PGL
2
(F
3
) ≃ S
4
every such representation is modular
by the theorem of Langlands and Tunnell mentioned above. In particular, ev-
ery representation into GL
2
(Z
3
) whose reduction satisfies the given conditions
is modular. We deduce:
Theorem 0.3. Suppose that E is an elliptic curve defined over Q and
that ρ
0
is the Galois action on the 3-division points. Suppose that E has the
following properties:
(i) E has good or multiplicative reduction at 3.
(ii) ρ
0
is absolutely irreducible when restricted to Q
√
−3
.
(iii) For any q ≡ −1 mod 3 either ρ
0
|
D
q
is reducible over the algebraic closure
or ρ
0
|I
q
is absolutely irreducible.
Then E should be modular.
We should point out that while the properties of the zeta function follow
directly from Theorem 0.2 the stronger version that E is covered by X
0
(N)
448 ANDREW JOHN WILES
requires also the isogeny theorem proved by Faltings (and earlier by Serre when
E has nonintegral j-invariant, a case which includes the semistable curves).
We note that if E is modular then so is any twist of E, so we could relax
condition (i) somewhat.
The important class of semistable curves, i.e., those with square-free con-
ductor, satisfies (i) and (iii) but not necessarily (ii). If (ii) fails then in fact ρ
0
is reducible. Rather surprisingly, Theorem 0.2 can often be applied in this case
also by showing that the representation on the 5-division points also occurs for
another elliptic curve which Theorem 0.3 has already proved modular. Thus
Theorem 0.2 is applied this time with p = 5. This argument, which is explained
in Chapter 5, is the only part of the paper which really uses deformations of
the elliptic curve rather than deformations of the Galois representation. The
argument works more generally than the semistable case but in this setting
we obtain the following theorem:
Theorem 0.4. Suppose that E is a semistable elliptic curve defined over
Q. Then E is modular.
More general families of elliptic curves which are modular are given in Chap-
ter 5.
In 1986, stimulated by an ingenious idea of Frey [Fr], Serre conjectured
and Ribet proved (in [Ri1]) a property of the Galois representation associated
to modular forms which enabled Ribet to show that Theorem 0.4 implies ‘Fer-
mat’s Last Theorem’. Frey’s suggestion, in the notation of the following theo-
rem, was to show that the (hypothetical) elliptic curve y
2
= x(x + u
p
)(x− v
p
)
could not be modular. Such elliptic curves had already been studied in [He]
but without the connection with modular forms. Serre made precise the idea
of Frey by proposing a conjecture on modular forms which meant that the rep-
resentation on the p-division points of this particular elliptic curve, if modular,
would be associated to a form of conductor 2. This, by a simple inspection,
could not exist. Serre’s conjecture was then proved by Ribet in the summer
of 1986. However, one still needed to know that the curve in question would
have to be modular, and this is accomplished by Theorem 0.4. We have then
(finally!):
Theorem 0.5. Suppose that u
p
+ v
p
+ w
p
= 0 with u, v, w ∈ Q and p ≥ 3,
then uvw = 0. (Equivalently - there are no nonzero integers a, b, c, n with n > 2
such that a
n
+ b
n
= c
n
.)
The second result we prove about the conjecture does not require the
assumption that ρ
0
be modular (since it is already known in this case).
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 449
Theorem 0.6. Suppose that ρ
0
is irreducible and satisfies the hypothesis
of the conjecture, including (I) above. Suppose further that
(i) ρ
0
= Ind
Q
L
κ
0
for a character κ
0
of an imaginary quadratic extension L
of Q which is unramified at p.
(ii) det ρ
0
|
I
p
= ω.
Then a representation ρ as in the conjecture does indeed come from a modular
form.
This theorem can also be used to prove that certain families of elliptic
curves are modular. In this summary we have only described the principal
theorems associated to Galois representations and elliptic curves. Our results
concerning generalized class groups are described in Theorem 3.3.
The following is an account of the origins of this work and of the more
specialized developments of the 1980’s that affected it. I began working on
these problems in the late summer of 1986 immediately on learning of Ribet’s
result. For several years I had been working on the Iwasawa conjecture for
totally real fields and some applications of it. In the process, I had been using
and developing results on ℓ-adic representations associated to Hilbert modular
forms. It was therefore natural for me to consider the problem of modularity
from the point of view of ℓ-adic representations. I began with the assumption
that the reduction of a given ordinary ℓ-adic representation was reducible and
tried to prove under this hypothesis that the representation itself would have
to be modular. I hoped rather naively that in this situation I could apply the
techniques of Iwasawa theory. Even more optimistically I hoped that the case
ℓ = 2 would be tractable as this would suffice for the study of the curves used
by Frey. From now on and in the main text, we write p for ℓ because of the
connections with Iwasawa theory.
After several months studying the 2-adic representation, I made the first
real breakthrough in realizing that I could use the 3-adic representation instead:
the Langlands-Tunnell theorem meant that ρ
3
, the mod 3 representation of any
given elliptic curve over Q, would necessarily be modular. This enabled me
to try inductively to prove that the GL
2
(Z/3
n
Z) representation would be
modular for each n. At this time I considered only the ordinary case. This led
quickly to the study of H
i
(Gal(F
∞
/Q), W
f
) for i = 1 and 2, where F
∞
is the
splitting field of the m-adic torsion on the Jacobian of a suitable modular curve,
m being the maximal ideal of a Hecke ring associated to ρ
3
and W
f
the module
associated to a modular form f described in Chapter 1. More specifically, I
needed to compare this cohomology with the cohomology of Gal(Q
Σ
/Q) acting
on the same module.
I tried to apply some ideas from Iwasawa theory to this problem. In my
solution to the Iwasawa conjecture for totally real fields [Wi4], I had introduced
450 ANDREW JOHN WILES
a new technique in order to deal with the trivial zeroes. It involved replacing
the standard Iwasawa theory method of considering the fields in the cyclotomic
Z
p
-extension by a similar analysis based on a choice of infinitely many distinct
primes q
i
≡ 1 mod p
n
i
with n
i
→ ∞ as i → ∞. Some aspects of this method
suggested that an alternative to the standard technique of Iwasawa theory,
which seemed problematic in the study of W
f
, might be to make a comparison
between the cohomology groups as Σ varies but with the field Q fixed. The
new principle said roughly that the unramified cohomology classes are trapped
by the tamely ramified ones. After reading the paper [Gre1]. I realized that the
duality theorems in Galois cohomology of Poitou and Tate would be useful for
this. The crucial extract from this latter theory is in Section 2 of Chapter 1.
In order to put ideas into practice I developed in a naive form the
techniques of the first two sections of Chapter 2. This drew in particular on
a detailed study of all the congruences between f and other modular forms
of differing levels, a theory that had been initiated by Hida and Ribet. The
outcome was that I could estimate the first cohomology group well under two
assumptions, first that a certain subgroup of the second cohomology group
vanished and second that the form f was chosen at the minimal level for m.
These assumptions were much too restrictive to be really effective but at least
they pointed in the right direction. Some of these arguments are to be found
in the second section of Chapter 1 and some form the first weak approximation
to the argument in Chapter 3. At that time, however, I used auxiliary primes
q ≡ −1 mod p when varying Σ as the geometric techniques I worked with did
not apply in general for primes q ≡ 1 mod p. (This was for much the same
reason that the reduction of level argument in [Ri1] is much more difficult
when q ≡ 1 mod p.) In all this work I used the more general assumption that
ρ
p
was modular rather than the assumption that p = −3.
In the late 1980’s, I translated these ideas into ring-theoretic language. A
few years previously Hida had constructed some explicit one-parameter fam-
ilies of Galois representations. In an attempt to understand this, Mazur had
been developing the language of deformations of Galois representations. More-
over, Mazur realized that the universal deformation rings he found should be
given by Hecke ings, at least in certain special cases. This critical conjecture
refined the expectation that all ordinary liftings of modular representations
should be modular. In making the translation to this ring-theoretic language
I realized that the vanishing assumption on the subgroup of H
2
which I had
needed should be replaced by the stronger condition that the Hecke rings were
complete intersections. This fitted well with their being deformation rings
where one could estimate the number of generators and relations and so made
the original assumption more plausible.
To be of use, the deformation theory required some development. Apart
from some special examples examined by Boston and Mazur there had been
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 451
little work on it. I checked that one could make the appropriate adjustments to
the theory in order to describe deformation theories at the minimal level. In the
fall of 1989, I set Ramakrishna, then a student of mine at Princeton, the task
of proving the existence of a deformation theory associated to representations
arising from finite flat group schemes over Z
p
. This was needed in order to
remove the restriction to the ordinary case. These developments are described
in the first section of Chapter 1 although the work of Ramakrishna was not
completed until the fall of 1991. For a long time the ring-theoretic version
of the problem, although more natural, did not look any simpler. The usual
methods of Iwasawa theory when translated into the ring-theoretic language
seemed to require unknown principles of base change. One needed to know the
exact relations between the Hecke rings for different fields in the cyclotomic
Z
p
-extension of Q, and not just the relations up to torsion.
The turning point in this and indeed in the whole proof came in the
spring of 1991. In searching for a clue from commutative algebra I had been
particularly struck some years earlier by a paper of Kunz [Ku2]. I had already
needed to verify that the Hecke rings were Gorenstein in order to compute the
congruences developed in Chapter 2. This property had first been proved by
Mazur in the case of prime level and his argument had already been extended
by other authors as the need arose. Kunz’s paper suggested the use of an
invariant (the η-invariant of the appendix) which I saw could be used to test
for isomorphisms between Gorenstein rings. A different invariant (the p/p
2
-
invariant of the appendix) I had already observed could be used to test for
isomorphisms between complete intersections. It was only on reading Section 6
of [Ti2] that I learned that it followed from Tate’s account of Grothendieck
duality theory for complete intersections that these two invariants were equal
for such rings. Not long afterwards I realized that, unlike though it seemed at
first, the equality of these invariants was actually a criterion for a Gorenstein
ring to be a complete intersection. These arguments are given in the appendix.
The impact of this result on the main problem was enormous. Firstly, the
relationship between the Hecke rings and the deformation rings could be tested
just using these two invariants. In particular I could provide the inductive ar-
gument of section 3 of Chapter 2 to show that if all liftings with restricted
ramification are modular then all liftings are modular. This I had been trying
to do for a long time but without success until the breakthrough in commuta-
tive algebra. Secondly, by means of a calculation of Hida summarized in [Hi2]
the main problem could be transformed into a problem about class numbers
of a type well-known in Iwasawa theory. In particular, I could check this in
the ordinary CM case using the recent theorems of Rubin and Kolyvagin. This
is the content of Chapter 4. Thirdly, it meant that for the first time it could
be verified that infinitely many j-invariants were modular. Finally, it meant
that I could focus on the minimal level where the estimates given by me earlier
452 ANDREW JOHN WILES
Galois cohomology calculations looked more promising. Here I was also using
the work of Ribet and others on Serre’s conjecture (the same work of Ribet
that had linked Fermat’s Last Theorem to modular forms in the first place) to
know that there was a minimal level.
The class number problem was of a type well-known in Iwasawa theory
and in the ordinary case had already been conjectured by Coates and Schmidt.
However, the traditional methods of Iwasawa theory did not seem quite suf-
ficient in this case and, as explained earlier, when translated into the ring-
theoretic language seemed to require unknown principles of base change. So
instead I developed further the idea of using auxiliary primes to replace the
change of field that is used in Iwasawa theory. The Galois cohomology esti-
mates described in Chapter 3 were now much stronger, although at that time
I was still using primes q ≡ −1 mod p for the argument. The main difficulty
was that although I knew how the η-invariant changed as one passed to an
auxiliary level from the results of Chapter 2, I did not know how to estimate
the change in the p/p
2
-invariant precisely. However, the method did give the
right bound for the generalised class group, or Selmer group as it often called
in this context, under the additional assumption that the minimal Hecke ring
was a complete intersection.
I had earlier realized that ideally what I needed in this method of auxiliary
primes was a replacement for the power series ring construction one obtains in
the more natural approach based on Iwasawa theory. In this more usual setting,
the projective limit of the Hecke rings for the varying fields in a cyclotomic
tower would be expected to be a power series ring, at least if one assumed
the vanishing of the µ-invariant. However, in the setting with auxiliary primes
where one would change the level but not the field, the natural limiting process
did not appear to be helpful, with the exception of the closely related and very
important construction of Hida [Hi1]. This method of Hida often gave one step
towards a power series ring in the ordinary case. There were also tenuous hints
of a patching argument in Iwasawa theory ([Scho], [Wi4, §10]), but I searched
without success for the key.
Then, in August, 1991, I learned of a new construction of Flach [Fl] and
quickly became convinced that an extension of his method was more plausi-
ble. Flach’s approach seemed to be the first step towards the construction of
an Euler system, an approach which would give the precise upper bound for
the size of the Selmer group if it could be completed. By the fall of 1992, I
believed I had achieved this and begun then to consider the remaining case
where the mod 3 representation was assumed reducible. For several months I
tried simply to repeat the methods using deformation rings and Hecke rings.
Then unexpectedly in May 1993, on reading of a construction of twisted forms
of modular curves in a paper of Mazur [Ma3], I made a crucial and surprising
breakthrough: I found the argument using families of elliptic curves with a
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 453
common ρ
5
which is given in Chapter 5. Believing now that the proof was
complete, I sketched the whole theory in three lectures in Cambridge, England
on June 21-23. However, it became clear to me in the fall of 1993 that the con-
struction of the Euler system used to extend Flach’s method was incomplete
and possibly flawed.
Chapter 3 follows the original approach I had taken to the problem of
bounding the Selmer group but had abandoned on learning of Flach’s paper.
Darmon encouraged me in February, 1994, to explain the reduction to the com-
plete intersection property, as it gave a quick way to exhibit infinite families
of modular j-invariants. In presenting it in a lecture at Princeton, I made,
almost unconsciously, critical switch to the special primes used in Chapter 3
as auxiliary primes. I had only observed the existence and importance of these
primes in the fall of 1992 while trying to extend Flach’s work. Previously, I had
only used primes q ≡ −1 mod p as auxiliary primes. In hindsight this change
was crucial because of a development due to de Shalit. As explained before, I
had realized earlier that Hida’s theory often provided one step towards a power
series ring at least in the ordinary case. At the Cambridge conference de Shalit
had explained to me that for primes q ≡ 1 mod p he had obtained a version of
Hida’s results. But excerpt for explaining the complete intersection argument
in the lecture at Princeton, I still did not give any thought to my initial ap-
proach, which I had put aside since the summer of 1991, since I continued to
believe that the Euler system approach was the correct one.
Meanwhile in January, 1994, R. Taylor had joined me in the attempt to
repair the Euler system argument. Then in the spring of 1994, frustrated in
the efforts to repair the Euler system argument, I begun to work with Taylor
on an attempt to devise a new argument using p = 2. The attempt to use p = 2
reached an impasse at the end of August. As Taylor was still not convinced that
the Euler system argument was irreparable, I decided in September to take one
last look at my attempt to generalise Flach, if only to formulate more precisely
the obstruction. In doing this I came suddenly to a marvelous revelation: I
saw in a flash on September 19th, 1994, that de Shalit’s theory, if generalised,
could be used together with duality to glue the Hecke rings at suitable auxiliary
levels into a power series ring. I had unexpectedly found the missing key to my
old abandoned approach. It was the old idea of picking q
i
’s with q
i
≡ 1mod p
n
i
and n
i
→ ∞ as i → ∞ that I used to achieve the limiting process. The switch
to the special primes of Chapter 3 had made all this possible.
After I communicated the argument to Taylor, we spent the next few days
making sure of the details. the full argument, together with the deduction of
the complete intersection property, is given in [TW].
In conclusion the key breakthrough in the proof had been the realization
in the spring of 1991 that the two invariants introduced in the appendix could
be used to relate the deformation rings and the Hecke rings. In effect the η-
454 ANDREW JOHN WILES
invariant could be used to count Galois representations. The last step after the
June, 1993, announcement, though elusive, was but the conclusion of a long
process whose purpose was to replace, in the ring-theoretic setting, the methods
based on Iwasawa theory by methods based on the use of auxiliary primes.
One improvement that I have not included but which might be used to
simplify some of Chapter 2 is the observation of Lenstra that the criterion for
Gorenstein rings to be complete intersections can be extended to more general
rings which are finite and free as Z
p
-modules. Faltings has pointed out an
improvement, also not included, which simplifies the argument in Chapter 3
and [TW]. This is however explained in the appendix to [TW].
It is a pleasure to thank those who read carefully a first draft of some of this
paper after the Cambridge conference and particularly N. Katz who patiently
answered many questions in the course of my work on Euler systems, and
together with Illusie read critically the Euler system argument. Their questions
led to my discovery of the problem with it. Katz also listened critically to my
first attempts to correct it in the fall of 1993. I am grateful also to Taylor for
his assistance in analyzing in depth the Euler system argument. I am indebted
to F. Diamond for his generous assistance in the preparation of the final version
of this paper. In addition to his many valuable suggestions, several others also
made helpful comments and suggestions especially Conrad, de Shalit, Faltings,
Ribet, Rubin, Skinner and Taylor.I am most grateful to H. Darmon for his
encouragement to reconsider my old argument. Although I paid no heed to his
advice at the time, it surely left its mark.
Table of Contents
Chapter 1 1. Deformations of Galois representations
2. Some computations of cohomology groups
3. Some results on subgroups of GL
2
(k)
Chapter 2 1. The Gorenstein property
2. Congruences between Hecke rings
3. The main conjectures
Chapter 3 Estimates for the Selmer group
Chapter 4 1. The ordinary CM case
2. Calculation of η
Chapter 5 Application to elliptic curves
Appendix
References
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 455
Chapter 1
This chapter is devoted to the study of certain Galois representations.
In the first section we introduce and study Mazur’s deformation theory and
discuss various refinements of it. These refinements will be needed later to
make precise the correspondence between the universal deformation rings and
the Hecke rings in Chapter 2. The main results needed are Proposition 1.2
which is used to interpret various generalized cotangent spaces as Selmer groups
and (1.7) which later will be used to study them. At the end of the section we
relate these Selmer groups to ones used in the Bloch-Kato conjecture, but this
connection is not needed for the proofs of our main results.
In the second section we extract from the results of Poitou and Tate on
Galois cohomology certain general relations between Selmer groups as Σ varies,
as well as between Selmer groups and their duals. The most important obser-
vation of the third section is Lemma 1.10(i) which guarantees the existence of
the special primes used in Chapter 3 and [TW].
1. Deformations of Galois representations
Let p be an odd prime. Let Σ be a finite set of primes including p and
let Q
Σ
be the maximal extension of Q unramified outside this set and ∞.
Throughout we fix an embedding of Q, and so also of Q
Σ
, in C. We will also
fix a choice of decomposition group D
q
for all primes q in Z. Suppose that k
is a finite field characteristic p and that
(1.1) ρ
0
: Gal(Q
Σ
/Q) → GL
2
(k)
is an irreducible representation. In contrast to the introduction we will assume
in the rest of the paper that ρ
0
comes with its field of definition k. Suppose
further that det ρ
0
is odd. In particular this implies that the smallest field of
definition for ρ
0
is given by the field k
0
generated by the traces but we will not
assume that k = k
0
. It also implies that ρ
0
is absolutely irreducible. We con-
sider the deformation [ρ] to GL
2
(A) of ρ
0
in the sense of Mazur [Ma1]. Thus
if W (k) is the ring of Witt vectors of k, A is to be a complete Noeterian local
W (k)-algebra with residue field k and maximal ideal m, and a deformation [ρ]
is just a strict equivalence class of homomorphisms ρ : Gal(Q
Σ
/Q) → GL
2
(A)
such that ρ mod m = ρ
0
, two such homomorphisms being called strictly equiv-
alent if one can be brought to the other by conjugation by an element of
ker : GL
2
(A) → GL
2
(k). We often simply write ρ instead of [ρ] for the
equivalent class.
456 ANDREW JOHN WILES
We will restrict our choice of ρ
0
further by assuming that either:
(i) ρ
0
is ordinary; viz., the restriction of ρ
0
to the decomposition group D
p
has (for a suitable choice of basis) the form
(1.2) ρ
0
|
D
p
≈
χ
1
∗
0 χ
2
where χ
1
and χ
2
are homomorphisms from D
p
to k
∗
with χ
2
unramified.
Moreover we require that χ
1
̸= χ
2
. We do allow here that ρ
0
|
D
p
be
semisimple. (If χ
1
and χ
2
are both unramified and ρ
0
|
D
p
is semisimple
then we fix our choices of χ
1
and χ
2
once and for all.)
(ii) ρ
0
is flat at p but not ordinary (cf. [Se1] where the terminology finite is
used); viz., ρ
0
|
D
p
is the representation associated to a finite flat group
scheme over Z
p
but is not ordinary in the sense of (i). (In general when we
refer to the flat case we will mean that ρ
0
is assumed not to be ordinary
unless we specify otherwise.) We will assume also that det ρ
0
|
I
p
= ω
where I
p
is an inertia group at p and ω is the Teichm¨uller character
giving the action on p
th
roots of unity.
In case (ii) it follows from results of Raynaud that ρ
0
|
D
p
is absolutely
irreducible and one can describe ρ
0
|
I
p
explicitly. For extending a Jordan-H¨older
series for the representation space (as an I
p
-module) to one for finite flat group
schemes (cf. [Ray 1]) we observe first that the trivial character does not occur on
a subquotient, as otherwise (using the classification of Oort-Tate or Raynaud)
the group scheme would be ordinary. So we find by Raynaud’s results, that
ρ
0
|
I
p
⊗
k
¯
k ≃ ψ
1
⊕ ψ
2
where ψ
1
and ψ
2
are the two fundamental characters of
degree 2 (cf. Corollary 3.4.4 of [Ray1]). Since ψ
1
and ψ
2
do not extend to
characters of Gal(
¯
Q
p
/Q
p
), ρ
0
|
D
p
must be absolutely irreducible.
We sometimes wish to make one of the following restrictions on the
deformations we allow:
(i) (a) Selmer deformations. In this case we assume that ρ
0
is ordinary, with no-
tion as above, and that the deformation has a representative
ρ : Gal(Q
Σ
/Q) → GL
2
(A) with the property that (for a suitable choice
of basis)
ρ|
D
p
≈
˜χ
1
∗
0 ˜χ
2
with ˜χ
2
unramified, ˜χ ≡ χ
2
mod m, and det ρ|
I
p
= εω
−1
χ
1
χ
2
where
ε is the cyclotomic character, ε : Gal(Q
Σ
/Q) → Z
∗
p
, giving the action
on all p-power roots of unity, ω is of order prime to p satisfying ω ≡ ε
mod p, and χ
1
and χ
2
are the characters of (i) viewed as taking values in
k
∗
↩→ A
∗
.
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