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n n n
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
n
H1(Qp, �p ) � H1(Qp, Z/pn) �! Z/pn
the orthogonal complement of the unramified homomorphisms is the image
n
n n
of the units Z�/(Z�)p �! H1(Qp, �p ). The proof for V� is essentially the
p p
same.
472 ANDREW JOHN WILES
2. Some computations of cohomology groups
We now make some comparisons of orders of cohomology groups using
the theorems of Poitou and Tate. We retain the notation and conventions of
Section 1 though it will be convenient to state the first two propositions in a
more general context. Suppose that
L = Lq �" H1(Qq, X)
p"�
is a subgroup, where X is a finite module for Gal(Q�/Q) of p-power order.
We define L" to be the orthogonal complement of L under the perfect pairing
(local Tate duality)
H1(Qq, X) � H1(Qq, X") �! Qp/Zp
q"� q"�
"
where X" = Hom(X, �p ). Let
�X : H1(Q�/Q, X) �! H1(Qq, X)
q"�
"
be the localization map and similarly �X for X". Then we set
1 1
HL(Q�/Q, X) =�-1(L), HL (Q�/Q, X") =�-1 (L").
"
X X"
The following result was suggested by a result of Greenberg (cf. [Gre1]) and
is a simple consequence of the theorems of Poitou and Tate. Recall that p is
always assumed odd and that p " �.
Proposition 1.6.
1 1
#HL(Q�/Q, X)/#HL (Q�/Q, X") =h" hq
"
q"�
where
hq =#H0(Qq, X")/[H1(Qq, X) : Lq]
h" =#H0(R, X")#H0(Q, X)/#H0(Q, X").
Proof.AdaptingtheexactsequenceproofofPoitouandTate(cf.[Mi2,Th.4.20])
we get a seven term exact sequence
1
0 -�! HL(Q�/Q, X) -�! H1(Q�/Q, X) -�! H1(Qq, X)/Lq
q"�
��
��
1
H2(Qq, X) �!- H2(Q�/Q, X) �!- HL (Q�/Q, X")'"
"
q"�
|�! H0(Q�/Q, X")'" 0,
-�!
MODULAR ELLIPTIC CURVES AND FERMAT S LAST THEOREM 473
where M'" = Hom(M, Qp/Zp). Now using local duality and global Euler char-
acteristics (cf. [Mi2, Cor. 2.3 and Th. 5.1]) we easily obtain the formula in the
proposition. We repeat that in the above proposition X can be arbitrary of
p-power order.
1
We wish to apply the proposition to investigate HD. Let D =(�, �, O, M)
be a standard deformation theory as in Section 1 and define a corresponding
group Ln = LD,n by setting
��
n
H1(Qq, V� ) for q = p and q "M
��
��
1
n
HD (Qq, V� ) for q = p and q "M
Ln,q =
q
��
ó�
n
H.1(Qp, V� ) for q = p.
1 1
n n
Then HD(Q�/Q, V� ) =HL (Q�/Q, V� ) and we also define
n
1 " 1 "
HD (Q�/Q, V� ) =HL (Q�/Q, V� ).
" n " n
n
We will adopt the convention implicit in the above that if we consider � �" �
1
n
then HD(Q� /Q, V� ) places no local restriction on the cohomology classes at
1 "
primes q " � - �. Thus in HD (Q� /Q, V� ) we will require (by duality) that
" n
the cohomology class be locally trivial at q " � - �.
We need now some estimates for the local cohomology groups. First we
consider an arbitrary finite Gal(Q�/Q)-module X:
Proposition 1.7. If q " �, and X is an arbitrary finite Gal(Q�/Q)-
module of p-power order,
1 1
#HL (Q�*"q/Q, X)/#HL(Q�/Q, X) d" #H0(Qq, X")
where L = L for " � and L = H (Qq, X).
q
Proof. Consider the short exact sequence of inflation-restriction:
1 1
�
0�!HL(Q�/Q, X)�!HL (Q�*"q/Q, X)�!Hom(Gal(Q�*"q/Q�), X)Gal(Q /Q)
�� ��
)"
�� ��
�� ��
�� ��
unr unr
q q
H1(Qunr, X)Gal(Q /Qq) �!H1(Qunr, X)Gal(Q /Qq)
q q
The proposition follows when we note that
unr
q
#H0(Qq, X") = #H1(Qunr, X)Gal(Q /Qq).
q
n n
Now we return to the study of V� and W� .
n
Proposition 1.8. If q "M(q = p) and X = V� then hq =1.
474 ANDREW JOHN WILES
Proof. This is a straightforward calculation. For example if q is of type
(A) then we have
n n
Ln,q =ker{H1(Qq, V� ) �! H1(Qq, W� /W� ) �" H1(Qunr, O/�n)}.
n
q
Using the long exact sequence of cohomology associated to
0 0
n n
0 �! W� �! W� �! W� /W� �! 0
n n
n
one obtains a formula for the order of Ln,q in terms of #H1(Qq, W� ),
n
#Hi(Qq, W� /W� ) etc. Using local Euler characteristics these are easily re-
n
"
duced to ones involving H0(Qq, W� ) etc. and the result follows easily.
n
The calculation of hp is more delicate. We content ourselves with an
inequality in some cases.
n
Proposition 1.9. (i) If X = V� then
" "
hph" =#(O/�)3n#H0(Qp, V� )/#H0(Q, V� )
n n
in the unrestricted case.
n
(ii) If X = V� then
ord "
hph" d" #(O/�)n#H0(Qp, (V� )")/#H0(Q, W� )
n n
in the ordinary case.
0 "
n n
(iii) If X = V� or W� then hph" d" #H0(Qp, (W� )")/#H0(Q, W� )
n n
in the Selmer case.
n n
(iv) If X = V� or W� then hph" =1 in the strict case.
n
(v) If X = V� then hph" =1 in the flat case.
"
n n
(vi) If X = V� or W� then hph" = 1/#H0(Q, V� ) if Ln,p =
n
1
HF (Qp, X) and �f,� arises from an ordinary p-divisible group.
n
Proof. Case (i) is trivial. Consider then case (ii) with X = V� . We have
a long exact sequence of cohomology associated to the exact sequence:
0 0
n n
(1.16) 0 �! W� �! V� �! V� /W� �! 0.
n n
In particular this gives the map u in the diagram
n
H1(Qp, V� )
|
��
�
��
u ��
��
0 0 0
n n n
1 �! Z = H1(Qunr/Qp,(V� /W� )H) �! H1(Qp,V� /W� ) �! H1(Qunr,V� /W� )G �! 1
n n n
p p
�
where G = Gal(Qunr/Qp), H = Gal(Qp/Qunr) and � is defined to make the
p p
triangle commute. Then writing hi(M) for #H1(Qp, M) we have that #Z =
MODULAR ELLIPTIC CURVES AND FERMAT S LAST THEOREM 475
n
h0(V� /W� ) and #im � e" (#im u)/(#Z). A simple calculation using the
n
long exact sequence associated to (1.16) gives
n n
h1(V� /W� )h2(V� )
n
(1.17) #im u = .
0 0
n
h2(W� )h2(V� /W� )
n n
Hence
" 0
n
[H1(Qp, V� ) : Ln,p] = #im� e" #(O/�)3nh0(V� )/h0(W� )".
n n
n n
The inequality in (iii) follows for X = V� and the case X = W� is similar.
Case (ii) is similar. In case (iv) we just need #im u which is given by (1.17)
n n
with W� replacing V� . In case (v) we have already observed in Section 1 that
"
Raynaud s results imply that #H0(Qp, V� ) = 1 in the flat case. Moreover
n
n
#Hf1(Qp, V� ) can be computed to be #(O/�)2n from
n n n
Hf1(Qp, V� ) Hf1(Qp, V)� HomO(pR/p2 , K/O)�
R
where R is the universal local flat deformation ring of �0 for O-algebras. Using
the relation R Rfl �" O where Rfl is the corresponding ring for W (k)-
W (k)
algebras, and the main theorem of [Ram] (Theorem 4.2) which computes Rfl,
we can deduce the result.
We now prove (vi). From the definitions
n
(#O/�n)r#H0(Qp, W� ) if �f,�|D does not split
p
1
n
#HF (Qp, V� ) =
(#O/�n)r if �f,�|D splits
p
1
where r = dimK HF (Qp, V). This we can compute using the calculations in
[BK, Cor. 3.8.4]. We find that r = 2 in the non-split case and r = 3 in the
split case and (vi) follows easily.
3. Some results on subgroups of GL2(k)
We now give two group-theoretic results which will not be used until
Chapter 3. Although these could be phrased in purely group-theoretic terms
it will be more convenient to continue to work in the setting of Section 1, i.e.,
with �0 as in (1.1) so that im �0 is a subgroup of GL2(k) and det �0 is assumed
odd.
Lemma 1.10. If im �0 has order divisible by p then:
(i) It contains an element �0 of order m e" 3 with (m, p) =1 and �0 trivial
on any abelian quotient of im �0.
(ii) It contains an element �0(�) with any prescribed image in the Sylow [ Pobierz całość w formacie PDF ]