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Algebraic Geometry

By Andrew Wiles

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2) by the theorem of Tate [Ta] as before. 6) together with the lemma shows that Tan(J1 (N, p))/Zp ⊗ Tm Tm . 7) together with this implies that as Tm -modules V := J1 (N, p)[p]t (Qp )m (Tm /p). 8) Tan(G/Fp ) G(Qp ) ⊗ Fp Fp for any multiplicative-type group scheme (finite and flat) G/Zp which is killed by p and moreover to give such an isomorphism that respects the action of endomorphism of G/Zp . 9) HomQp (µp , G) ⊗ Fp Fp HomFp (µp , G) ⊗ Fp Fp Hom Tan(µp /Fp ), Tan(G/Fp ) where HomQp denotes homomorphisms of the group schemes viewed over Qp and similarly for HomFp .

1) we can define a principal ideal (∆q ) of TH (N, q)m by (∆q ) = (α ◦ α) where α : TH (N q, q 2 )mq S1,m1 T(N, q)m is the restriction map induced (q) by the restriction map on m -localizations described above. 10. Suppose that m is a maximal ideal of TH (N, q) associated to an irreducible m of type (A). Then (∆q ) = (q − 1)2 (q + 1). Proof. 6. We let S2 = TH (N, q)[U2 ]/U2 (U2 − Uq ) be the ring of endomorphisms of JH (N, q)2 where U2 is given by the matrix Uq 0 q . 0 This satisfies the compatability ξ3 U2 = Uq ξ3 .

Remark. By a well-known result on the finite subgroups of PGL2 (Fp ) this lemma covers all ρ0 whose images are absolutely irreducible and for which ρ0 is not dihedral. Let K1 be the splitting field of ρ0 . Then we can view Wλ and Wλ∗ as Gal(K1 (ζp )/Q)-modules. We need to analyze their cohomology. Recall that we are assuming that ρ0 is absolutely irreducible. Let ρ0 be the associated projective representation to PGL2 (k). The following proposition is based on the computations in [CPS]. 11. Suppose that ρ0 is absolutely irreducible.

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