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most conveniently defined using an extension from characteristic zero. This is
explained below. We will first show that dimT/m H0(X1(N, p)/F , &!)[m] d" 1,
p
this being the essential step. If we embed T/m ’! Fp and then set
m = ker : T —" Fp ’! Fp (the map given by t —" a ’! at mod m) then it is
enough to show that dimF H0(X1(N, p)/F , &!)[m ] d" 1. First we will suppose
p p
486 ANDREW JOHN WILES
that there is no nonzero holomorphic differential in H0(X1(N, p)/F , &!)[m ],
p
i.e., no differential form which pulls back to holomorphic differentials on £ét
and £µ. Then if É1 and É2 are two differentials in H0(X1(N, p)/F , &!)[m ],
p
the q-expansion principle shows that µÉ1 - »É2 has zero q-expansion at " for
2
some pair (µ, ») =(0, 0) in Fp and thus is zero on £µ. As µÉ1 - »É2 =0 on
£µ it is holomorphic on £ét. By our hypothesis it would then be zero which
shows that É1 and É2 are linearly dependent.
This use of the q-expansion principle in characteristic p is crucial and due
to Mazur [Ma2]. The point is simply that all the coefficients in the q-expansion
are determined by elementary formulae from the coefficient of q provided that
É is an eigenform for all the Hecke operators. The formulae for the action of
these operators in characteristic p follow from the formulae in characteristic
zero. To see this formally (especially for the Up operator) one checks first
that H0(X1(N, p)/Z , &!), where &! denotes the sheaf of regular differentials on
p
X1(N, p)/Z , behaves well under the base changes Zp ’! Zp and Zp ’! Qp;
p
cf. [Ma2, §II.3] or [Wi3, Prop. 6.1]. The action of the Hecke operators on
J1(N, p) induces an action on the connected component of the Neron model of
J1(N, p)/Q , so also on its tangent space and cotangent space. By Grothendieck
p
duality the cotangent space is isomorphic to H0(X1(N, p)/Z , &!); see (2.5)
p
below. (For a summary of the duality statements used in this context, see
[Ma2, §II.3]. For explicit duality over fields see [AK, Ch. VIII].) This then
defines an action of the Hecke operators on this group. To check that over Qp
this gives the standard action one uses the commutativity of the diagram after
Proposition 2.2 in [Mi1].
Now assume that there is a nonzero holomorphic differential in
H0(X1(N, p)/F , &!)[m ].
p
We claim that the space of holomorphic differentials then has dimension 1 and
that any such differential É = 0 is actually nonzero on £µ. The dimension
claim follows from the second assertion by using the q-expansion principle. To
prove that É = 0 on £µ we use the formula
Up"(x, y) =(Fx, y )
for (x, y) " (Pic0£ét × Pic0£µ)(Fp), where F denotes the Frobenius endo-
morphism. The value of y will not be needed. This formula is a variant
on the second part of Theorem 5.3 of [Wi3] where the corresponding re-
sult is proved for X1(Np). (A correction to the first part of Theorem 5.3
was noted in [MW1, p. 188].) One check then that the action of Up on
X0 = H0(£µ, &!1) •" H0(£ét, £1) viewed as a subspace of H0(X1(N, p)/F , &!)
p
is the same as the action on X0 viewed as the cotangent space of Pic0£µ ×
É
Pic0£ét. From this we see that if É = 0 on £µ then Up = 0 on £ét. But Up
MODULAR ELLIPTIC CURVES AND FERMAT S LAST THEOREM 487
acts as a nonzero scalar which gives a contradiction if É = 0. We can thus as-
sume that the space of m -torsion holomorphic differentials has dimension 1 and
is generated by É. So if É2 is now any differential in H0(X1(N, p)/F , &!)[m ]
p
then É2 -»É has zero q-expansion at " for some choice of ». Then É2 -»É =0
on £µ whence É2 - »É is holomorphic and so É2 = »É. We have now shown
in general that dim(H0(X1(N, p)/F , &!)[m ]) d" 1.
p
The singularities of X1(N, p)/Z at the supersingular points are formally
p
isomorphic over Zunr to Zunr[[X, Y ]]/(XY - pk) with k =1, 2 or 3 [cf. [DR,
p p
Ch. 6, Th. 6.9]). If we consider a minimal regular resolution M1(N, p)/Z
p
then H0(M1(N, p)/F , &!) H0(X1(N, p)/F , &!) (see the argument in [Ma2,
p p
Prop. 3.4]), and a similar isomorphism holds for H0(M1(N, p)/Z , &!).
p
As M1(N, p)/Z is regular, a theorem of Raynaud [Ray2] says that the
p
connected component of the Neron model of J1(N, p)/Q is J1(N, p)0
p /Zp
Pic0(M1(N, p)/Z ). Taking tangent spaces at the origin, we obtain
p
(2.5) Tan(J1(N, p)0 ) H1(M1(N, p)/Z , OM (N,p)).
/Zp p 1
Reducing both sides mod p and applying Grothendieck duality we get an iso-
morphism
(2.6) Tan(J1(N, p)0 ) ’! Hom(H0(X1(N, p)/F , &!), Fp).
/Fp p
(To justify the reduction in detail see the arguments in [Ma2, §II. 3]). Since
Tan(J1(N, p)0 ) is a faithful T —" Zp-module it follows that
/Zp
H0(X1(N, p)/F , &!)[m]
p
is nonzero. This completes the proof of the lemma.
To complete the proof of the theorem we choose an abelian subvariety
A of J1(N, p) with multiplicative reduction at p. Specifically let A be the
connected part of the kernel of J1(N, p) ’! J1(N) × J1(N) under the natural
map Õ described in Section 2 (see (2.10)). Then we have an exact sequence
Æ
0 ’! A ’! J1(N, p) ’! B ’! 0
and J1(N, p) has semistable reduction over Qp and B has good reduction.
By Proposition 1.3 of [Ma3] the corresponding sequence of connected group
schemes
0 ’! A[p]0 ’! J1(N, p)[p]0 ’! B[p]0 ’! 0
/Zp /Zp /Zp
is also exact, and by Corollary 1.1 of the same proposition the corresponding
sequence of tangent spaces of Neron models is exact. Using this we may check
that the natural map
(2.7) Tan(J1(N, p)[p]t ) —" Tm ’! Tan(J1(N, p)/F ) —" Tm
/Fp Tp p
Tp
488 ANDREW JOHN WILES
is an isomorphism, where t denotes the maximal multiplicative-type subgroup
scheme (cf. [Ma3, §1]). For it is enough to check such a relation on A and B
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