《费尔马大定理证明》Modular elliptic curves and Fermat’s Last Theorem（英文版）

ANDREW JOHN WILES where R is the universal deformation ring of Po viewed as a representation of Gal(Qp/Q) on O-algebras and pR is the ideal of R corresponding to pp (i. e, its inverse ima age in R). Since a#0, associated to (a) is a quotient pR/PR,a)of pR/pR which is a free O-module of rank one. We then obtain a homomorphism Pa: Gal(Qp /Q)+GL2(R/ (Pi, a) induced from the universal deformation(we pick a representation in the uni- versal class ). This is associated to an O-module of rank 4 which tensored with K gives a K-vector space Ec(K) which is an extension 0→→E→l→0 where u a k has the Galis representation pf, a(viewed locally) In the first construction a E HF(Qp, v) if and only if the extension(1.9 )is crystalline, as the extension given in(1.9)is a sum of copies of the more usual extension where Qp replaces K in(1.9). On the other hand a)CH(Qp, v)if and only if the second construction can be made through R, or equivalently if and only if E is the representation associated to a p-divisible group a priori, the representation associated to Pa only has the property that on all finite quotients it comes from a finite flat group scheme. However a theorem of Raynaud Rayl says that then Pa comes from a p-divisible group. For more details on R", the universal fat deformation ring of the local representation Po, see Ram. )Now the extension E comes from a p-divisible group if and ly if it is crystalline; cf. Fo, $6 at(1.9) if and only if(1.10) is crystalline One obtains(1.10) from(1.)as follows. We view V as Homk(u, U)and X=ker:(Homk(u, 7ou-u1 where the map is the natural one f k f(w).(All tensor products in this proof will be as K-vector spaces. )Then as KDpI-modules E"c(E⑧)/X To check this, one calculates explicitly with the definition of the action on (given above on e) and on e(given in the proof of Proposition 1.1). It follows from standard properties of crystalline representations that if E is crystalline, so is E &U and also E. Conversely, we can recover E from E as follows E"⑧lc(E⑧W⑧l1/(X⑧D). Then there is p: E 8(det)-E'ou induced by the direct sum decomposition u u2 ( det) Symu. Here det denotes a 1-dimensional vector space over K with Galois action via det Pf, A. Now we claim that p is injective on ve(det). For

468 ANDREW JOHN WILES where R is the universal deformation ring of ρ0 viewed as a representation of Gal(Q¯ p/Q) on O-algebras and pR is the ideal of R corresponding to pD (i.e., its inverse image in R). Since α = 0, associated to #α$ is a quotient pR/(p2 R, a) of pR/p2 R which is a free O-module of rank one. We then obtain a homomorphism ρα : Gal(Q¯ p/Qp) → GL2  R/(p2 R, a)  induced from the universal deformation (we pick a representation in the uni￾versal class). This is associated to an O-module of rank 4 which tensored with K gives a K-vector space E (K)4 which is an extension (1.10) 0 →U→ E →U→ 0 where U K2 has the Galis representation ρf,λ (viewed locally). In the first construction α ∈ H1 F (Qp, V) if and only if the extension (1.9) is crystalline, as the extension given in (1.9) is a sum of copies of the more usual extension where Qp replaces K in (1.9). On the other hand #α$ ⊆ H1 f (Qp, V) if and only if the second construction can be made through Rfl, or equivalently if and only if E is the representation associated to a p-divisible group. A priori, the representation associated to ρα only has the property that on all finite quotients it comes from a finite flat group scheme. However a theorem of Raynaud [Ray1] says that then ρα comes from a p-divisible group. For more details on Rfl, the universal flat deformation ring of the local representation ρ0, see [Ram].) Now the extension E comes from a p-divisible group if and only if it is crystalline; cf. [Fo, §6]. So we have to show that (1.9) is crystalline if and only if (1.10) is crystalline. One obtains (1.10) from (1.9) as follows. We view V as HomK(U, U) and let X = ker : {HomK(U, U) ⊗U → U} where the map is the natural one f ⊗ w → f(w). (All tensor products in this proof will be as K-vector spaces.) Then as K[Dp]-modules E (E ⊗ U)/X. To check this, one calculates explicitly with the definition of the action on E (given above on e) and on E (given in the proof of Proposition 1.1). It follows from standard properties of crystalline representations that if E is crystalline, so is E ⊗ U and also E . Conversely, we can recover E from E as follows. Consider E ⊗U (E ⊗U⊗U)/(X ⊗ U). Then there is a natural map ϕ : E ⊗ (det) → E ⊗ U induced by the direct sum decomposition U⊗U (det) ⊕ Sym2U. Here det denotes a 1-dimensional vector space over K with Galois action via det ρf,λ. Now we claim that ϕ is injective on V ⊗ (det). For

MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM 469 if f e v then p(f)=fo(w1 8w2-w20w1)where w1, w2 are a basis for l for which wl∧2=1 in detN K. So if e(f)∈ Xou then f(w1ow2-f(w2)0w1=0 inuou But this is false unless f(wn)= f(2)=0 whence f=0. So p is injective on v@ det and if y itself were not injective then E would split contradicting at0. So p is injective and we have exhibited E o(det)as a subrepresentation of Eol which is crystalline. We deduce that E is crystalline if E is. This completes the proof of (i To prove(i)we check first that Hse(Q, Van)=jn(Hse(Qp, v)(this was already used in(1.7). We next have to show that HF(Qp, v)cHSe(Qp,v) where the latter is defined by HSe(Qp, v)=ker: H(Qp, v)-H(Qp, v/o) with yo the subspace of v on which I, acts via E. But this follows from the computations in Corollary 3.8.4 of BK]. Finally we observe that (HSe(Qp, v)sHSe(Q, v) lthough the inclusion may be strict, and pr(Hf(Qp, v))=Hf(Qp, v) by definition. This completes the proof These groups have the property that for s>r, 1.11 H(Qp,VA)nJr (HF(Q V)=HF(Q Vr) where j r, s: VAr -Vs is the natural injection. The same holds for VA and VAs in place of VAr and Vis where Var is defined by Hom(Vx;μpr and similarly for VAs. Both results are immediate from the definition(and indeed were part of the motivation for the definition) We also give a finite level version of a result of Bloch-Kato which deduced from the vector space version. As before let T be a galois stable lattice so that Ta 04. Define HF(Qp,T)=i(HF(Qp, v under the natural inclusion i: Ta v. and likewise for the dual lattice T* Homz,(V,(Qp/Zp)(1) in V".(Here V*=Hom(v, Qp(1); throughout this paper we use M* to denote a dual of M with a Cartier twist. Also write

MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 469 if f ∈ V then ϕ(f) = f ⊗ (w1 ⊗ w2 − w2 ⊗ w1) where w1, w2 are a basis for U for which w1 ∧ w2 = 1 in det K. So if ϕ(f) ∈ X ⊗ U then f(w1) ⊗ w2 − f(w2) ⊗ w1 = 0 in U⊗U. But this is false unless f(w1) = f(w2) = 0 whence f = 0. So ϕ is injective on V ⊗ det and if ϕ itself were not injective then E would split contradicting α = 0. So ϕ is injective and we have exhibited E ⊗(det) as a subrepresentation of E ⊗ U which is crystalline. We deduce that E is crystalline if E is. This completes the proof of (i). To prove (ii) we check first that H1 Se(Qp, Vλn ) = j−1 n  H1 Se(Qp, V )  (this was already used in (1.7)). We next have to show that H1 F (Qp,V) ⊆ H1 Se(Qp,V) where the latter is defined by H1 Se(Qp, V) = ker : H1(Qp, V) → H1(Qunr p , V/V0) with V0 the subspace of V on which Ip acts via ε. But this follows from the computations in Corollary 3.8.4 of [BK]. Finally we observe that pr H1 Se(Qp, V)  ⊆ H1 Se(Qp, V ) although the inclusion may be strict, and pr H1 F (Qp, V)  = H1 F (Qp, V ) by definition. This completes the proof.
These groups have the property that for s ≥ r, (1.11) H1(Qp, V r λ ) ∩ j−1 r,s  H1 F (Qp, Vλs )  = H1 F (Qp, Vλr ) where jr,s : Vλr → Vλs is the natural injection. The same holds for V ∗ λr and V ∗ λs in place of Vλr and Vλs where V ∗ λr is defined by V ∗ λr = Hom(Vλr , µpr ) and similarly for V ∗ λs . Both results are immediate from the definition (and indeed were part of the motivation for the definition). We also give a finite level version of a result of Bloch-Kato which is easily deduced from the vector space version. As before let T ⊂ V be a Galois stable lattice so that T O4. Define H1 F (Qp, T) = i −1  H1 F (Qp, V)  under the natural inclusion i : T ,→ V, and likewise for the dual lattice T ∗ = HomZp (V,(Qp/Zp)(1)) in V∗. (Here V∗ = Hom(V, Qp(1)); throughout this paper we use M∗ to denote a dual of M with a Cartier twist.) Also write

470 ANDREW JOHN WILES prn: T-T/A" for the natural projection map, and for the mapping it induces on cohomolo PROPOSITION 1.4. If pf a is associated to a p-divisible group( the ordi- nary case is allowed) then (i)prn(HF(Qp, T)=HF(Qp, T/A")and similarly for T*,T*/An (ii)HF(Qp, VAn)is the orthogonal complement of F(Qp, v*) under Tate local duality between H (Qp, VAn)and H(Qp, Van)and similarly for WAr and wan replacing VAn and Van More generally these results hold for any crystalline representation v'in place ofv and x' a uniformize in K where K is any finite extension of Qp aith KC EndGal( Q/Q)V Proof. We first observe that prn(HF(Qp, r))C HF(Qp, T/A").Now from the construction we may identify T/An with VAn. A result of Bloch- Kato(BK, Prop. 3.8)says that HF(Qp, v) and HF(Qp, v*)are orthogonal complements under Tate local duality. It follows formally that HF(Qp, VAn) and pr,(HF(Qp, T)are orthogonal complements, so to prove the proposition it is enough to show that (1.12) #HF(Qp, VAn)#HF(Qp, VAn)=#h(Qp, V Now if r=dimk H(Qp, v)and s= dimK HF(Qp, v")then (1.13 r+s= dimK H(Qp,1)+ dimK H°(Qp,y*)+dimk少 From the definition (1. 14) #HF(Qp, VAn)=#(O/A").#ker(H(Qp, VAn)-+H(Qp, V) The second factor is equal to #V(Qp)/Anv(Qp). When we write V(Qp)div for the maximal divisible subgroup of V(Qp) this is the same as +(V(Qp)/V(Qp)/"=#(V(Qp)/V(Qp))a #V(Qp)an/#(v(Qp Combining this with(1.14) gives (1.15)#H(Qp,Vx)=#(O/A) #H(Qp,Vn)/#(O/A)山 This, together with an analogous formula for #HF(Qp, VAn)and(1. 13), gives #HF(Qp,VA)#HF(Qp, VAn)=#(O/A).#H(Qp, VAn)#H(Qp, VAn

470 ANDREW JOHN WILES prn : T → T /λn for the natural projection map, and for the mapping it induces on cohomology. Proposition 1.4. If ρf,λ is associated to a p-divisible group (the ordi￾nary case is allowed) then (i) prn  H1 F (Qp, T)  = H1 F (Qp, T /λn) and similarly for T ∗, T ∗/λn. (ii) H1 F (Qp, Vλn ) is the orthogonal complement of H1 F (Qp, V ∗ λn ) under Tate local duality between H1(Qp, Vλn ) and H1(Qp, V ∗ λn ) and similarly for Wλn and W∗ λn replacing Vλn and V ∗ λn . More generally these results hold for any crystalline representation V in place of V and λ a uniformizer in K where K is any finite extension of Qp with K ⊂ EndGal(Qp/Qp)V . Proof. We first observe that prn(H1 F (Qp, T)) ⊂ H1 F (Qp, T /λn). Now from the construction we may identify T /λn with Vλn . A result of Bloch￾Kato ([BK, Prop. 3.8]) says that H1 F (Qp, V) and H1 F (Qp, V∗) are orthogonal complements under Tate local duality. It follows formally that H1 F (Qp, V ∗ λn ) and prn(H1 F (Qp, T)) are orthogonal complements, so to prove the proposition it is enough to show that (1.12) #H1 F (Qp, V ∗ λn )#H1 F (Qp, Vλn )=#H1(Qp, Vλn ). Now if r = dimK H1 F (Qp, V) and s = dimK H1 F (Qp, V∗) then (1.13) r + s = dimK H0(Qp, V) + dimK H0(Qp, V∗) + dimK V. From the definition, (1.14) #H1 F (Qp, Vλn ) = #(O/λn) r · # ker{H1(Qp, Vλn ) → H1(Qp, V )}. The second factor is equal to #{V (Qp)/λnV (Qp)}. When we write V (Qp)div for the maximal divisible subgroup of V (Qp) this is the same as #(V (Qp)/V (Qp) div)/λn = #(V (Qp)/V (Qp) div)λn = #V (Qp)λn /#(V (Qp) div)λn . Combining this with (1.14) gives #H1 F (Qp, Vλn ) = #(O/λn) r (1.15) · #H0(Qp, Vλn )/#(O/λn) dimKH0(Qp,V) . This, together with an analogous formula for #H1 F (Qp, V ∗ λn ) and (1.13), gives #H1 F (Qp, V λn )#H1 F (Qp, V ∗ λn ) = #(O/λn)4 · #H0(Qp, Vλn )#H0(Qp, V ∗ λn )

MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM As#H(Qp,v*An)=#H(Qp, VAn)the assertion of (1.12)now follows from the formula for the Euler characteristic of VA The proof for WAn, or indeed more generally for any crystalline repr tation, is the same We also give a characterization of the orthogonal complements of HSe(Qp, Win)and HSe(Qp, VAn), under Tate's local duality. We write these duals as HSe(Qp, Wan)and HSe-(Qp, VAn)respectively. Let Pu: H(Qp, win)-(Qp, Wan/Win) be the natural map where(Win)' is the orthogonal complement of WAn in An, and let Xn, i be defined as the image under the composite map Xn, i=im: Zp/(Zp)P8O/A-H(Qp, Hp"8O/A") →H1(Qp,Wn/(Wxn)) where in the middle term upn8O/An is to be identified with(WAn)/(WAn )o Similarly if we replace Win by VAn we let Yn, i be the image of Z/(Z)p8 (o/An)2in H (Qp, V*/(Wan)), and we replace u by the analogous map p. PROPOSITION 1.5 HSe-(Qp, WAn)=pa(Xn. i) HSe(Qp, VAn)=Pu (Yn, i Proof. This can be checked by dualizing the sequence 0→H(Qp,Wx)→He(Qp,Wxn) →ker:{H(Qp,Wx/(Wxn)→H2(Qm,Wx/(Wx)9}, where Hstr(Qp, WAn)=ker: H(Qp, WAn)-H(Qp, WAn/(WAn)).The term is orthogonal to ker: H(Qp, Win)-H(Qp, WA/(Win))).By naturality of the cup product pairing with respect to quotients and subgroups the claim then reduces to the well known fact that under the cup product paIring H(Qp,p)×H(Qp,Z/p)→Z/p the orthogonal complement of the unramified homomorphisms is the image of the units ZP/(Zp)P-H(Qp, Apm). The proof for VAn is essentially the same

MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 471 As #H0(Qp, V ∗λn)=#H2(Qp, Vλn ) the assertion of (1.12) now follows from the formula for the Euler characteristic of Vλn . The proof for Wλn , or indeed more generally for any crystalline represen￾tation, is the same.
We also give a characterization of the orthogonal complements of H1 Se(Qp, Wλn ) and H1 Se(Qp, Vλn ), under Tate’s local duality. We write these duals as H1 Se∗ (Qp, W∗ λn ) and H1 Se∗ (Qp, V ∗ λn ) respectively. Let ϕw : H1(Qp, W∗ λn ) → (Qp, W∗ λn /(W∗ λn ) 0) be the natural map where (W∗ λn )i is the orthogonal complement of W1−i λn in W∗ λn , and let Xn,i be defined as the image under the composite map Xn,i = im : Z× p /(Z× p ) pn ⊗ O/λn → H1(Qp, µpn ⊗ O/λn) → H1(Qp, W∗ λn /(W∗ λn ) 0) where in the middle term µpn ⊗ O/λn is to be identified with (W∗ λn )1/(W∗ λn )0. Similarly if we replace W∗ λn by V ∗ λn we let Yn,i be the image of Z× p /(Z× p )pn ⊗ (O/λn)2 in H1(Qp, V ∗ λn /(W∗ λn )0), and we replace ϕw by the analogous map ϕv. Proposition 1.5. H1 Se∗ (Qp, W∗ λn ) = ϕ−1 w (Xn,i), H1 Se∗ (Qp, V ∗ λn ) = ϕ−1 v (Yn,i). Proof. This can be checked by dualizing the sequence 0 → H1 Str(Qp, Wλn ) → H1 Se(Qp, Wλn ) → ker : {H1(Qp, Wλn /(Wλn ) 0) → H1(Qunr p , Wλn /(Wλn ) 0}, where H1 str(Qp, Wλn ) = ker : H1(Qp, Wλn ) → H1(Qp, Wλn /(Wλn )0). The first term is orthogonal to ker : H1(Qp, W∗ λn ) → H1(Qp, W∗ λn /(W∗ λn )1). By the naturality of the cup product pairing with respect to quotients and subgroups the claim then reduces to the well known fact that under the cup product pairing H1(Qp, µpn ) × H1(Qp, Z/pn) → Z/pn the orthogonal complement of the unramified homomorphisms is the image of the units Z× p /(Z× p )pn → H1(Qp, µpn ). The proof for Vλn is essentially the same.

472 ANDREW JOHN WILES 2. Some computations of cohomology groups We now make some comparisons of orders of cohomology groups usin the theorems of poitou and tate. We retain the notation and conventions of Section 1 though it will be convenient to state the first two propositions in a more general context. Suppose that L=ILn∈IH(Q,X p∈∑ is a subgroup, where X is a finite module for Gal(Qz/ Q)of p-power order We define I' to be the orthogonal complement of L under the perfect pairin (local Tate duality ∏m(Q2x×Ⅱ理2(Q,x)→Qn/Zn where X*= Hom(X, Hpoo) Let Ax:F(Q/Q,x)→ⅡH(Qx) q∈∑ be the localization map and similarly Ax. for X*. Then we set H(Q/Q,X)=入x(L,H、(Q/Q,X)=x(L) The following result was suggested by a result of Greenberg(cf. [ Grel])and is a simple consequence of the theorems of Poitou and Tate. Recall that p is always assumed odd and that p∈∑ PROPOSITION 1. 6 并H(QQ,X)/并H(Q/Q,x)=bxIh ∈∑ he hg =#H(Qq, X*/H(Qq, X): Lal hoo =#H(R, X*)#Ho(Q, X)/#H(Q, X*) Proof. Adapting theexact sequence proofof Poitou and Tate(cf [ Mi2, Th. 4.20) we get a seven term exact sequence 0→H(Qx/Q,X) H(Qz/Q,X) II H(Qq, X)/Lg q∈ II H(Qq, X) H(Qz/Q, X) Hi.Qz/Q,X*) ∑ Lm(QQ,x)→0

472 ANDREW JOHN WILES 2. Some computations of cohomologygroups We now make some comparisons of orders of cohomology groups using the theorems of Poitou and Tate. We retain the notation and conventions of Section 1 though it will be convenient to state the first two propositions in a more general context. Suppose that L = Lq ⊆  p∈Σ H1(Qq, X) is a subgroup, where X is a finite module for Gal(QΣ/Q) of p-power order. We define L∗ to be the orthogonal complement of L under the perfect pairing (local Tate duality)  q∈Σ H1(Qq, X) ×  q∈Σ H1(Qq, X∗) → Qp/Zp where X∗ = Hom(X, µp∞). Let λX : H1(QΣ/Q, X) →  q∈Σ H1(Qq, X) be the localization map and similarly λX∗ for X∗. Then we set H1 L(QΣ/Q, X) = λ−1 X (L), H1 L∗ (QΣ/Q, X∗) = λ−1 X∗ (L∗). The following result was suggested by a result of Greenberg (cf. [Gre1]) and is a simple consequence of the theorems of Poitou and Tate. Recall that p is always assumed odd and that p ∈ Σ. Proposition 1.6. #H1 L(QΣ/Q, X)/#H1 L∗ (QΣ/Q, X∗) = h∞  q∈Σ hq where  hq = #H0(Qq, X∗)/[H1(Qq, X) : Lq] h∞ = #H0(R, X∗)#H0(Q, X)/#H0(Q, X∗). Proof.AdaptingtheexactsequenceproofofPoitouandTate(cf.[Mi2,Th.4.20]) we get a seven term exact sequence 0 −→ H1 L(QΣ/Q, X) −→ H1(QΣ/Q, X) −→  q∈Σ H1(Qq, X)/Lq     q∈Σ H2(Qq, X) ←− H2(QΣ/Q, X) ←− H1 L∗ (QΣ/Q, X∗)∧ | → H0(QΣ/Q, X∗)∧ −→ 0