MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM 463 o[el/("E, a2))which is an O-algebra deformation of Po(see the proof of Propo- sition 1. 1 below). Let E=On[e]2 where the Galois action is via Pa. Then there is an exact sequence 0→E/M→E/A (E/e)mn→0 and hence an extension ks now that (1.8 is a map of O-modules ExtA(UAm, UA,)under(1.8),i.e, those extensions which ready extensions in the category of finite fat group schemes Zp. Observe that ExtA(UA, UA)n ExtOL,(UAm, UAn) is an O-module, So HF(Qp, VAn)is seen to be an O-sub- module of H( Qp, VA,). We observe that our definition is equivalent to requir- ing that the classes in HF( Qp, VAn)map under(1.8)to ExtA(UAm, UAn)for all m>n. For if em is the extension class in Ext(UAm, UAn)then em c+ OUAm as Galois-modules and we can apply results of [Rayl] to see that em comes from a finite fat group scheme over Zp if en does In the fat(non-ordinary) case polI, is determined by Raynaud s results as mentioned at the beginning of the chapter. It follows in particular that, since PolDp is absolutely irreducible, V(Qp=h(Qp, v) is divisible in this case (in fact V(Qp)a kT/0). This H(Qp, VAn)cH(Qp, V)An and hence we can define H(Q,V)=∪m(Qn,Vm), and we claim that Hf( Qp, V)An e H (Qp, VAn). To see this we have to compare representations for m >n Pn,m: Gal(Qp/Qp)- GL2(On(E/Am) Gal(Qp/Qp)- GL2([el/Am) where Pn, m and Pm, m are obtained from an E H(Qp, VXAn) and im(an ( Qp, VAm)and m, n: a+be -a+Am-nbE By [Ram, Prop 1. 1 and Lemma 2.1]if Pn. m comes from a finite flat group scheme then so does Pm. m. C Pm.n is injective and so Pn, m comes from a finite fat group scheme if pm. m does cf [Rayl]. The definitions of HD(Qz/Q, VAn)and HD(Qx/Q,V)now extend to the flat case and we note that(1.7) is also valid in the flat cas Still in the fat(non-ordinary) case we can again use the determination of polI, to see that H(Qp, v) is divisible. For it is enough to check that H(Qp, VA)=0 and this follows by duality from the fact that H(Qp,V)=0
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 463 O[ε]/(λnε, ε2)) which is an O-algebra deformation of ρ0 (see the proof of Proposition 1.1 below). Let E = On[ε] 2 where the Galois action is via ρα. Then there is an exact sequence 0 −→ εE/λm −→ E/λm −→ (E/ε)/λm −→ 0 | | Uλn Uλm and hence an extension class in Ext1(Uλm, Uλn ). One checks now that (1.8) is a map of O-modules. We define H1 f (Qp, Vλn ) to be the inverse image of Ext1 fl(Uλn , Uλn ) under (1.8), i.e., those extensions which are already extensions in the category of finite flat group schemes Zp. Observe that Ext1 fl(Uλn , Uλn )∩ Ext1 O[Dp](Uλn , Uλn ) is an O-module, so H1 f (Qp, Vλn ) is seen to be an O-submodule of H1(Qp, Vλn ). We observe that our definition is equivalent to requiring that the classes in H1 f (Qp, Vλn ) map under (1.8) to Ext1 fl(Uλm, Uλn ) for all m ≥ n. For if em is the extension class in Ext1(Uλm, Uλn ) then em ,→ en ⊕Uλm as Galois-modules and we can apply results of [Ray1] to see that em comes from a finite flat group scheme over Zp if en does. In the flat (non-ordinary) case ρ0|Ip is determined by Raynaud’s results as mentioned at the beginning of the chapter. It follows in particular that, since ρ0|Dp is absolutely irreducible, V (Qp = H0(Qp, V ) is divisible in this case (in fact V (Qp) KT/O). This H1(Qp, Vλn ) H1(Qp, V )λn and hence we can define H1 f (Qp, V ) = ∞ n=1 H1 f (Qp, Vλn ), and we claim that H1 f (Qp, V )λn H1 f (Qp, Vλn ). To see this we have to compare representations for m ≥ n, ρn,m : Gal(Q¯ p/Qp) −→ GL2(On[ε]/λm) ϕm,n ρm,m : Gal(Q¯ p/Qp) −→ GL2(Om[ε]/λm) where ρn,m and ρm,m are obtained from αn ∈ H1(Qp,VXλn ) and im(αn) ∈ H1(Qp, Vλm) and ϕm,n : a+bε → a+λm−nbε. By [Ram, Prop 1.1 and Lemma 2.1] if ρn,m comes from a finite flat group scheme then so does ρm,m. Conversely ϕm,n is injective and so ρn,m comes from a finite flat group scheme if ρm,m does; cf. [Ray1]. The definitions of H1 D(QΣ/Q, Vλn ) and H1 D(QΣ/Q, V ) now extend to the flat case and we note that (1.7) is also valid in the flat case. Still in the flat (non-ordinary) case we can again use the determination of ρ0|Ip to see that H1(Qp, V ) is divisible. For it is enough to check that H2(Qp, Vλ) = 0 and this follows by duality from the fact that H0(Qp, V ∗ λ )=0
ANDREW JOHN WILES where V*= Hom(VA, up) and up is the group of pth roots of unity.(Again this follows from the explicit form of polp. Much subtler is the fact that HF(Qp, V) is divisible. This result is essentially due to Ra using a local version of Proposition 1. 1 below we have that makrishna.For Homo(pr/pa, K/0)cH(Qp,V where R is the universal local fat deformation ring for PolD, and O-algebras (This exists by Theorem 1.1 of Ram] because polD is absolutely irreducible. Since R Rn O where r is the corresponding ring for W(k-algebras w(k) the main theorem of Ram, Th. 4.2 shows that R is a power series ring and the divisibility of Hf(Qp, v)then follows. We refer to Ram for more details Next we need an analogue of(1.5)for V. Again this is a variant of standard results in deformation theory and is given(at least for D=(ord, >,W(k),o with some restriction on x1, x2 in i(a) in MT, Prop 25 PROPOSITION 1. 2. Suppose that pf a is a deformation of po of type D=( 2,O, M)withO an unramified ertension of Of, A. Then as O-modules Homo(pp/p3,K/0)cHD(Qx/Q,v) Remark. The isomorphism is functorial in an obvious way if one changes Proof. We will just describe the Selmer case with M= o as the other cases use similar arguments. Suppose that a is a cocycle which represents a cohomology class in HSe(Q>/Q, VAn). Let O,[e] denote the ring O[el/(A"E, E2) We can associate to a a representation Pa: Gal(Qx/Q)+GL2(OnE) follows: set Pa(g)=a(g)p (9), a priori in GL2(O), is viewed GL2(OnE) via the natural mapping 0-OnE. Here a basis for O is chosen so that the representation Pf, A on the decomposition group Dp C Gal(Qz/Q) has the upper triangular form of (i)(a), and then a(g)E VAn is viewed in GL2(One) by identifying 1+ye Iker: GL2(On[E)-GL2(O) en
464 ANDREW JOHN WILES where V ∗ λ = Hom(Vλ, µp) and µp is the group of pth roots of unity. (Again this follows from the explicit form of ρ0| Dp .) Much subtler is the fact that H1 f (Qp, V ) is divisible. This result is essentially due to Ramakrishna. For, using a local version of Proposition 1.1 below we have that HomO(pR/p2 R, K/O) H1 f (Qp, V ) where R is the universal local flat deformation ring for ρ0|Dp and O-algebras. (This exists by Theorem 1.1 of [Ram] because ρ0|Dp is absolutely irreducible.) Since R Rfl ⊗ W(k) O where Rfl is the corresponding ring for W(k)-algebras the main theorem of [Ram, Th. 4.2] shows that R is a power series ring and the divisibility of H1 f (Qp, V ) then follows. We refer to [Ram] for more details about Rfl. Next we need an analogue of (1.5) for V . Again this is a variant of standard results in deformation theory and is given (at least for D = (ord, Σ, W(k), φ) with some restriction on χ1, χ2 in i(a)) in [MT, Prop 25]. Proposition 1.2. Suppose that ρf,λ is a deformation of ρ0 of type D = (·, Σ, O,M) with O an unramified extension of Of,λ. Then as O-modules HomO(pD/p2 D, K/O) H1 D(QΣ/Q, V ). Remark. The isomorphism is functorial in an obvious way if one changes D to a larger D . Proof. We will just describe the Selmer case with M = φ as the other cases use similar arguments. Suppose that α is a cocycle which represents a cohomology class in H1 Se(QΣ/Q, Vλn ). Let On[ε] denote the ring O[ε]/(λnε, ε2). We can associate to α a representation ρα : Gal(QΣ/Q) → GL2(On[ε]) as follows: set ρα(g) = α(g)ρf,λ(g) where ρf,λ(g), a priori in GL2(O), is viewed in GL2(On[ε]) via the natural mapping O→On[ε]. Here a basis for O2 is chosen so that the representation ρf,λ on the decomposition group Dp ⊂ Gal(QΣ/Q) has the upper triangular form of (i)(a), and then α(g) ∈ Vλn is viewed in GL2(On[ε]) by identifying Vλn 1 + yε xε zε 1 − tε = {ker : GL2(On[ε]) → GL2(O)}. Then W0 λn = 1 xε 1 ,
MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM 1+ ye 1+ 1 d 1-te One checks readily that Pa is a continuous homomorphism and that the defor- mation Pa] is unchanged if we add a coboundary to a We need to check that [ pal is a Selmer deformation. Let H Gal(QpQun)and g= Gal(Qpn/ Qp). Consider the exact sequence of O[g] 0→(V/wn)→(Vx/n)→X→0 where X is a submodule of (VAn/VAn). Since the action of p on VAn/Van via a character which is nontrivial mod A (it equals x2XI mod A and x1 x2), we see that X9=0 and H(g, X)=0. Then we have an exact diagram of O-modules HH(G,、(V/WQn)x)≈H1(9,(Vx/Wn)+) H(Qp, VAn/won) H(Qpn, VAn/won)s By hypothesis the image of a is zero in H(Qunr, VAn/won).Hence it is in the image of H(g, (VAn/WAn)H). Thus we can assume that it is rep resented in H(Qp, VAn/won) by a cocycle, which maps g to Vn /win; i.e f(Dp)cVn/ won, f(Ip)=0. The difference between f and the image of a is a coboundary oHoii-i for some u E VAn By subtracting the coboundary toHou-u from a globally we get a new a such that a= f as cocycles mapping g to VAn /wAn. Thus a(Dp)C Vn, a(Ip)c won and it is now easy to check that [pal is a Selmer deformation of Po Since [pal is a Selmer deformation there is a unique map of local O algebras Pa:RD→Onle] inducing it.(IfM≠φ we must check the
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 465 W1 λn = 1 + yε xε 1 − yε , Wλn = 1 + yε xε zε 1 − yε , and V 1 λn = 1 + yε xε 1 − tε . One checks readily that ρα is a continuous homomorphism and that the deformation [ρα] is unchanged if we add a coboundary to α. We need to check that [ρα] is a Selmer deformation. Let H = Gal(Q¯ p/Qunr p ) and G = Gal(Qunr p /Qp). Consider the exact sequence of O[G]- modules 0 → (V 1 λn /W0 λn ) H → (Vλn /W0 λn ) H → X → 0 where X is a submodule of (Vλn /V 1 λn )H. Since the action of p on Vλn /V 1 λn is via a character which is nontrivial mod λ (it equals χ2χ−1 1 mod λ and χ1 ≡ χ2), we see that XG = 0 and H1(G, X)=0. Then we have an exact diagram of O-modules 0 H1(G,(V 1 λn /W0 λn )H) H1(G,(Vλn /W0 λn )H) H1(Qp, Vλn /W0 λn ) H1(Qunr p , Vλn /W0 λn )G. By hypothesis the image of α is zero in H1(Qunr p , Vλn /W0 λn )G. Hence it is in the image of H1(G,(V 1 λn /W0 λn )H). Thus we can assume that it is represented in H1(Qp, Vλn /W0 λn ) by a cocycle, which maps G to V 1 λn /W0 λn ; i.e., f(Dp) ⊂ V 1 λn /W0 λn , f(Ip)=0. The difference between f and the image of α is a coboundary {σ → σµ¯ −µ¯} for some u ∈ Vλn . By subtracting the coboundary {σ → σu − u} from α globally we get a new α such that α = f as cocycles mapping G to V 1 λn /W0 λn . Thus α(Dp) ⊂ V 1 λn , α(Ip) ⊂ W0 λn and it is now easy to check that [ρα] is a Selmer deformation of ρ0. Since [ρα] is a Selmer deformation there is a unique map of local Oalgebras ϕα : RD → On[ε] inducing it. (If M = φ we must check the
ANDREW JOHN WILES other conditions also. )Since Pa= Pf A mod e we see that restricting a to pD ives a homomorphism of O-modules O/入 such that Pa(pD)=0. Thus we have defined a map p: c-a p: HSe(Q2/Q, VAn)- Homo(pD/p2,O/A") It is straightforward to check that this is a map of O-modules. To check the injectivity of p suppose that Pa(pD)=0 Then Pa factors through RD/pD 2 O and being an O-algebra homomorphism this determines Pa. Thus [ Pf, x]=[pa] If A- PaA=Pf a then A mod e is seen to be central by Schur's lemma and so may be taken to be I. a simple calculation now shows that a is a coboundary To see that yp is surjective choose 业∈Homo(pn/p2,O/A2) Then Py: Gal(Qz/Q)-GL2(RD/(pz, ker y))is induced by a representative of the universal deformation(chosen to equal pf, a when reduced mod pp)and we define a map ay: Gal( Qz/Q)-Van by 1+pp/(pz, ker Y) pp/(p3, ker y) av(g)=p(g)p,x(9)-1∈ CAN D/(p2, ker y) 1+pp/(p2, ker Y) where Pf,A(g)is viewed in GL2(RD/(pD, ker y)) via the structural InaD rD(RD being an O-algebra and the structural map being local because of the existence of a section). The right-hand inclusion comes from pD/(p2,kery)o/x=(/Ax)·∈ 1 Then ay is really seen to be a continuous cocycle whose cohomology class lies in HSe(Qz/Q, VAn). Finally p(ay)=Y. Moreover, the constructions are compatible with change of n, i.e., for VAn<VAn+I and A: O/AncO/An+l. D We now relate the local cohomology groups we have defined of Fontaine and in particular to the groups of Bloch-Kato BK will dis- Hl for the cohomolo os of bloch-Kato. non of the results described in the rest of this section are used in the rest of the paper. They serve only to relate the Selmer groups we have defined(and later compute)to the more standard versions. Using the lattice associated to pf, we obtain also a lattice T=O with Galois action via Ad pf, A. Let V=TOzp Qp be associated vector space and identify V with V/T. Let pr: V-V be
466 ANDREW JOHN WILES other conditions also.) Since ρα ≡ ρf,λ mod ε we see that restricting ϕα to pD gives a homomorphism of O-modules, ϕα : pD → ε.O/λn such that ϕα(p2 D)=0. Thus we have defined a map ϕ : α → ϕα, ϕ : H1 Se(QΣ/Q, Vλn ) → HomO(pD/p2 D, O/λn). It is straightforward to check that this is a map of O-modules. To check the injectivity of ϕ suppose that ϕα(pD)=0. Then ϕα factors through RD/pD O and being an O-algebra homomorphism this determines ϕα. Thus [ρf,λ]=[ρα]. If A−1ραA = ρf,λ then A mod ε is seen to be central by Schur’s lemma and so may be taken to be I. A simple calculation now shows that α is a coboundary. To see that ϕ is surjective choose Ψ ∈ HomO(pD/p2 D, O/λn). Then ρΨ : Gal(QΣ/Q) → GL2(RD/(p2 D, ker Ψ)) is induced by a representative of the universal deformation (chosen to equal ρf,λ when reduced mod pD) and we define a map αΨ : Gal(QΣ/Q) → Vλn by αΨ(g) = ρΨ(g)ρf,λ(g)−1 ∈ 1 + pD/(p2 D, ker Ψ) pD/(p2 D, ker Ψ) pD/(p2 D, ker Ψ) 1 + pD/(p2 D, ker Ψ) ⊆ Vλn where ρf,λ(g) is viewed in GL2(RD/(p2 D, ker Ψ)) via the structural map O → RD (RD being an O-algebra and the structural map being local because of the existence of a section). The right-hand inclusion comes from pD/(p2 D, ker Ψ) Ψ ,→ O/λn ∼ → (O/λn) · ε 1 → ε. Then αΨ is really seen to be a continuous cocycle whose cohomology class lies in H1 Se(QΣ/Q, Vλn ). Finally ϕ(αΨ)=Ψ. Moreover, the constructions are compatible with change of n, i.e., for Vλn ,→Vλn+1 and λ:O/λn ,→ O/λn+1. We now relate the local cohomology groups we have defined to the theory of Fontaine and in particular to the groups of Bloch-Kato [BK]. We will distinguish these by writing H1 F for the cohomology groups of Bloch-Kato. None of the results described in the rest of this section are used in the rest of the paper. They serve only to relate the Selmer groups we have defined (and later compute) to the more standard versions. Using the lattice associated to ρf,λ we obtain also a lattice T O4 with Galois action via Ad ρf,λ. Let V = T ⊗Zp Qp be associated vector space and identify V with V/T. Let pr : V → V be
MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM 467 the natural projection and define cohomology modules b H}(Qp,少)=ker:H2(Q,V)→H( Qp, v8 Borys), HF(Qp,v)=pr(HF( Qp, v)cH'(Qp, V). HE(Qp, Vw)=(n )-(HF(Qp, v)cHI where in: VAn -V is the natural map and the two groups in the definition of HF(Qp,v) are defined using continuous cochains. Similar definitions apply to V*= HomQ, (v, Qp(1) and indeed to any finite-dimensional continuous p-adic representation space. The reader is cautioned that the definition of HF(Qp, VAn)is dependent on the lattice T(or equivalently on V). Under certainly conditions Bloch and Kato show, using the theory of Fontaine and Lafaille, that this is independent of the lattice(see BK, Lemmas 4.4 and 4.5). In any case we will consider in what follows a fixed lattice associated to p=Pf, A, Ad p, etc. Henceforth we will only use the notation HF(Qp,-)when the underlying vector space is crystalline PROPOSITION 1.3. (i) If po is flat but ordinary and pf, a is associated to a p-divisible group then for all n HF(Qp, VAn)=HF(Qp, VAn (i)Ifps A is ordinary, det Ps. A pe and pf, A is associated to a p-divisible group, then for all n, HF(Qp, VAn)C HSe(Qp, VAn Proof. Beginning with(i), we define Hf(Qp,v)=aE H(Qp, v) K(a/An)E H(Qp, V) for all n) where K: H(Qp, v)+H(Qp, V).Then we see that in case(i), H(Qp, v) is divisible. So it is enough to how that HF(Qp,V)=HF(Qp,V) We have to compare two constructions associated to a nonzero element a of H(Qp,v). The first is to associate an extension 0→1→E→K→0 of K-vector spaces with commuting continuous Galois action. If we fix an e with S(e)= l the action on e is defined by ae =e+a(o) with aa cocycle representing a. The second construction begins with the image of the subspace (a) in H(Qp, V). By the analogue of Proposition 1.2 in the local case, there is an O-module isomorphism H(Qp, V)= Homo(pr/pR, K/o
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 467 the natural projection and define cohomology modules by H1 F (Qp, V) = ker : H1(Qp, V) → H1(Qp, V ⊗ Qp Bcrys), H1 F (Qp, V ) = pr H1 F (Qp, V) ⊂ H1(Qp, V ), H1 F (Qp, Vλn )=(jn) −1 H1 F (Qp, V ) ⊂ H1(Qp, Vλn ), where jn : Vλn → V is the natural map and the two groups in the definition of H1 F (Qp, V) are defined using continuous cochains. Similar definitions apply to V∗ = HomQp (V, Qp(1)) and indeed to any finite-dimensional continuous p-adic representation space. The reader is cautioned that the definition of H1 F (Qp, Vλn ) is dependent on the lattice T (or equivalently on V ). Under certainly conditions Bloch and Kato show, using the theory of Fontaine and Lafaille, that this is independent of the lattice (see [BK, Lemmas 4.4 and 4.5]). In any case we will consider in what follows a fixed lattice associated to ρ = ρf,λ, Ad ρ, etc. Henceforth we will only use the notation H1 F (Qp, −) when the underlying vector space is crystalline. Proposition 1.3. (i) If ρ0 is flat but ordinary and ρf,λ is associated to a p-divisible group then for all n H1 f (Qp, Vλn ) = H1 F (Qp, Vλn ). (ii) If ρf,λ is ordinary, det ρf,λ Ip = ε and ρf,λ is associated to a p-divisible group, then for all n, H1 F (Qp, Vλn ) ⊆ H1 Se(Qp, Vλn . Proof. Beginning with (i), we define H1 f (Qp, V) = {α ∈ H1(Qp, V) : κ(α/λn) ∈ H1 f (Qp, V ) for all n} where κ : H1(Qp, V) → H1(Qp, V ). Then we see that in case (i), H1 f (Qp, V ) is divisible. So it is enough to how that H1 F (Qp, V) = H1 f (Qp, V). We have to compare two constructions associated to a nonzero element α of H1(Qp, V). The first is to associate an extension (1.9) 0 →V→ E δ → K → 0 of K-vector spaces with commuting continuous Galois action. If we fix an e with δ(e) = 1 the action on e is defined by σe = e + ˆα(σ) with ˆα a cocycle representing α. The second construction begins with the image of the subspace #α$ in H1(Qp, V ). By the analogue of Proposition 1.2 in the local case, there is an O-module isomorphism H1(Qp, V ) HomO(pR/p2 R, K/O)