Modular Elliptic Curves and Fermat's Last Theorem

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T,f
2
where cp =#(O/Up - p )
p
a finite flat group scheme over Zp and det �0|I = �, and cp =1 otherwise;
p
(ii) if TD is a complete intersection over O then (i) is an equality, RD
TD and TD is a complete intersection.
In general, for any (not necessarily minimal) D of Selmer, strict or flat
1
type, and any �f,� of type D, #HD(Q�/Q, Vf )
Remarks. The finiteness was proved by Flach in [Fl] under some restric-
tions on f, p and D by a different method. In particular, he did not consider
the strict case. The bound we obtain in (i) is in fact the actual order of
1
HD(Q�/Q, Vf ) as follows from the main result of [TW] which proves the hy-
pothesis of part (ii). Then applying Theorem 2.17 we obtain the order of this
group for more general D s associated to �0 under the condition that a minimal
D exists associated to �0. This is stated in Theorem 3.3.
MODULAR ELLIPTIC CURVES AND FERMAT S LAST THEOREM 519
The case where the projective representation associated to �0 is dihedral
does not always have the property that a twist of it has an associated minimal
D. In the case where the associated quadratic field is imaginary we will give a
different argument in Chapter 4.
Proof. We will assume throughout the proof that D is minimal, indicating
only at the end the slight changes needed fot the final assertion of the theorem.
Let Q be a finite set of primes disjoint from � satisfying q a" 1(p) and �0(Frobq)
having distinct eigenvalues for each q " Q. For the minimal deformation
problem D =(�, �, O, M), let DQ be the deformation problem described before
(2.34); i.e., it is the refinement of (�, � *" Q, O, M) obtained by imposing the
additional restriction (2.34) at each q " Q. (We will assume for the proof that
O is chosen so O/� = k contains the eigenvalues of �0(Frob q) for each q " Q.)
We set
T = TD , R = RD
and recall the definition of TQ and RQ from Chapter 2, �3 (cf. (2.35)). We
write V for Vf and recall the definition of V(q) following (2.38). Also remember
that mQ is a maximal ideal of TH(Nq1 . . . qr) as in (2.35) for which �m �0
Q
�
over Fp (recall that this uses the same choice of embedding km -�! k as in
Q
the definition of TQ). We use mQ also to denote the maximal ideal of TQ if
the context makes this reasonable.
Consider the exact and commutative diagram
�Q unr
1 1 (q)
0 �! HD(Q�/Q, V) �! HDQ (Q�*"Q/Q, V) �! H1(Qunr, V )Gal(Qq /Qq)
q
q"Q
| |
��
��
0 �! (pR/p2 )" �! (pRQ /p2 )" ��
Q
R RQ
��
�!�!
uQ
0 �! (pT/p2 )" �! (pTQ /p2 )" �! KQ �! 0
T TQ
where KQ is by definition the cokernel in the horizontal sequence and " denotes
HomO( , K/O) for K the field of fractions of O. The key result is:
Lemma 3.2. The map �Q is injective for any finite set of primes Q
satisfying
2
q a" 1(p), Tq a" q (1 + q)2mod m f or all q " Q.
Proof. Note that the hypotheses of the lemma ensure that �0(Frob q) has
distinct eigenvaluesw for each q " Q. First, consider the ideal aQ of RQ defined
520 ANDREW JOHN WILES
by
ai bi
(3.4) aQ = ai -1, bi, ci, di -1 : = �D (�i) with �i " Iq , qi " Q .
Q i
ci di
Then the universal property of RQ shows that RQ/aQ R. This permits us
to identify (pR/p2 )" as
R
(pR/p2 )" = {f " (pR /p2 )" : f(aQ) =0}.
R Q RQ
If we prove the same relation for the Hecke rings, i.e., with T and TQ replacing
R and RQ then we will have the injectivity of �Q. We will write Q for the
image of aQ in TQ under the map �Q of (2.37).
It will be enough to check that for any q " Q , Q a subset of Q, TQ /q
TQ where aq is defined as in (3.4) but with Q replaced by q. Let
-{q}
N = N(�0)p�(� ) � qi where �(�0) is as defined in Theorem 2.14.
qi"Q -{q}
�
Then take an element � " Iq �" Gal(Qq/Qq) which restricts to a generator
of Gal(Q(�N q/Q(�N )). Then det(�) = tq " TQ in the representation to
GL2(TQ ) defined after Theorem 2.1. (Thus tq a" 1(N ) and tq is a primitive
root mod q.) It is easily checked that
� �
(3.5) JH(N .q)m (Q) JH(N q)m (Q)[ tq -1].
Q Q
Here H is still a subgroup of (Z/M0Z)". (We use here that �0 is not reducible
"
for the injectivity and also that �0 is not induced from a character of Q( -3)
for the surjectivity when p = 3. The latter is to avoid the ramification points of
the covering XH(N q) �! XH(N , q) of order 3 which can give rise to invariant
divisors of XH(N q) which are not the images of divisors on XH(N , q).)
Now by Corollary 1 to Theorem 2.1 the Pontrjagin duals of the modules
in (3.5) are free of rank two. It follows that
(3.6) (TH(N q)m )2/( tq -1) (TH(N , q)m )2.
Q Q
The hypotheses of the lemma imply the condition that �0(Frob q) has distinct
eigenvalues. So applying Proposition 2.4 (at the end of �2) and the remark
following it (or using the fact remarked in Chapter 2, �3 that this condition
implies that �0 does not occur as the residual representation associated to any
form which has the special representation at q) we see that after tensoring over
W (km ) with O, the right-hand side of (3.6) can be replaced by T2 thus
Q Q -{q}
giving
T2 T2 ,
Q /q Q -{q}
since tq -1 " q. Repeated inductively this gives the desired relation
TQ/Q T, and completes the proof of the lemma.
MODULAR ELLIPTIC CURVES AND FERMAT S LAST THEOREM 521
Suppose now that Q is a finite set of primes chosen as in the lemma. Recall
that from the theory of congruences (Prop. 2.4 at the end of �2)
�T /�T,f = (q - 1),
Q,f
q"Q
2
the factors (�q - q ) being units by our hypotheses on q " Q. (We only need
that the right-hand side divides the left which is somewhat easier.) Also, from
the theory of Fitting ideals (see the proof of (2.44))
#(pT/p2 ) e" #(O/�T )
T f
#(pT /p2 e" #(O/�T ).
Q TQ Q,f
We dedeuce that
#KQ e" # O (q - 1) � t-1
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