/qa/ - Question & Answer

Bump

>>903095 (OP)
This has no solutions due to Fermat's Last Theorem which states

> no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

.

>>903120
Nuqafagcacas doesn't even know… There is a proof of the solution but I can't write it here because of the character limit

im trans btw

>>903124
nusois…

Introduction

An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form X₀(N). Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a functional equation of the standard type. If an elliptic curve over Q with a given j-invariant is modular then it is easy to see that all elliptic curves with the same j-invariant are modular (in which case we say that the j-invariant is modular). A well-known conjecture which grew out of the work of Shimura and Taniyama in the 1950’s and 1960’s asserts that every elliptic curve over Q is modular. However, it only became widely known through its publication in a paper of Weil in 1967 [We] (as an exercise for the interested reader!), in which, moreover, Weil gave conceptual evidence for the conjecture. Although it had been numerically verified in many cases, prior to the results described in this paper it had only been known that finitely many j-invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should
imply Fermat’s Last Theorem. The precise mechanism relating the two was formulated by Serre as the ε-conjecture and this was then proved by Ribet in the summer of 1986. Ribet’s result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat’s Last Theorem.

Our approach to the study of elliptic curves is via their associated Galois representations. Suppose that ρₚ is the representation of Gal(Q¯ /Q) on the p-division points of an elliptic curve over Q, and suppose for the moment that ρ₃ is irreducible. The choice of 3 is critical because a crucial theorem of Langlands and Tunnell shows that if ρ₃ is irreducible then it is also modular. We then proceed by showing that under the hypothesis that ρ3 is semistable at 3, together with some milder restrictions on the ramification of ρ3 at the other primes, every suitable lifting of ρ3 is modular. To do this we link the problem, via some novel arguments from commutative algebra, to a class number problem of a well-known type. This we then solve with the help of the paper [TW]. This suffices to prove the modularity of E as it is known that E is modular if and only if the associated 3-adic representation is modular.

nusois

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 ε-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 formula 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 ρ₃ be irreducible at 3 is bypassed by means of an intriguing argument with families of elliptic curves which share a common ρ₅. 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.

Let f be an eigenform associated to the congruence subgroup Γ₁(N) of SL₂(Z) of weight k ≥ 2 and character χ. Thus if Tₙ is the Hecke operator associated to an integer n there is an algebraic integer c(n, f) such that Tₙf = c(n, f)f for each n. We let Kf be the number field generated over Q by the {c(n, f)} together with the values of χ and let Of be its ring of integers. For any prime λ of Of let Of,λ be the completion of Of at λ. The following theorem is due to Eichler and Shimura (for k = 2) and Deligne (for k > 2). The analogous result when k = 1 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 0.1. For each prime p ∈ Z and each prime λ|p of Of there is a continuous representation
ρf,λ : Gal(Q¯ /Q) → GL₂(Of,λ)
which is unramified outside the primes dividing N p and such that for all primes
q ∤ N p,
trace ρf,λ(Frob q) = c(q, f), det ρf,λ(Frob q) = χ(q)qᵏ⁻¹.

We will be concerned with trying to prove results in the opposite direction,
that is to say, with establishing criteria under which a λ-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 λ-adic
representations except that the proof may be easier for some λ than for others.
Assume
ρ0 : Gal(Q¯ /Q) −→ GL2(F¯ p)
is a continuous representation with values in the algebraic closure of a finite
field of characteristic p and that det ρ0 is odd. We say that ρ0 is modular
if ρ0 and ρf,λ mod λ are isomorphic over F¯ p for some f and λ and some
embedding of Of /λ in F¯ p. Serre has conjectured that every irreducible ρ0 of
odd determinant is modular. Very little is known about this conjecture except
when the image of ρ0 in PGL2(F¯ p) is dihedral, A4 or S4. In the dihedral case
it is true and due (essentially) to Hecke, and in the A4 and S4 cases it is again
true and due primarily to Langlands, with one important case due to Tunnell
(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
ρf,λ = ρf,λ mod λ is independent of the choice of lattice in K2
f,λ.

If O is the ring of integers of a local field (containing Qp) we will say that
ρ : Gal(Q¯ /Q) −→ GL2(O) is a lifting of ρ0 if, for a specified embedding of the
residue field of O in F¯ p, ρ¯ and ρ0 are isomorphic over F¯ p. Our point of view
will be to assume that ρ0 is modular and then to attempt to give conditions
under which a representation ρ lifting ρ0 comes from a modular form in the
sense that ρ
ρf,λ over Kf,λ for some f, λ. We will restrict our attention to
two cases:
(I) ρ0 is ordinary (at p) by which we mean that there is a one-dimensional
subspace of F¯ 2
p, 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.
(II) ρ0 is flat (at p), meaning that as a representation of a decomposition
group at p, ρ0 is equivalent to one that arises from a finite flat group
scheme over Zp, and det ρ0 restricted to an inertia group at p is the
cyclotomic character.
We say similarly that ρ is ordinary (at p), if viewed as a representation to Q¯ 2
p,
there is a one-dimensional subspace of Q¯ 2
p stable under a decomposition group
at p and such that the action on the quotient space is unramified.
Let ε : Gal(Q¯ /Q) −→ Z×
p 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 Drinfield 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 p 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 ρ0 is modular.)
Conjecture. Suppose that ρ : Gal(Q¯ /Q) −→ GL2(O) is an irreducible
lifting of ρ0 and that ρ is unramified outside of a finite set of primes. There
are two cases:
(i) Assume that ρ0 is ordinary. Then if ρ is ordinary and det ρ = εk−1χ for
some integer k ≥ 2 and some χ of finite order, ρ comes from a modular
form.
(ii) Assume that ρ0 is flat and that p is odd. Then if ρ restricted to a decomposition group at p is equivalent to a representation on a p-divisible
group, again ρ comes from a modular form.

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 GL2.
We prove two results concerning this conjecture. The first includes the
hypothesis that ρ0 is modular. Here and for the rest of this paper we will
assume that p is an odd prime.
Theorem 0.2. Suppose that ρ0 is irreducible and satisfies either (I) or
(II) above. Suppose also that ρ0 is modular and that
(i) ρ0 is absolutely irreducible when restricted to Q

(−1) p−1
2 p
�
.
(ii) If q ≡ −1 mod p is ramified in ρ0 then either ρ0|Dq is reducible over
the algebraic closure where Dq is a decomposition group at q or ρ0|Iq is
absolutely irreducible where Iq is an inertia group at q.
Then any representation ρ as in the conjecture does indeed come from a modular form.
The only condition which really seems essential to our method is the requirement that ρ0 be modular.
The most interesting case at the moment is when p = 3 and ρ0 can be defined over F3. Then since PGL2(F3)
S4 every such representation is modular
by the theorem of Langlands and Tunnell mentioned above. In particular, every representation into GL2(Z3) 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 ρ0 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) ρ0 is absolutely irreducible when restricted to Q�√−3 �
.
(iii) For any q ≡ −1 mod 3 either ρ0|Dq is reducible over the algebraic closure
or ρ0|Iq 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 X0(N)

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 conductor, satisfies (i) and (iii) but not necessarily (ii). If (ii) fails then in fact ρ0
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 Chapter 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 ‘Fermat’s Last Theorem’. Frey’s suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve y2 = x(x + up)(x − vp)
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 representation 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 up +vp +wp = 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 an + bn = cn.)
The second result we prove about the conjecture does not require the
assumption that ρ0 be modular (since it is already known in this case).

Theorem 0.6. Suppose that ρ0 is irreducible and satisfies the hypothesis
of the conjecture, including (I) above. Suppose further that
(i) ρ0 = IndQ
L κ0 for a character κ0 of an imaginary quadratic extension L
of Q which is unramified at p.
(ii) det ρ0|Ip = 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 #-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 #-adic representations. I began with the assumption
that the reduction of a given ordinary #-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
# = 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 # 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 ρ3, 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 GL2(Z/3nZ) representation would be
modular for each n. At this time I considered only the ordinary case. This led
quickly to the study of Hi
(Gal(F∞/Q), Wf ) 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 ρ3 and Wf 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(QΣ/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

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
Zp-extension by a similar analysis based on a choice of infinitely many distinct
primes qi ≡ 1 mod pni with ni → ∞ 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 Wf , 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
ρp 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 families of Galois representations. In an attempt to understand this, Mazur had
been developing the language of deformations of Galois representations. Moreover, 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 H2 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

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 Zp. 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
Zp-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/p2-
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 argument 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 commutative 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

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 sufficient in this case and, as explained earlier, when translated into the ringtheoretic 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 estimates 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/p2-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 plausible. 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

common ρ5 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 construction 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 complete 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 approach, 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 qi’s with qi ≡ 1mod pni
and ni → ∞ 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-

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 Zp-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. Katzwho 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. Katzalso 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.

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 observation 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 QΣ be the maximal extension of Q unramified outside this set and ∞.
Throughout we fix an embedding of Q, and so also of QΣ, in C. We will also
fix a choice of decomposition group Dq for all primes q in Z. Suppose that k
is a finite field characteristic p and that
(1.1) ρ0 : Gal(QΣ/Q) → GL2(k)
is an irreducible representation. In contrast to the introduction we will assume
in the rest of the paper that ρ0 comes with its field of definition k. Suppose
further that det ρ0 is odd. In particular this implies that the smallest field of
definition for ρ0 is given by the field k0 generated by the traces but we will not
assume that k = k0. It also implies that ρ0 is absolutely irreducible. We consider the deformation [ρ] to GL2(A) of ρ0 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(QΣ/Q) → GL2(A)
such that ρ mod m = ρ0, two such homomorphisms being called strictly equivalent if one can be brought to the other by conjugation by an element of
ker : GL2(A) → GL2(k). We often simply write ρ instead of [ρ] for the
equivalent class.

We will restrict our choice of ρ0 further by assuming that either:
(i) ρ0 is ordinary; viz., the restriction of ρ0 to the decomposition group Dp
has (for a suitable choice of basis) the form
(1.2) ρ0|Dp ≈
χ1 ∗
0 χ2
�
where χ1 and χ2 are homomorphisms from Dp to k∗ with χ2 unramified.
Moreover we require that χ1 �= χ2. We do allow here that ρ0|Dp be
semisimple. (If χ1 and χ2 are both unramified and ρ0|Dp is semisimple
then we fix our choices of χ1 and χ2 once and for all.)
(ii) ρ0 is flat at p but not ordinary (cf. [Se1] where the terminology finite is
used); viz., ρ0|Dp is the representation associated to a finite flat group
scheme over Zp but is not ordinary in the sense of (i). (In general when we
refer to the flat case we will mean that ρ0 is assumed not to be ordinary
unless we specify otherwise.) We will assume also that det ρ0|Ip = ω
where Ip is an inertia group at p and ω is the Teichm¨uller character
giving the action on pth roots of unity.
In case (ii) it follows from results of Raynaud that ρ0|Dp is absolutely
irreducible and one can describe ρ0|Ip explicitly. For extending a Jordan-H¨older
series for the representation space (as an Ip-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
ρ0|Ip ⊗
k
¯
k
ψ1 ⊕ ψ2 where ψ1 and ψ2 are the two fundamental characters of
degree 2 (cf. Corollary 3.4.4 of [Ray1]). Since ψ1 and ψ2 do not extend to
characters of Gal(Q¯ p/Qp), ρ0|Dp 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 ρ0 is ordinary, with notion as above, and that the deformation has a representative
ρ : Gal(QΣ/Q) → GL2(A) with the property that (for a suitable choice
of basis)
ρ|Dp ≈
χ˜1 ∗
0 ˜χ2
�
with ˜χ2 unramified, ˜χ ≡ χ2 mod m, and det ρ|Ip = εω−1χ1χ2 where
ε is the cyclotomic character, ε : Gal(QΣ/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 χ1 and χ2 are the characters of (i) viewed as taking values in
k∗ ,→ A∗.

(i) (b) Ordinary deformations. The same as in (i)(a) but with no condition on
the determinant.
(i) (c) Strict deformations. This is a variant on (i) (a) which we only use when
ρ0|Dp is not semisimple and not flat (i.e. not associated to a finite flat
group scheme). We also assume that χ1χ−1
2 = ω in this case. Then a
strict deformation is as in (i)(a) except that we assume in addition that
(˜χ1/χ˜2)|Dp = ε.
(ii) Flat (at p) deformations. We assume that each deformation ρ to GL2(A)
has the property that for any quotient A/a of finite order ρ|Dp mod a
is the Galois representation associated to the Q¯ p-points of a finite flat
group scheme over Zp.
In each of these four cases, as well as in the unrestricted case (in which we
impose no local restriction at p) one can verify that Mazur’s use of Schlessinger’s
criteria [Sch] proves the existence of a universal deformation
ρ : Gal(QΣ/Q) → GL2(R).
In the ordinary and restricted case this was proved by Mazur and in the
flat case by Ramakrishna [Ram]. The other cases require minor modifications
of Mazur’s argument. We denote the universal ring RΣ in the unrestricted
case and Rse
Σ , Rord
Σ , Rstr
Σ , Rf
Σ in the other four cases. We often omit the Σ if the
context makes it clear.
There are certain generalizations to all of the above which we will also
need. The first is that instead of considering W(k)-algebras A we may consider
O-algebras for O the ring of integers of any local field with residue field k. If
we need to record which O we are using we will write RΣ,O etc. It is easy to
see that the natural local map of local O-algebras
RΣ,O → RΣ ⊗
W(k)
O
is an isomorphism because for functorial reasons the map has a natural section
which induces an isomorphism on Zariski tangent spaces at closed points, and
one can then use Nakayama’s lemma. Note, however, hat if we change the
residue field via i :,→ k� then we have a new deformation problem associated
to the representation ρ�
0 = i ◦ ρ0. There is again a natural map of W(k�
)-
algebras
R(ρ�
0) → R ⊗
W(k)
W(k�
)
which is an isomorphism on Zariski tangent spaces. One can check that this
is again an isomorphism by considering the subring R1 of R(ρ�
0) defined as the
subring of all elements whose reduction modulo the maximal ideal lies in k.
Since R(ρ�
0) is a finite R1-module, R1 is also a complete local Noetherian ring

with residue field k. The universal representation associated to ρ�
0 is defined
over R1 and the universal property of R then defines a map R → R1. So we
obtain a section to the map R(ρ�
0) → R ⊗
W(k)
W(k�
) and the map is therefore
an isomorphism. (I am grateful to Faltings for this observation.) We will also
need to extend the consideration of O-algebras tp the restricted cases. In each
case we can require A to be an O-algebra and again it is easy to see that
R·
Σ,O
R·
Σ ⊗
W(k)
O in each case.
The second generalization concerns primes q �= p which are ramified in ρ0.
We distinguish three special cases (types (A) and (C) need not be disjoint):
(A) ρ0|Dq = ( χ1 ∗
χ2 ) for a suitable choice of basis, with χ1 and χ2 unramified,
χ1χ−1
2 = ω and the fixed space of Iq of dimension 1,
(B) ρ0|Iq = ( χq
0
0
1 ), χq �= 1, for a suitable choice of basis,
(C) H1(Qq, Wλ) = 0 where Wλ is as defined in (1.6).
Then in each case we can define a suitable deformation theory by imposing
additional restrictions on those we have already considered, namely:
(A) ρ|Dq = ( ψ1 ∗
ψ2 ) for a suitable choice of basis of A2 with ψ1 and ψ2 unramified and ψ1ψ−1
2 = ε;
(B) ρ|Iq = ( χq
0
0
1 ) for a suitable choice of basis (χq of order prime to p, so the
same character as above);
(C) det ρ|Iq = det ρ0|Iq , i.e., of order prime to p.
Thus if M is a set of primes in Σ distinct from p and each satisfying one of
(A), (B) or (C) for ρ0, we will impose the corresponding restriction at each
prime in M.
Thus to each set of data D = {·, Σ, O,M} where · is Se, str, ord, flat or
unrestricted, we can associate a deformation theory to ρ0 provided
(1.3) ρ0 : Gal(QΣ/Q) → GL2(k)
is itself of type D and O is the ring of integers of a totally ramified extension
of W(k); ρ0 is ordinary if · is Se or ord, strict if · is strict and flat if · is fl
(meaning flat); ρ0 is of type M, i.e., of type (A), (B) or (C) at each ramified
primes q �= p, q ∈ M. We allow different types at different q’s. We will refer
to these as the standard deformation theories and write RD for the universal
ring associated to D and ρD for the universal deformation (or even ρ if D is
clear from the context).
We note here that if D = (ord, Σ, O,M) and D� = (Se, Σ, O,M) then
there is a simple relation between RD and RD . Indeed there is a natural map

RD → RD by the universal property of RD, and its kernel is a principal ideal
generated by T = ε−1(γ) det ρD(γ) − 1 where γ ∈ Gal(QΣ/Q) is any element
whose restriction to Gal(Q∞/Q) is a generator (where Q∞ is the Zp-extension
of Q) and whose restriction to Gal(Q(ζNp )/Q) is trivial for any N prime to p
with ζN ∈ QΣ, ζN being a primitive Nth root of 1:
(1.4) RD/T
R�
D.
It turns out that under the hypothesis that ρ0 is strict, i.e. that ρ0|Dp
is not associated to a finite flat group scheme, the deformation problems in
(i)(a) and (i)(c) are the same; i.e., every Selmer deformation is already a strict
deformation. This was observed by Diamond. the argument is local, so the
decomposition group Dp could be replaced by Gal(Q¯ p/Q).
Proposition 1.1 (Diamond). Suppose that π : Dp → GL2(A) is a continuous representation where A is an Artinian local ring with residue field k, a
finite field of characteristic p. Suppose π ≈ ( χ1ε
0
∗
χ2 ) with χ1 and χ2 unramified
and χ1 �= χ2. Then the residual representation π¯ is associated to a finite flat
group scheme over Zp.
Proof (taken from [Dia, Prop. 6.1]). We may replace π by π ⊗ χ−1
2 and
we let ϕ = χ1χ−1
2 . Then π ∼= ( ϕε
0
t
1 ) determines a cocycle t : Dp → M(1) where
M is a free A-module of rank one on which Dp acts via ϕ. Let u denote the
cohomology class in H1(Dp, M(1)) defined by t, and let u0 denote its image
in H1(Dp, M0(1)) where M0 = M/mM. Let G = ker ϕ and let F be the fixed
field of G (so F is a finite unramified extension of Qp). Choose n so that pnA
= 0. Since H2(G, µpr → H2(G, µps ) is injective for r ≤ s, we see that the
natural map of A[Dp/G]-modules H1(G, µpn ⊗Zp M) → H1(G, M(1)) is an
isomorphism. By Kummer theory, we have H1(G, M(1)) ∼= F ×/(F ×)pn
⊗Zp M
as Dp-modules. Now consider the commutative diagram
H1(G, M(1))Dp ∼ −−−−→((F ×/(F ×)pn
⊗Zp M)Dp−−−−→MDp




�




�




� ,
H1(G, M0(1)) ∼ −−−−→ (F ×/(F ×)p) ⊗Fp M0 −−−−→ M0
where the right-hand horizontal maps are induced by vp : F × → Z. If ϕ �= 1,
then MDp ⊂ mM, so that the element res u0 of H1(G, M0(1)) is in the image
of (O×
F /(O×
F )p) ⊗Fp M0. But this means that ¯π is “peu ramifi´e” in the sense of
[Se] and therefore ¯π comes from a finite flat group scheme. (See [E1, (8.20].)
Remark. Diamond also observes that essentially the same proof shows
that if π : Gal(Q¯ q/Qq) → GL2(A), where A is a complete local Noetherian

ring with residue field k, has the form π|Iq
∼= ( 1
0
∗
1 ) with ¯π ramified then π is
of type (A).
Globally, Proposition 1.1 says that if ρ0 is strict and if D = (Se, Σ, O,M)
and D� = (str, Σ, O,M) then the natural map RD → RD is an isomorphism.
In each case the tangent space of RD may be computed as in [Ma1]. Let
λ be a uniformizer for O and let Uλ
k2 be the representation space for ρ0.
(The motivation for the subscript λ will become apparent later.) Let Vλ be the
representation space of Gal(QΣ/Q) on Adρ0 = Homk(Uλ, Uλ)
M2(k). Then
there is an isomorphism of k-vector spaces (cf. the proof of Prop. 1.2 below)
(1.5) Homk(mD/(m2
D, λ), k)
H1
D(QΣ/Q, Vλ)
where H1
D(QΣ/Q, Vλ) is a subspace of H1(QΣ/Q, Vλ) which we now describe
and mD is the maximal ideal of RC alD. It consists of the cohomology classes
which satisfy certain local restrictions at p and at the primes in M. We call
mD/(m2
D, λ) the reduced cotangent space of RD.
We begin with p. First we may write (since p �= 2), as k[Gal(QΣ/Q)]-
modules,
(1.6) Vλ = Wλ ⊕ k, where Wλ = {f ∈ Homk(Uλ, Uλ) : tracef = 0}

(Sym2 ⊗ det−1)ρ0
and k is the one-dimensional subspace of scalar multiplications. Then if ρ0
is ordinary the action of Dp on Uλ induces a filtration of Uλ and also on Wλ
and Vλ. Suppose we write these 0 ⊂ U0
λ ⊂ Uλ, 0 ⊂ W0
λ ⊂ W1
λ ⊂ Wλ and
0 ⊂ V 0
λ ⊂ V 1
λ ⊂ Vλ. Thus U0
λ is defined by the requirement that Dp act on it
via the character χ1 (cf. (1.2)) and on Uλ/U0
λ via χ2. For Wλ the filtrations
are defined by
W1
λ = {f ∈ Wλ : f(U0
λ) ⊂ U0
λ},
W0
λ = {f ∈ W1
λ : f = 0 on U0
λ},
and the filtrations for Vλ are obtained by replacing W by V . We note that
these filtrations are often characterized by the action of Dp. Thus the action
of Dp on W0
λ is via χ1/χ2; on W1
λ/W0
λ it is trivial and on Qλ/W1
λ it is via
χ2/χ1. These determine the filtration if either χ1/χ2 is not quadratic or ρ0|Dp
is not semisimple. We define the k-vector spaces
V ord
λ = {f ∈ V 1
λ : f = 0 in Hom(Uλ/U0
λ, Uλ/U0
λ)},
H1
Se(Qp, Vλ) = ker{H1(Qp, Vλ) → H1(Qunr
p , Vλ/W0
λ)},
H1
ord(Qp, Vλ) = ker{H1(Qp, Vλ) → H1(Qunr
p , Vλ/V ord
λ )},
H1
str(Qp, Vλ) = ker{H1(Qp, Vλ) → H1(Qp, Wλ/W0
λ) ⊕ H1(Qunr
p , k)}.

In the Selmer case we make an analogous definition for H1
Se(Qp, Wλ) by
replacing Vλ by Wλ, and similarly in the strict case. In the flat case we use
the fact that there is a natural isomorphism of k-vector spaces
H1(Qp, Vλ) → Ext1
k[Dp](Uλ, Uλ)
where the extensions are computed in the category of k-vector spaces with local
Galois action. Then H1
f (Qp, Vλ) is defined as the k-subspace of H1(Qp, Vλ)
which is the inverse image of Ext1
fl(G, G), the group of extensions in the category of finite flat commutative group schemes over Zp killed by p, G being the
(unique) finite flat group scheme over Zp associated to Uλ. By [Ray1] all such
extensions in the inverse image even correspond to k-vector space schemes. For
more details and calculations see [Ram].
For q different from p and q ∈ M we have three cases (A), (B), (C). In
case (A) there is a filtration by Dq entirely analogous to the one for p. We
write this 0 ⊂ W0,q
λ ⊂ W1,q
λ ⊂ Wλ and we set
H1
Dq (Qq, Vλ) =



ker : H1(Qq, Vλ
→ H1(Qq, Wλ/W0,q
λ ) ⊕ H1(Qunr
q , k) in case (A)
ker : H1(Qq, Vλ)
→ H1(Qunr
q , Vλ) in case (B) or (C).
Again we make an analogous definition for H1
Dq (Qq, Wλ) by replacing Vλ
by Wλ and deleting the last term in case (A). We now define the k-vector
space H1
D(QΣ/Q, Vλ) as
H1
D(QΣ/Q, Vλ) = {α ∈ H1(QΣ/Q, Vλ) : αq ∈ H1
Dq (Qq, Vλ) for all q ∈ M,
αq ∈ H1
∗ (Qp, Vλ)}
where ∗ is Se, str, ord, fl or unrestricted according to the type of D. A similar
definition applies to H1
D(QΣ/Q, Wλ) if · is Selmer or strict.
Now and for the rest of the section we are going to assume that ρ0 arises
from the reduction of the λ-adic representation associated to an eigenform.
More precisely we assume that there is a normalized eigenform f of weight 2
and level N, divisible only by the primes in Σ, and that there ia a prime λ
of Of such that ρ0 = ρf,λ mod λ. Here Of is the ring of integers of the field
generated by the Fourier coefficients of f so the fields of definition of the two
representations need not be the same. However we assume that k ⊇ Of,λ/λ
and we fix such an embedding so the comparison can be made over k. It will
be convenient moreover to assume that if we are considering ρ0 as being of
type D then D is defined using O-algebras where O⊇Of,λ is an unramified
extension whose residue field is k. (Although this condition is unnecessary, it
is convenient to use λ as the uniformizer for O.) Finally we assume that ρf,λ

itself is of type D. Again this is a slight abuse of terminology as we are really
considering the extension of scalars ρf,λ ⊗
Of,λ
O and not ρf,λ itself, but we will
do this without further mention if the context makes it clear. (The analysis of
this section actually applies to any characteristic zero lifting of ρ0 but in all
our applications we will be in the more restrictive context we have described
here.)
With these hypotheses there is a unique local homomorphism RD → O
of O-algebras which takes the universal deformation to (the class of) ρf,λ. Let
pD = ker : RD → O. Let K be the field of fractions of O and let Uf = (K/O)2
with the Galois action taken from ρf,λ. Similarly, let Vf = Adρf,λ ⊗O K/O

(K/O)4 with the adjoint representation so that
Vf
Wf ⊕ K/O
where Wf has Galois action via Sym2ρf,λ ⊗ det ρ−1
f,λ and the action on the
second factor is trivial. Then if ρ0 is ordinary the filtration of Uf under the
Adρ action of Dp induces one on Wf which we write 0 ⊂ W0
f ⊂ W1
f ⊂ Wf .
Often to simplify the notation we will drop the index f from W1
f , Vf etc. There
is also a filtration on Wλn = {ker λn : Wf → Wf } given by Wi
λn = Wλn
∩ Wi
(compatible with our previous description for n = 1). Likewise we write Vλn
for {ker λn : Vf → Vf }.
We now explain how to extend the definition of H1
D to give meaning to
H1
D(QΣ/Q, Vλn ) and H1
D(QΣ/Q, V ) and these are O/λn and O-modules, respectively. In the case where ρ0 is ordinary the definitions are the same with
Vλn or V replacing Vλ and O/λn or K/O replacing k. One checks easily that
as O-modules
(1.7) H1
D(QΣ/Q, Vλn )
H1
D(QΣ/Q, V )λn ,
where as usual the subscript λn denotes the kernel of multiplication by λn.
This just uses the divisibility of H0(QΣ/Q, V ) and H0(Qp, W/W0) in the
strict case. In the Selmer case one checks that for m>n the kernel of
H1(Qunr
p , Vλn /W0
λn ) → H1(Qunr
p , Vλm/W0
λm)
has only the zero element fixed under Gal(Qunr
p /Qp) and the ord case is similar.
Checking conditions at q ∈ M is dome with similar arguments. In the Selmer
and strict cases we make analogous definitions with Wλn in place of Vλn and
W in place of V and the analogue of (1.7) still holds.
We now consider the case where ρ0 is flat (but not ordinary). We claim
first that there is a natural map of O-modules
(1.8) H1(Qp, Vλn ) → Ext1
O[Dp](Uλm, Uλn )
for each m ≥ n where the extensions are of O-modules with local Galois
action. To describe this suppose that α ∈ H1(Qp, Vλn ). Then we can associate to α a representation ρα : Gal(Q¯ p/Qp) → GL2(On[ε]) (where On[ε] =

O[ε]/(λnε, ε2)) which is an O-algebra deformation of ρ0 (see the proof of Proposition 1.1 below). Let E = On[ε]
2 where the Galois action is via ρα. Then there
is an exact sequence
0 −→ εE/λm −→ E/λm −→ (E/ε)/λm −→ 0
|� |�
Uλn Uλm
and hence an extension class in Ext1(Uλm, Uλn ). One checks now that (1.8)
is a map of O-modules. We define H1
f (Qp, Vλn ) to be the inverse image of
Ext1
fl(Uλn , Uλn ) under (1.8), i.e., those extensions which are already extensions
in the category of finite flat group schemes Zp. Observe that Ext1
fl(Uλn , Uλn )∩
Ext1
O[Dp](Uλn , Uλn ) is an O-module, so H1
f (Qp, Vλn ) is seen to be an O-submodule of H1(Qp, Vλn ). We observe that our definition is equivalent to requiring that the classes in H1
f (Qp, Vλn ) map under (1.8) to Ext1
fl(Uλm, Uλn ) for all
m ≥ n. For if em is the extension class in Ext1(Uλm, Uλn ) then em ,→ en ⊕Uλm
as Galois-modules and we can apply results of [Ray1] to see that em comes
from a finite flat group scheme over Zp if en does.
In the flat (non-ordinary) case ρ0|Ip is determined by Raynaud’s results as
mentioned at the beginning of the chapter. It follows in particular that, since
ρ0|Dp is absolutely irreducible, V (Qp = H0(Qp, V ) is divisible in this case
(in fact V (Qp)
KT/O). This H1(Qp, Vλn )
H1(Qp, V )λn and hence we can
define
H1
f (Qp, V ) =

∞
n=1
H1
f (Qp, Vλn ),
and we claim that H1
f (Qp, V )λn
H1
f (Qp, Vλn ). To see this we have to compare
representations for m ≥ n,
ρn,m : Gal(Q¯ p/Qp) −→ GL2(On[ε]/λm)
�
�
�


�ϕm,n
ρm,m : Gal(Q¯ p/Qp) −→ GL2(Om[ε]/λm)
where ρn,m and ρm,m are obtained from αn ∈ H1(Qp,VXλn ) and im(αn) ∈
H1(Qp, Vλm) and ϕm,n : a+bε → a+λm−nbε. By [Ram, Prop 1.1 and Lemma
2.1] if ρn,m comes from a finite flat group scheme then so does ρm,m. Conversely
ϕm,n is injective and so ρn,m comes from a finite flat group scheme if ρm,m does;
cf. [Ray1]. The definitions of H1
D(QΣ/Q, Vλn ) and H1
D(QΣ/Q, V ) now extend
to the flat case and we note that (1.7) is also valid in the flat case.
Still in the flat (non-ordinary) case we can again use the determination
of ρ0|Ip to see that H1(Qp, V ) is divisible. For it is enough to check that
H2(Qp, Vλ) = 0 and this follows by duality from the fact that H0(Qp, V ∗
λ )=0

where V ∗
λ = Hom(Vλ, µp) and µp is the group of pth roots of unity. (Again
this follows from the explicit form of ρ0|
Dp .) Much subtler is the fact that
H1
f (Qp, V ) is divisible. This result is essentially due to Ramakrishna. For,
using a local version of Proposition 1.1 below we have that
HomO(pR/p2
R, K/O)
H1
f (Qp, V )
where R is the universal local flat deformation ring for ρ0|Dp and O-algebras.
(This exists by Theorem 1.1 of [Ram] because ρ0|Dp is absolutely irreducible.)
Since R
Rfl ⊗
W(k)
O where Rfl is the corresponding ring for W(k)-algebras
the main theorem of [Ram, Th. 4.2] shows that R is a power series ring and
the divisibility of H1
f (Qp, V ) then follows. We refer to [Ram] for more details
about Rfl.
Next we need an analogue of (1.5) for V . Again this is a variant of standard
results in deformation theory and is given (at least for D = (ord, Σ, W(k), φ)
with some restriction on χ1, χ2 in i(a)) in [MT, Prop 25].
Proposition 1.2. Suppose that ρf,λ is a deformation of ρ0 of type
D = (·, Σ, O,M) with O an unramified extension of Of,λ. Then as O-modules
HomO(pD/p2
D, K/O)
H1
D(QΣ/Q, V ).
Remark. The isomorphism is functorial in an obvious way if one changes
D to a larger D�
.
Proof. We will just describe the Selmer case with M = φ as the other
cases use similar arguments. Suppose that α is a cocycle which represents a
cohomology class in H1
Se(QΣ/Q, Vλn ). Let On[ε] denote the ring O[ε]/(λnε, ε2).
We can associate to α a representation
ρα : Gal(QΣ/Q) → GL2(On[ε])
as follows: set ρα(g) = α(g)ρf,λ(g) where ρf,λ(g), a priori in GL2(O), is viewed
in GL2(On[ε]) via the natural mapping O→On[ε]. Here a basis for O2
is chosen so that the representation ρf,λ on the decomposition group Dp ⊂
Gal(QΣ/Q) has the upper triangular form of (i)(a), and then α(g) ∈ Vλn is
viewed in GL2(On[ε]) by identifying
Vλn

� 1 + yε xε
zε 1 − tε � � = {ker : GL2(On[ε]) → GL2(O)}.
Then
W0
λn =
� 1 xε
1
� �,

W1
λn =
� 1 + yε xε
1 − yε � �,
Wλn =
� 1 + yε xε
zε 1 − yε � �,
and
V 1
λn =
� 1 + yε xε
1 − tε � �.
One checks readily that ρα is a continuous homomorphism and that the deformation [ρα] is unchanged if we add a coboundary to α.
We need to check that [ρα] is a Selmer deformation. Let H =
Gal(Q¯ p/Qunr
p ) and G = Gal(Qunr
p /Qp). Consider the exact sequence of O[G]-
modules
0 → (V 1
λn /W0
λn )
H → (Vλn /W0
λn )
H → X → 0
where X is a submodule of (Vλn /V 1
λn )H. Since the action of p on Vλn /V 1
λn is
via a character which is nontrivial mod λ (it equals χ2χ−1
1 mod λ and χ1 �≡ χ2),
we see that XG = 0 and H1(G, X)=0. Then we have an exact diagram of
O-modules

0



�
H1(G,(V 1
λn /W0
λn )H)
H1(G,(Vλn /W0
λn )H)




�
H1(Qp, Vλn /W0
λn )




�
H1(Qunr
p , Vλn /W0
λn )G.
By hypothesis the image of α is zero in H1(Qunr
p , Vλn /W0
λn )G. Hence it
is in the image of H1(G,(V 1
λn /W0
λn )H). Thus we can assume that it is represented in H1(Qp, Vλn /W0
λn ) by a cocycle, which maps G to V 1
λn /W0
λn ; i.e.,
f(Dp) ⊂ V 1
λn /W0
λn , f(Ip)=0. The difference between f and the image of α is
a coboundary {σ → σµ¯ −µ¯} for some u ∈ Vλn . By subtracting the coboundary
{σ → σu − u} from α globally we get a new α such that α = f as cocycles
mapping G to V 1
λn /W0
λn . Thus α(Dp) ⊂ V 1
λn , α(Ip) ⊂ W0
λn and it is now easy
to check that [ρα] is a Selmer deformation of ρ0.
Since [ρα] is a Selmer deformation there is a unique map of local Oalgebras ϕα : RD → On[ε] inducing it. (If M �= φ we must check the

other conditions also.) Since ρα ≡ ρf,λ mod ε we see that restricting ϕα to pD
gives a homomorphism of O-modules,
ϕα : pD → ε.O/λn
such that ϕα(p2
D)=0. Thus we have defined a map ϕ : α → ϕα,
ϕ : H1
Se(QΣ/Q, Vλn ) → HomO(pD/p2
D, O/λn).
It is straightforward to check that this is a map of O-modules. To check the
injectivity of ϕ suppose that ϕα(pD)=0. Then ϕα factors through RD/pD
O
and being an O-algebra homomorphism this determines ϕα. Thus [ρf,λ]=[ρα].
If A−1ραA = ρf,λ then A mod ε is seen to be central by Schur’s lemma and so
may be taken to be I. A simple calculation now shows that α is a coboundary.
To see that ϕ is surjective choose
Ψ ∈ HomO(pD/p2
D, O/λn).
Then ρΨ : Gal(QΣ/Q) → GL2(RD/(p2
D, ker Ψ)) is induced by a representative
of the universal deformation (chosen to equal ρf,λ when reduced mod pD) and
we define a map αΨ : Gal(QΣ/Q) → Vλn by
αΨ(g) = ρΨ(g)ρf,λ(g)−1 ∈



1 + pD/(p2
D, ker Ψ) pD/(p2
D, ker Ψ)
pD/(p2
D, ker Ψ) 1 + pD/(p2
D, ker Ψ)



⊆ Vλn
where ρf,λ(g) is viewed in GL2(RD/(p2
D, ker Ψ)) via the structural map O →
RD (RD being an O-algebra and the structural map being local because of
the existence of a section). The right-hand inclusion comes from
pD/(p2
D, ker Ψ) Ψ
,→ O/λn ∼
→ (O/λn) · ε
1 → ε.
Then αΨ is really seen to be a continuous cocycle whose cohomology class
lies in H1
Se(QΣ/Q, Vλn ). Finally ϕ(αΨ)=Ψ. Moreover, the constructions are
compatible with change of n, i.e., for Vλn ,→Vλn+1 and λ:O/λn ,→ O/λn+1. �
We now relate the local cohomology groups we have defined to the theory
of Fontaine and in particular to the groups of Bloch-Kato [BK]. We will distinguish these by writing H1
F for the cohomology groups of Bloch-Kato. None
of the results described in the rest of this section are used in the rest of the
paper. They serve only to relate the Selmer groups we have defined (and later
compute) to the more standard versions. Using the lattice associated to ρf,λ we
obtain also a lattice T
O4 with Galois action via Ad ρf,λ. Let V = T ⊗Zp Qp
be associated vector space and identify V with V/T. Let pr : V → V be

the natural projection and define cohomology modules by
H1
F (Qp, V) = ker : H1(Qp, V) → H1(Qp, V ⊗
Qp
Bcrys),
H1
F (Qp, V ) = pr�
H1
F (Qp, V)
�
⊂ H1(Qp, V ),
H1
F (Qp, Vλn )=(jn)
−1
�
H1
F (Qp, V )
�
⊂ H1(Qp, Vλn ),
where jn : Vλn → V is the natural map and the two groups in the definition
of H1
F (Qp, V) are defined using continuous cochains. Similar definitions apply
to V∗ = HomQp (V, Qp(1)) and indeed to any finite-dimensional continuous
p-adic representation space. The reader is cautioned that the definition of
H1
F (Qp, Vλn ) is dependent on the lattice T (or equivalently on V ). Under
certainly conditions Bloch and Kato show, using the theory of Fontaine and
Lafaille, that this is independent of the lattice (see [BK, Lemmas 4.4 and
4.5]). In any case we will consider in what follows a fixed lattice associated to
ρ = ρf,λ, Ad ρ, etc. Henceforth we will only use the notation H1
F (Qp, −) when
the underlying vector space is crystalline.
Proposition 1.3. (i) If ρ0 is flat but ordinary and ρf,λ is associated
to a p-divisible group then for all n
H1
f (Qp, Vλn ) = H1
F (Qp, Vλn ).
(ii) If ρf,λ is ordinary, det ρf,λ
�
�
�
Ip
= ε and ρf,λ is associated to a p-divisible
group, then for all n,
H1
F (Qp, Vλn ) ⊆ H1
Se(Qp, Vλn .
Proof. Beginning with (i), we define H1
f (Qp, V) = {α ∈ H1(Qp, V) :
κ(α/λn) ∈ H1
f (Qp, V ) for all n} where κ : H1(Qp, V) → H1(Qp, V ). Then
we see that in case (i), H1
f (Qp, V ) is divisible. So it is enough to how that
H1
F (Qp, V) = H1
f (Qp, V).
We have to compare two constructions associated to a nonzero element α of
H1(Qp, V). The first is to associate an extension
(1.9) 0 →V→ E δ
→ K → 0
of K-vector spaces with commuting continuous Galois action. If we fix an e
with δ(e) = 1 the action on e is defined by σe = e + ˆα(σ) with ˆα a cocycle
representing α. The second construction begins with the image of the subspace
#α$ in H1(Qp, V ). By the analogue of Proposition 1.2 in the local case, there
is an O-module isomorphism
H1(Qp, V )
HomO(pR/p2
R, K/O)

where R is the universal deformation ring of ρ0 viewed as a representation
of Gal(Q¯ p/Q) on O-algebras and pR is the ideal of R corresponding to pD
(i.e., its inverse image in R). Since α �= 0, associated to #α$ is a quotient
pR/(p2
R, a) of pR/p2
R which is a free O-module of rank one. We then obtain a
homomorphism
ρα : Gal(Q¯ p/Qp) → GL2
�
R/(p2
R, a)
�
induced from the universal deformation (we pick a representation in the universal class). This is associated to an O-module of rank 4 which tensored with
K gives a K-vector space E�
(K)4 which is an extension
(1.10) 0 →U→ E� →U→ 0
where U
K2 has the Galis representation ρf,λ (viewed locally).
In the first construction α ∈ H1
F (Qp, V) if and only if the extension (1.9) is
crystalline, as the extension given in (1.9) is a sum of copies of the more usual
extension where Qp replaces K in (1.9). On the other hand #α$ ⊆ H1
f (Qp, V) if
and only if the second construction can be made through Rfl, or equivalently if
and only if E� is the representation associated to a p-divisible group. A priori,
the representation associated to ρα only has the property that on all finite
quotients it comes from a finite flat group scheme. However a theorem of
Raynaud [Ray1] says that then ρα comes from a p-divisible group. For more
details on Rfl, the universal flat deformation ring of the local representation
ρ0, see [Ram].) Now the extension E� comes from a p-divisible group if and
only if it is crystalline; cf. [Fo, §6]. So we have to show that (1.9) is crystalline
if and only if (1.10) is crystalline.
One obtains (1.10) from (1.9) as follows. We view V as HomK(U, U) and
let
X = ker : {HomK(U, U) ⊗U → U}
where the map is the natural one f ⊗ w → f(w). (All tensor products in this
proof will be as K-vector spaces.) Then as K[Dp]-modules
E�
(E ⊗ U)/X.
To check this, one calculates explicitly with the definition of the action on E
(given above on e) and on E� (given in the proof of Proposition 1.1). It follows
from standard properties of crystalline representations that if E is crystalline,
so is E ⊗ U and also E�
. Conversely, we can recover E from E� as follows.
Consider E� ⊗U
(E ⊗U⊗U)/(X ⊗ U). Then there is a natural map
ϕ : E ⊗ (det) → E� ⊗ U induced by the direct sum decomposition U⊗U

(det) ⊕ Sym2U. Here det denotes a 1-dimensional vector space over K with
Galois action via det ρf,λ. Now we claim that ϕ is injective on V ⊗ (det). For

if f ∈ V then ϕ(f) = f ⊗ (w1 ⊗ w2 − w2 ⊗ w1) where w1, w2 are a basis for U
for which w1 ∧ w2 = 1 in det
K. So if ϕ(f) ∈ X ⊗ U then
f(w1) ⊗ w2 − f(w2) ⊗ w1 = 0 in U⊗U.
But this is false unless f(w1) = f(w2) = 0 whence f = 0. So ϕ is injective
on V ⊗ det and if ϕ itself were not injective then E would split contradicting
α �= 0. So ϕ is injective and we have exhibited E ⊗(det) as a subrepresentation
of E� ⊗ U which is crystalline. We deduce that E is crystalline if E� is. This
completes the proof of (i).
To prove (ii) we check first that H1
Se(Qp, Vλn ) = j−1 n
�
H1
Se(Qp, V )
�
(this
was already used in (1.7)). We next have to show that H1
F (Qp,V) ⊆ H1
Se(Qp,V)
where the latter is defined by
H1
Se(Qp, V) = ker : H1(Qp, V) → H1(Qunr
p , V/V0)
with V0 the subspace of V on which Ip acts via ε. But this follows from the
computations in Corollary 3.8.4 of [BK]. Finally we observe that
pr�
H1
Se(Qp, V)
�
⊆ H1
Se(Qp, V )
although the inclusion may be strict, and
pr�
H1
F (Qp, V)
�
= H1
F (Qp, V )
by definition. This completes the proof. �
These groups have the property that for s ≥ r,
(1.11) H1(Qp, V r
λ ) ∩ j−1
r,s �
H1
F (Qp, Vλs )
�
= H1
F (Qp, Vλr )
where jr,s : Vλr → Vλs is the natural injection. The same holds for V ∗
λr and
V ∗
λs in place of Vλr and Vλs where V ∗
λr is defined by
V ∗
λr = Hom(Vλr , µpr )
and similarly for V ∗
λs . Both results are immediate from the definition (and
indeed were part of the motivation for the definition).
We also give a finite level version of a result of Bloch-Kato which is easily
deduced from the vector space version. As before let T ⊂ V be a Galois stable
lattice so that T
O4. Define
H1
F (Qp, T) = i
−1
�
H1
F (Qp, V)
�
under the natural inclusion i : T ,→ V, and likewise for the dual lattice T ∗ =
HomZp (V,(Qp/Zp)(1)) in V∗. (Here V∗ = Hom(V, Qp(1)); throughout this
paper we use M∗ to denote a dual of M with a Cartier twist.) Also write

prn : T → T /λn for the natural projection map, and for the mapping it
induces on cohomology.
Proposition 1.4. If ρf,λ is associated to a p-divisible group (the ordinary case is allowed) then
(i) prn
�
H1
F (Qp, T)
�
= H1
F (Qp, T /λn) and similarly for T ∗, T ∗/λn.
(ii) H1
F (Qp, Vλn ) is the orthogonal complement of H1
F (Qp, V ∗
λn ) under Tate
local duality between H1(Qp, Vλn ) and H1(Qp, V ∗
λn ) and similarly for Wλn
and W∗
λn replacing Vλn and V ∗
λn .
More generally these results hold for any crystalline representation V� in
place of V and λ� a uniformizer in K� where K� is any finite extension of Qp
with K� ⊂ EndGal(Qp/Qp)V�
.
Proof. We first observe that prn(H1
F (Qp, T)) ⊂ H1
F (Qp, T /λn). Now
from the construction we may identify T /λn with Vλn . A result of BlochKato ([BK, Prop. 3.8]) says that H1
F (Qp, V) and H1
F (Qp, V∗) are orthogonal
complements under Tate local duality. It follows formally that H1
F (Qp, V ∗
λn )
and prn(H1
F (Qp, T)) are orthogonal complements, so to prove the proposition
it is enough to show that
(1.12) #H1
F (Qp, V ∗
λn )#H1
F (Qp, Vλn )=#H1(Qp, Vλn ).
Now if r = dimK H1
F (Qp, V) and s = dimK H1
F (Qp, V∗) then
(1.13) r + s = dimK H0(Qp, V) + dimK H0(Qp, V∗) + dimK V.
From the definition,
(1.14) #H1
F (Qp, Vλn ) = #(O/λn)
r · # ker{H1(Qp, Vλn ) → H1(Qp, V )}.
The second factor is equal to #{V (Qp)/λnV (Qp)}. When we write V (Qp)div
for the maximal divisible subgroup of V (Qp) this is the same as
#(V (Qp)/V (Qp)
div)/λn = #(V (Qp)/V (Qp)
div)λn
= #V (Qp)λn /#(V (Qp)
div)λn .
Combining this with (1.14) gives
#H1
F (Qp, Vλn ) = #(O/λn)
r (1.15)
· #H0(Qp, Vλn )/#(O/λn)
dimKH0(Qp,V)
.
This, together with an analogous formula for #H1
F (Qp, V ∗
λn ) and (1.13), gives
#H1
F (Qp, V λn
)#H1
F (Qp, V ∗
λn ) = #(O/λn)4 · #H0(Qp, Vλn )#H0(Qp, V ∗
λn ).

As #H0(Qp, V ∗λn)=#H2(Qp, Vλn ) the assertion of (1.12) now follows from
the formula for the Euler characteristic of Vλn .
The proof for Wλn , or indeed more generally for any crystalline representation, is the same. �
We also give a characterization of the orthogonal complements of
H1
Se(Qp, Wλn ) and H1
Se(Qp, Vλn ), under Tate’s local duality. We write these
duals as H1
Se∗ (Qp, W∗
λn ) and H1
Se∗ (Qp, V ∗
λn ) respectively. Let
ϕw : H1(Qp, W∗
λn ) → (Qp, W∗
λn /(W∗
λn )
0)
be the natural map where (W∗
λn )i is the orthogonal complement of W1−i
λn in
W∗
λn , and let Xn,i be defined as the image under the composite map
Xn,i = im : Z×
p /(Z×
p )
pn
⊗ O/λn → H1(Qp, µpn ⊗ O/λn)
→ H1(Qp, W∗
λn /(W∗
λn )
0)
where in the middle term µpn ⊗ O/λn is to be identified with (W∗
λn )1/(W∗
λn )0.
Similarly if we replace W∗
λn by V ∗
λn we let Yn,i be the image of Z×
p /(Z×
p )pn
⊗
(O/λn)2 in H1(Qp, V ∗
λn /(W∗
λn )0), and we replace ϕw by the analogous map ϕv.
Proposition 1.5.
H1
Se∗ (Qp, W∗
λn ) = ϕ−1
w (Xn,i),
H1
Se∗ (Qp, V ∗
λn ) = ϕ−1
v (Yn,i).
Proof. This can be checked by dualizing the sequence
0 → H1
Str(Qp, Wλn ) → H1
Se(Qp, Wλn )
→ ker : {H1(Qp, Wλn /(Wλn )
0) → H1(Qunr
p , Wλn /(Wλn )
0},
where H1
str(Qp, Wλn ) = ker : H1(Qp, Wλn ) → H1(Qp, Wλn /(Wλn )0). The first
term is orthogonal to ker : H1(Qp, W∗
λn ) → H1(Qp, W∗
λn /(W∗
λn )1). By the
naturality of the cup product pairing with respect to quotients and subgroups
the claim then reduces to the well known fact that under the cup product
pairing
H1(Qp, µpn ) × H1(Qp, Z/pn) → Z/pn
the orthogonal complement of the unramified homomorphisms is the image
of the units Z×
p /(Z×
p )pn
→ H1(Qp, µpn ). The proof for Vλn is essentially the
same. �

The proof is like 100 pages just have alink to it
https://www.mat.uniroma2.it/~eal/Wiles-Fermat.pdf

evdoe only smart nusois

none of this math is used anywhere irl btw