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removed the last restrictions referred to in Ribet s survey article [Ri3]. The
following is a minor adaptation of the epsilon conjecture to our situation which
can be found in [Dia, Th. 6.4]. (We wish to use weight 2 only.) Let N(Á0) be
the prime to p part of the conductor of Á0 as defined for example in [Se].
Theorem 2.14. Suppose that Á0 is modular and satisfies (1.1) (so in
particular is irreducible) and is of type D =(·, £, O, M) with · =Se, str or fl.
Suppose that at least one of the following conditions holds (i) p >3 or (ii) Á0
"
is not induced from a character of Q( -3). Then there exists a newform f
of weight 2 and a prime » of Of such that Áf,» is of type D =(·, £, O , M)
for some O , and such that (Áf,» mod ») Á0 over Fp. Moreover we can
assume that f has character Çf of order prime to p and has level N(Á0)p´(Á )
504 ANDREW JOHN WILES
where ´(Á0) = 0 if Á0|D is associated to a finite flat group scheme over Zp
p
and det Á0 = É, and ´(Á0) =1 otherwise. Furthermore in the Selmer case
Ip
we can assume that ap(f) a" Ç2(Frob p)mod» in the notation of (1.2) where
ap(f) is the eigenvalue of Up.
For the rest of this chapter we will assume that Á0 is modular and that
"
if p = 3 then Á0 is not induced from a character of Q( -3). Here and in the
rest of the paper we use the term induced to signify that the representation
is induced after an extension of scalars to the algebraic closure.
For each D = {·, £, O, M} we will now define a Hecke ring TD except
where · is unrestricted. Suppose first that we are in the flat, Slemer or strict
cases. Recall that when referring to the flat case we assume that Á0 is not
ordinary and that det Á0|I = É. Suppose that £ = {qi} and that N(Á0) =
p
si
qi with si e" 0. If U» k2 is the representation space of Á0 we set nq =
q
dimk(U»)I where Iq in the inertia group at q. Define M0 and M by
2
(2.24) M0 = N(Á0) qi · qi , M = M0pÄ (Á0)
nqi =1
nqi =2
qi"M*"{p}
where Ä(Á0) = 1 if Á0 is ordinary and Ä(Á0) = 0 otherwise. Let H be the
subgroup of (Z/MZ)" generated by the Sylow p-subgroup of (Z/qiZ)" for each
qi "Mas well as by all of (Z/qiZ)" for each qi "Mof type (A). Let T (M)
H
denote the ring generated by the standard Hecke operators {Tl for l Mp, a
for (a, Mp) =1}. Let m denote the maximal ideal of T (M) associated to the
H
f and » given in the theorem and let km be the residue field T (M)/m. Note
H
that m does not depend on the particular choice of pair (f, ») in theorem 2.14.
Then km k0 where k0 is the smallest possible field of definition for Á0 because
km is generated by the traces. Henceforth we will identify k0 with km . There
is one exceptional case where Á0 is ordinary and Á0|D is isomorphic to a sum
p
of two distinct unramified characters (Ç1 and Ç2 in the notation of Chapter 1,
§1). If Á0 is not exceptional we define
(2.25(a)) TD = T (M)m —" O.
H
W (k0)
If Á0 is exceptional we let T (M) denote the ring generated by the operators
H
{Tl for l Mp, a for (a, Mp) = 1, Up}. We choose m to be a maximal
ideal of T (M) lying above m for which there is an embedding km ’! k (over
H
k0 = km ) satisfying Up ’! Ç2(Frob p). (Note that Ç2 is specified by D.) Then
in the exceptional case km is either k0 or its quadratic extension and we define
(2.25(b)) TD = T (M)m —" O.
H
W (km )
The omission of the Hecke operators Uq for q|M0 ensures that TD is reduced.
MODULAR ELLIPTIC CURVES AND FERMAT S LAST THEOREM 505
We need to relate TD to a Hecke ring with no missing operators in order
to apply the results of Section 1.
Proposition 2.15. In the nonexceptional case there is a maximal ideal
m for TH(M) with m )" (M) =m and k0 = km, and such that the natural
H
map T (M)m ’! TH(M)m is an isomorphism, thus given
H
TD TH(M)m —" O.
W (k0)
In the exceptional case the same statements hold with m replacing m , T (M)
H
replacing T (M) and km replacing k0.
H
Proof. For simplicity we describe the nonexceptional case indicating where
appropriate the slight modifications needed in the exceptional case. To con-
struct m we take the eigenform f0 obtain from the newform f of Theorem 2.14
by removing the Euler factors at all primes q " £ -{M*"p}. If Á0 is ordinary
and f has level prime to p we also remove the Euler factor (1 - ²p · p-s) where
²p is the non-unit eigenvalue in Of». (By removing Euler factors we mean
take the eigenform whose L-series is that of f with these Euler factors re-
moved.) Then f0 is an eigenform of weight 2 on “H(M) (this is ensured by the
choice of f) with Of,» coefficients. We have a corresponding homomorphism
-1
Àf : TH(M) ’!Of,» and we let m = Àf (»).
Since the Hecke operators we have used to generate T (M) are prime to
H
the level these is an inclusion with finite index
T (M) ’! Og
H
where g runs over representatives of the Galois conjugacy classes of newforms
associated to “H(M) and where we note that by multiplicity one Og can also be
described as the ring of integers generated by the eigenvalues of the operators
in T (M) acting on g. If we consider TH(M) in place of T (M) we get a
H H
similar map but we have to replace the ring Og by the ring
Sg = Og[Xq , . . . , Xq , Xp]/{Yi, Zp}r
1 r i=1
where {p, p1, . . . , qr} are the distinct primes dividing Mp. Here
ñø
ri-1
òø
Xq Xq - ±q (g) Xq - ²q (g) if qi level(g)
i i i i
i
(2.26) Yi =
óø
ri
Xq Xq - aq (g) if qi| level(g),
i i
i
where the Euler factor of g at qi (i.e., of its associated L-series) is
-s -s -s
(1-±q (g)qi )(1-²q (g)qi ) in the first cases and (1-aq (g)qi ) in the second
i i i
ri
case, and qi || M/level(g) . (We allow aq (g) to be zero here.) Similarly Zp is
i
506 ANDREW JOHN WILES
defined by
ñø
2
Xp
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