A motivic conjecture of Milne

Adrian Vasiu

We disprove the following conjecture due to Vı́ctor Neumann-Lara: for every pair (r, s) of integers such that r > s > 2, there is an infinite set of circulant tournaments T such that the dichromatic number and the cyclic triangle free disconnection of T are equal to r and s, respectively. Let Fr,s denote the set of circulant tournaments T with dc(T ) = r and − →ω 3 (T ) = s. We show that for every integer s > 4 there exists a lower bound b(s) for the dichromatic number r such that Fr,s = ∅ for every r < b(s). We construct an infinite set of circulant tournaments T such that dc(T ) = b(s) and − →ω 3(T ) = s and give an upper bound B(s) for the dichromatic number r such that for every r > B(s) there exists an infinite set Fr,s of circulant tournaments. Some infinite sets Fr,s of circulant tournaments are given for b(s) < r < B(s).

A motivic conjecture of Milne arXiv:math/0308202v4 [math.NT] 17 Nov 2008 Adrian Vasiu, Binghamton University November 17, 2008 ABSTRACT. Let k be an algebraically closed field of characteristic p > 0. Let W (k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates the étale cohomology with Zp coefficients to the crystalline cohomology with integral coefficients, in the wider context of p-divisible groups endowed with families of crystalline tensors over a finite, discrete valuation ring extension of W (k). The result extends work of Faltings. As a main new tool we construct global deformations of pdivisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W (k). Key words: étale and crystalline cohomologies with integral coefficients, p-divisible groups, affine group schemes, and deformations. MSC 2000: 11G10, 11G18, 11S25, 14F30, 14G35, 14L05, and 20G25. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Preliminaries 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Global deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4. Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 33 5. The ramified context . . . . . . . . . . . . . . . . . . . . . . . . . . 45 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1. Introduction Let p ∈ N be a prime. Let k be a perfect field of characteristic p. Let W (k) be the ring of Witt vectors with coefficients in k. In this paper we study p-divisible groups endowed with families of étale and crystalline tensors over (finite discrete valuation ring extensions of) W (k). We begin by introducing families of tensors. Let Spec(R) be an affine scheme. For a finitely generated, projective R-module N , let N ∗ := HomR (N, R) and let GL N be the reductive group scheme over Spec(R) of automorphisms of N . For s ∈ N ∪ {0}, let N ⊗s := N ⊗R · · · ⊗R N , the number of copies of N being s. By the essential tensor algebra of N we mean T(N ) := ⊕s,t∈N∪{0} N ⊗s ⊗R N ∗⊗t . 1 Let xR ∈ R be a non-divisor of 0. A family of tensors of T(N [ x1R ]) = T(N )[ x1R ] is denoted (wα )α∈J , with J as the set of indices. The scalar extensions of wα are also denoted by wα . Let N1 be another finitely generated, projective R-module. Let (w1,α )α∈J be a family of tensors of T(N1 [ x1R ]) indexed by the same set J. By an isomorphism ∼ f : (N, (wα )α∈J ) → (N1 , (w1,α )α∈J ) ∼ we mean an R-linear isomorphism f : N → N1 that extends naturally to an R[ x1R ]-linear ∼ T(N1 [ x1R ]) that takes wα to w1,α for all α ∈ J. If E is a smooth, isomorphism T(N [ x1R ]) → closed subgroup scheme of GL N , let Lie(E) be its Lie algebra over R. If † is a Spec(R)-scheme (or a morphism of Spec(R)-schemes or a p-divisible group over Spec(R)) and if Spec(R̃) → Spec(R) is an affine morphism, we define †R̃ := † ×Spec(R) Spec(R̃). Similarly we define †R̃ (resp. †∗,R̃ ), starting from †R (resp. †∗ with ∗ an index). Let σ := σk be the Frobenius automorphism of W (k) and B(k) induced from k. We fix an algebraic closure B(k) of B(k). For K a subfield of B(k) that contains B(k), let Gal(K) := Gal(B(k)/K). 1.1. On p-divisible groups over Spec(W (k)) endowed with tensors. Let D be a p-divisible group over Spec(W (k)). Let (M, φ) be the contravariant Dieudonné module of Dk . Thus M is a free W (k)-module whose rank rkW (k) (M ) equals to the height of D and φ is a σ-linear endomorphism of M such that we have pM ⊆ φ(M ). Let F 1 be the direct summand of M that is the Hodge filtration defined by D. We have φ(M + p1 F 1 ) = M . The rank of F 1 is the dimension of Dk . We refer to (M, F 1 , φ) as the filtered Dieudonné module of D. Let (F i (M ))i∈{0,1,2} be the decreasing, exhaustive, and separated filtration of M defined by F 2 (M ) := 0, F 1 (M ) := F 1 , and F 0 (M ) := M . Let (F i (M ∗ ))i∈{−1,0,1} be the decreasing, exhaustive, and separated filtration of M ∗ defined by F −1 (M ∗ ) = M ∗ , F 0 (M ∗ ) := {x ∈ M ∗ |x(F 1 ) = 0}, and F 1 (M ∗ ) = 0. We endow T(M ) with the tensor product filtration F i (T(M ))i∈Z defined by (F i (M ))i∈{0,1,2} and (F i (M ∗ ))i∈{−1,0,1} . The decreasing, exhaustive, and separated filtration F i (T(M ))i∈Z of T(M ) depends only on F 1 and therefore we also call it the filtration of T(M ) defined by F 1 . For ‡ ∈ M ∗ [ p1 ] let φ(‡) := σ ◦ ‡ ◦ φ−1 ∈ M ∗ [ p1 ]. Thus φ acts in the usual tensor product way on T(M [ p1 ]). Under the identification EndW (k) (M ) = M ⊗W (k) M ∗ , φ maps △∈ EndB(k) (M [ p1 ]) to φ◦ △ ◦φ−1 ∈ EndB(k) (M [ p1 ]). Let (tα )α∈J be a family of tensors of F 0 (T(M ))[ p1 ] ⊆ T(M [ p1 ]) which are fixed by φ; thus tα ∈ {x ∈ F 0 (T(M ))[ p1 ]|φ(x) = x}). Let G be the schematic closure in GLM of the subgroup of GLM [ p1 ] that fixes tα for all α ∈ J. It is a flat, closed subgroup scheme of GL M such that we have φ(Lie(GB(k) )) = Lie(GB(k) ). Thus the pair (Lie(GB(k) ), φ) is a Lie F -subisocrystal of (EndB(k) (M [ p1 ]), φ). Quadruples of the form (M, φ, G, (tα)α∈J ) play key roles in the study of special fibres of good integral models of Shimura varieties of Hodge type in mixed characteristic (0, p) (see [LR], [Ko], and [Va1, Sect. 5] for concrete situations with G a reductive group scheme). t Let Dt be the Cartier dual of Dt . Let H 1 (D) := Tp (DB(k) )(−1) be the dual of the 1 Tate-module Tp (DB(k) ) of DB(k) . Thus H (D) is a free Zp -module of rank rkW (k) (M ) and Gal(B(k)) acts on it. We disregard the Galois action whenever we consider GL H 1 (D)⊗Zp R 2 or pairs of the form (H 1 (D)⊗Zp R, (wα )α∈J ), with R as a Zp -algebra. Let vα ∈ T(H 1 (D)[ p1 ]) be the Gal(B(k))-invariant tensor that corresponds to tα via Fontaine comparison theory ´ (see [Fo3, Subsect. 5.5], [Fa2, Sect. 6], Subsubsection 2.2.2, etc.). Let Get Zp be the schematic closure in GL H 1 (D) of the subgroup of GL H 1 (D)[ p1 ] that fixes vα for all α ∈ J. The next definition plays a key role when p = 2. 1.1.1. Definition. We say the property (C) holds (for (M, φ, G)) if there exists no element h ∈ G(W (k̄)) such that the Dieudonné module (M ⊗W (k) W (k̄), h(φ ⊗ σk̄ )) over k̄ has both Newton polygon slopes 0 and 1 with positive multiplicities. The main goal of this paper is to construct global deformations of the pairs (D, (tα )α∈J ) and (D, (vα )α∈J ) and to use them to prove the following Main Theorem. ´ 1.2. Main Theorem. Suppose that k = k̄. If p = 2 we also assume that either Get Zp is a torus or the property (C) holds. Then there exists an isomorphism ∼ ρ : (M, (tα )α∈J ) → (H 1 (D) ⊗Zp W (k), (vα )α∈J ). ´ 1.3. Corollary. If p = 2, we assume that either Get Zp is a torus or the property (C) holds. ´ (a) Then the pulls back to Spec(W (k̄)) of G and Get Zp are isomorphic. In particular, ´ G is smooth (resp. reductive) if and only if Get Zp is smooth (resp. reductive). (b) Let α ∈ J. Then the tensor tα belongs to T(M ) (resp. to T(M ) \ pT(M )) if and only if the tensor vα belongs to T(H 1 (D)) (resp. to T(H 1 (D)) \ pT(H 1 (D))). Let i ∈ Z. If t ∈ F i (T(M ))[ p1 ] is such that φi (t) = pi t, then Fontaine comparison theory associates to t an étale Tate-cycle v ∈ T(H 1 (D)[ p1 ]) whose Zp -span is a Gal(B(k))module isomorphic to Zp (−i). Strictly speaking v is unique up to i-th powers of units of Zp i.e., up to a choice of a generator β0 of a suitable free Zp -submodule of B + (W (k)) of rank 1 (see Subsection 2.2 for the integral crystalline Fontaine ring B + (W (k)) and for its Zp -submodule Zp β0 ). We have the following equivalent form of the Main Theorem which involves also Tate-twists, in which all étale Tate-cycles are defined using a fixed generator β0 (see Subsubsection 2.2.2), and which is proved in Subsection 4.4. ´ 1.4. Corollary. We assume that k = k̄. If p = 2 we also assume that either Get Zp is a torus or the property (C) holds. Let (tα )α∈Jtwist be the family of all tensors in the set {x ∈ F i (T(M ))[ p1 ]|φ(x) = pi x}. Let W be the family of all direct summands W of the W (k)-module T(M ) which have finite ranks and for which we have an equality P ´ W = i∈Z p1i φ(W ∩ F i (T(M ))). For α ∈ Jtwist (resp. for W ∈ W) let vα (resp. W et ) be the tensor of T(H 1 (D))[ p1 ] (resp. the direct summand of T(H 1 (D))) that corresponds to tα (resp. to W ) via Fontaine comparison theory. Then there exists an isomorphism ρtwist : ∼ (M, (tα )α∈Jtwist ) → (H 1 (D) ⊗Zp W (k), (vα )α∈Jtwist ) with the property that the isomorphism ´ ∼ T(M ) → T(H 1 (D) ⊗Zp W (k)) induced by ρ maps each W ∈ W onto W et ⊗Zp W (k). ∼ The existence of ρ : (M, (tα )α∈J ) → (H 1 (D) ⊗Zp W (k), (vα )α∈J ) was conjectured by ´ Milne if Get Zp is a reductive group scheme over Spec(Zp ), D is the p-divisible group of 3 an abelian scheme A over Spec(W (k)), and each tα and vα are the de Rham component and the p-component of the étale component (respectively) of a Hodge cycle on AB(k) (see [De2, Sect. 2] for Hodge cycles). References to Milne’s conjecture in the context of abelian schemes can be found in [Mi3, property (4.16.3) and Subsect. 4.23], an unpublished manuscript of Milne dated Aug. 1995 which worked with p >> 0 (see also [Mi4]), [Va1, Rm. 5.6.5 and Conj. 5.6.6], [Va5], and [Va6]. In fact the conjecture [Va1, Conj. 5.6.6] is a slight restatement of Milne’s conjecture of 1995. In [Va5] and [Va6] we proved [Va1, Conj. 5.6.6] in many cases. The methods used in [Va5] and [Va6] in connection to Milne’s conjecture are very short and simple and of purely reductive group scheme theoretical nature. These methods can be viewed as refinements of [Ko, Lem. 7.2] and can be used to regain the Main Theorem only if the following two conditions hold: ´ – Get Zp and G are reductive group schemes and G is generated by cocharacters of G that act on M via the trivial and the identical character of Gm ; – the maximal torus of the center of G is the maximal torus of the center of the double centralizer of G in either GL M or GSp GSp(M, λM ) (resp. or GSO GSO(M, λM )), where λM : M ⊗W (k) M → W (k) is an alternating bilinear form (resp. a symmetric bilinear form ∼ which modulo 2 is alternating) normalized by G and defined by an isomorphism D → Dt . In Milne’s motivic approach to the proof of the Langlands–Rapoport conjecture (of [Mi2, Conj. 4.4]) for Shimura varieties of Hodge type, Milne’s conjecture plays a key role (see [Mi4], [Va6], and [Va8]). We recall that Shimura varieties of Hodge type are moduli spaces of polarized abelian schemes endowed with level structures and with (specializations of) Hodge cycles and, due to this, they are also the main testing ground for many parts of the Langlands program (like zeta functions, local correspondences). Thus the importance of Milne’s conjecture stems mainly from its meaningful applications to the Langlands program in general and to Shimura varieties in particular. Though for p ≥ 5, [Va1] and [Va6] combined can suffice for applications to the Langlands–Rapoport conjecture for Shimura varieties of Hodge type, for refined applications one needs the much stronger result of the Main Theorem. In future works we will use the Main Theorem as follows: (i) To prove the existence of integral canonical models in mixed characteristics (0, 3) and (0, 2) of Shimura varieties of abelian type (see [Va1] for definitions and for the analogue result in mixed characteristic (0, p) with p ≥ 5). See already [Va7]. (ii) To study different stratifications of special fibres of good integral models of Shimura varieties of abelian type (see already [Va3, Sect. 8] and [Va4]). (iii) To construct a comprehensive theory of automorphic vector bundles on integral canonical models of Shimura varieties of abelian type (presently this theory is not complete even for Siegel modular varieties, cf. end of [FC, Ch. VI, p. 238]). (iv) To get analogues of the Main Theorem and Corollary 1.3 for different motives associated to classes of polarized projective schemes whose moduli spaces are related to 2 Shimura varieties (see [An] for such classes; we have in mind mainly the case of Hcrys and 2 Het ´ groups of hyperkähler schemes). Corollary 1.3 (b) is implied by Faltings’ results [Fa2, Thm. 5 and Cor. 9] if and only if we have tα ∈ ⊕s,t∈{0,... ,p−2} (M ⊗s ⊗W (k) M ∗⊗t )[ p1 ]. 4 Both the Main Theorem and the Corollary 1.3 do not hold in general for p = 2 (like when D is isogenous but not isomorphic to Q2 /Z2 ⊕ µ2∞ and G is isomorphic to Gm ). Thus the Main Theorem and the different variants of it one can get based on its proof, are in essence the maximum one can prove for p = 2. 1.5. On contents. The proof of the Main Theorem involves two steps. The first step shows that it is enough to prove the Main Theorem if G is a torus (see Subsection 3.5) and the second step proves the Main Theorem if G is a torus (see Subsections 4.1 and 4.2). The first step relies on the existence of global deformations of the pairs (D, (tα )α∈J ) and (D, (vα )α∈J ) over certain regular, formally smooth Spec(W (k))-schemes whose special fibres are connected and have a Zariski dense set of k̄-valued points. Based on [Fa1, Thm. 7.1] and on a variant of it for p = 2, the existence of such deformations boils down to proving the existence of certain integrable and nilpotent modulo p connections (see Subsection 3.4). See Lemma 3.1.3 and Theorem 3.2 for our basic results which pertain to connections and see Subsection 3.3 for their proofs. Though these global deformations are of interest in their own (for instance, they can be used to solve essentially any specialization problem which pertains to pairs of the form (Dk , (tα )α∈J )), here we will use them only to accomplish the first step. The second step relies on the integral version of Fontaine comparison theory studied in [Fa2, Sect. 4] and it involves a non-trivial reduction to the Lubin–Tate case in which k = k̄, F 1 has rank 1, the F -isocrystal (M [ p1 ], φ) over k is simple, and G is a torus. It is easy to prove Theorem 1.2 in the Lubin–Tate case, cf. Example 4.1.3 and Lemma 4.2.1. Section 2 gathers prerequisites for Sections 3 to 5. Corollary 1.4 is proved in Subsection 4.4. See Subsections 4.3 and 4.5 and Section 5 for variants and complements to the Main Theorem and Corollary 1.3. Here we only add that any natural variant of the Main Theorem which is over the spectrum of a finite, totally ramified discrete valuation ring extension of W (k) and which involves a setting one expects to be deformable to Spec(W (k)), follows from the Main Theorem and from a refinement of the deformation theory of [Fa2, Ch. 7] (see Subsections 5.2 to 5.4). Analogues of the Main Theorem do not exist in general for those ramified settings that are not deformable to Spec(W (k)), cf. Subsection 5.5. 2. Preliminaries See Subsection 2.1 for conventions and notations to be used throughout the paper. In Subsections 2.2 and 2.3 we recall some facts which pertain to Fontaine comparison theories and to Faltings–Fontaine categories. In Subsection 2.4 we introduce Artin–Schreier systems of equations. They play a key role in Section 3. In Subsections 2.5 to 2.7 we include few simple group scheme theoretical properties. 2.1. Conventions and notations. If f1 and f2 are endomorphisms of a group, we often denote f1 ◦ f2 by f1 f2 . For n ∈ N let Wn (k) := W (k)/pn W (k). Let R∧ be the p-adic completion of a commutative, flat W (k)-algebra R. If Y = Spec(R) let Y ∧ := Spec(R∧ ). By a Frobenius lift of R∧ (resp. of Y ∧ ) we mean an endomorphism of R∧ (resp. of Y ∧ ) that lifts the Frobenius endomorphism of R/pR (resp. of Yk∧ = Yk ). If mY : Ỹ → Y is a morphism of Spec(W (k))-schemes, then by its special fibre we mean Ỹk → Yk , by its 5 generic fibre we mean ỸB(k) → YB(k) , and by mY modulo pn we mean ỸWn (k) → YWn (k) . If Spec(R̃) → Spec(R) is an affine morphism between affine, flat W (k)-schemes, then whenever we say that it (or the induced W (k)-homomorphism R → R̃) is formally smooth (or étale), both the W (k)-algebras R and R̃ are endowed with the p-adic topology. Let R/pR(p) be R/pR but viewed as an R/pR-algebra via the Frobenius endomorphism of R/pR. If R̂ is the completion of R with respect to an ideal I of R that contains p and if Spf(R̂) is the corresponding formal scheme, then we identify canonically the Zp -linear categories of p-divisible groups over Spec(R̂) and respectively over Spf(R̂) (cf. [Me, Ch. II, Lem. 4.16]); thus we will often use the Grothendieck–Messing deformation theory of [Me, Ch. 5] to lift p-divisible groups over Spec(R/I) to p-divisible groups over Spec(R̂). Let δ(p) be the natural divided power structure from characteristic 0 of the ideal (p) of either R or R/pn R. Let CRIS(Yk /Spec(W (k))) be the Berthelot crystalline site of [Be, Ch. III, Sect. 4]. In what follows, it is convenient to call both pairs (Yk ֒→ YWn (k) , δ(p)) and (Yk ֒→ Y ∧ , δ(p)) as thickenings, even though only the first pair is a thickening (i.e., an épaississement à puissances diviséss) in the sense of [Be, Ch. III, Def. 1.1.1]. For Dieudonné crystals on CRIS(Yk /Spec(W (k))) of either p-divisible groups or finite, flat, commutative group schemes annihilated by a power of p over Yk , we refer to [BBM, Ch. 3] and [BM, Chs. 2 and 3]. If DYk is a p-divisible group over Yk and if ΦR is a fixed Frobenius lift of R∧ compatible with σ, then by the evaluation (N, ΦN , ∇N ) of the Dieudonné crystal D(DYk ) on CRIS(Yk /Spec(W (k))) at the thickening (Yk ֒→ Y ∧ , δ(p)) we mean the projective limit indexed by n of the evaluation of D(DYk ) at the thickening (Yk ֒→ YWn (k) , δ(p)) of CRIS(Yk /Spec(W (k))), the Verschiebung maps being disregarded. Thus N is a projective R∧ -module, ΦN is a ΦR -linear endomorphism of N , and ∇N is an integrable, nilpotent modulo p connection on N . By the filtered Dieudonné crystal of a p-divisible group D over Y we mean D(DYk ) endowed with the Hodge filtration defined by D; thus the evaluation of the mentioned filtered Dieudonné crystal at the thickening (Yk ֒→ Y ∧ , δ(p)) is the quadruple (N, FN1 , ΦN , ∇N ), where FN1 ⊆ N is the Hodge filtration defined by D. We recall that N/pn N is the Lie algebra of the universal vector extension of the Cartier dual DtYW (k) of DYWn (k) , cf. [BM, Cor. 3.2.11] and [Me, Ch. 4]. n Let KA be the field of fractions of a Z-algebra A that is an integral domain. Let ∧ W (k) be the normalization of W (k) in B(k). Let V (k) := W (k) . Let K(k) := V (k)[ p1 ]. ´ The notations D, (M, φ), F 1 , (tα )α∈J , G, (vα )α∈J , and Get Zp will be as in Subsection 1.1. Let µ : Gm → G be a cocharacter that produces a direct sum decomposition M = F 1 ⊕ F 0 such that δ ∈ Gm (W (k)) acts through µ on F i as the multiplication with δ −i , i ∈ {0, 1}. For instance, we can take µ to be the factorization through G of the inverse of the canonical split cocharacter µcan : Gm → GL M of (M, F 1 , φ) defined in [Wi, p. 512] (the cocharacter µcan fixes each tα , cf. the functorial aspects of [Wi, p. 513]). 2.2. Fontaine comparison theory. For the review included in this Subsection we refer to [Fo2] and [Fa2, Sect. 2]. Let K be a finite, totally ramified field extension of B(k). Let V be the ring of integers of K i.e., the normalization of W (k) in K. Let e := [V : W (k)]. Let πV be a uniformizer of V . Let X be a free variable. The minimal polynomial fe ∈ W (k)[X] of πV over W (k) is an Eisenstein polynomial. Let Se be the en W (k)[[X]]-subalgebra of B(k)[[X]] generated by all Xn! with n ∈ N ∪ {0}; it is the divided 6 power hull of any one of the ideals (X e ), (fe ), or (p, X e ) = (p, fe ) of W (k)[[X]]. Let Je en feen (resp. Ke ) be the ideal of Se generated by all n! (resp. by all Xn! ) with n ∈ N. Let Re := Se∧ . By mapping X to πV , we get identifications V = W (k)[X]/(fe) = Se /Je and ee be the completion of Se with respect to the a W (k)-epimorphism eV : Re ։ V . Let R [n] ee = proj.lim.n∈N Se /Ke[n] . decreasing filtration given by its ideals Ke , n ∈ N ∪ {0}. Thus R [0] [n] We recall that Ke := Se and that for n ≥ 1 the ideal Ke of Se is generated by all products a1 am δ1 δm a1 ! · · · am ! with δ1 , . . . , δm ∈ Ke and with m, a1 , . . . , am ∈ N ∪ {0} such that we have em [n] ee is also ∈ Ke . If p > 2, then R a1 + · · · + am ≥ n. If m ∈ N, m ≥ n, then we have X m! [n] the completion of Se with respect to its decreasing filtration (Je )n∈N∪{0} ; thus for p > 2 ee ։ V that takes X to πV . we have as well a W (k)-epimorphism ẽV : R ee (resp. Re ) consists of formal power series Σn ≥ 0 an X n such The W (k)-algebra R that the sequence ([ ne ]!an )n∈N∩{0} is formed by elements of W (k) (resp. is formed by ee that elements of W (k) and converges to 0). Let Φk be the Frobenius lift of Se , Re , or R np [ ]! is compatible with σ and such that Φk (X) = X p . The sequence ( [ ne ]! )n∈N of integers in e ee ) ⊆ Re . We have W (k) converges to 0 in the p-adic topology and thus we have Φk (R ee onto W (k) defined by the rule Σn≥0 an X n → a0 . W (k)-epimorphisms from Se , Re , and R Let Ak be the perfect integral domain of sequences (xn )n∈N∪{0} of V (k)/pV (k) which satisfy the identity xn−1 = xpn for all n ∈ N. The Galois group Gal(B(k)) acts on V (k) and thus also on Ak . The Gal(B(k))-module Qp (1) can be identified with sequences (ηn )n∈N∪{0} of p-power roots of unity of V (k) which satisfy the identity ηn−1 = ηnp for all n ∈ N. Taking such sequences modulo p, we get a group homomorphism γk : Qp (1) → Gm (Ak ) that respects the Galois actions. For an element z ∈ V (k), we choose a sequence (z(n))n∈N∪{0} of elements of V (k) with the properties that z(0) = z and that z(n − 1) = z(n)p for all n ∈ N. Taking this sequence modulo p we obtain an element z ∈ Ak , well defined by z up to multiplication with an element of γk (Zp (1)). If x ∈ V (k)/pV (k) let x̃ ∈ V (k) be a lift of it. As Ak is an integral domain, the ring W (Ak ) of Witt vectors with coefficients in Ak is also an integral domain. Let sk : W (Ak ) ։ V (k) be the W (k)-epimorphism defined by the rule X pn x(n) sk ((x0 , x1 , . . . )) = n , n≥0 where (xn,m )m∈N∪{0} is the sequence of elements of V (k)/pV (k) that defines xn ∈ Ak and m (n) where xn is the p-adic limit in V (k) of the sequence (x̃pn,n+m )m∈N∪{0} (this limit does not depend on the choice of the lifts x̃n,n+m ’s). We have sk (f (πV , 0, 0, . . . )) = f (πV ) = 0. Let ξ0 := (p, −1p , 0, . . . ) ∈ W (Ak ). We have sk (ξ0 ) = p−p = 0. Thus both ξ0 and f (πV , 0, 0, . . . ) belong to Ker(sk ). Obviously the kernel of sk modulo p is generated by either ξ0 modulo p or by f (πV , 0, 0, . . . ) modulo p. From the last two sentences we get that either ξ0 or f (πV , 0, 0, . . . ) generates Ker(sk ). Let W + (Ak ) be the divided power hull of the ideal Ker(sk ) of W (Ak ) i.e., the W (Ak )ξn subalgebra of the field of fractions of W (Ak ) generated by all n!0 with n ∈ N ∪ {0}. 7 Let B + (W (k)) := W + (Ak )∧ be the integral crystalline Fontaine ring; it is a p-adically complete integral domain that is a W + (Ak )-algebra and thus also a W (k)-algebra. The W (k)-algebra B + (W (k)) has a decreasing filtration (F n (B + (W (k))))n∈N∪{0} by ideals, where F n (B + (W (k))) is the p-adic completion of Ker(sk )[n] . The group Gal(B(k)) acts on the filtered W (k)-algebra B + (W (k)). We denote also by sk the W (k)-epimorphism sk : B + (W (k)) ։ B + (W (k))/F 1 (B + (W (k))) = V (k). induced naturally by sk : W (Ak ) ։ V (k). As Ak is a perfect integral domain, we have a Teichmüller monomorphism Gm (Ak ) ֒→ Gm (W (Ak )). Composing the restriction of γk to Zp (1) with this Teichmüller monomorphism, we get a homomorphism tk : Zp (1) → Gm (W (Ak )). Let vk : Zp (1) → Gm (B + (W (k))) be the composite of tk with the natural monomorphism Gm (W (Ak )) ֒→ Gm (B + (W (k))). The composite sk ◦ vk is the trivial homomorphism Zp (1) → Gm (V (k)). Thus let βk : Zp (1) → F 1 (B + (W (k))) be the homomorphism obtained by taking log of vk . Throughout the paper we fix a generator β0 of the free Zp -module Im(βk ) of rank one. As Ker(sk ) has a divided power βp structure, we have p0 ∈ B + (W (k)). Therefore B + (W (k))[ β10 ] is a B(k)-algebra. There exists a W (k)-monomorphism ie : Re ֒→ B + (W (k)) defined by the rule: X → (πV , 0, 0, . . . ) ∈ W (Ak ). As Φk (ξ0 ) ∈ pW (Ak ), the canonical Frobenius lift of W (Ak ) extends to a Frobenius lift Φk of B + (W (k)) (it makes sense to denote it also by Φk , as all W (k)-monomorphisms ie respect Frobenius lifts). We have Φk ◦ βk = pβk . As B + (W (k))/F n (B + (W (k))) is p-adically complete, the W (Ak )-algebras B ++ (W (k)) := proj.lim.n∈N B + (W (k))/F n(B + (W (k))) and proj.lim.n∈N W + (Ak )/F n (B + (W (k))) ∩ W + (Ak ) are naturally identified. The W (Ak )-monomorphism B + (W (k)) ֒→ B ++ (W (k)) gives birth for each n ∈ N to an identification F n (B + (W (k))) = B + (W (k)) ∩ Ker(B ++ (W (k)) ։ B + (W (k))/F n (B + (W (k)))). 2.2.1. Key facts. Let ξ̄0 ∈ Ak be the reduction modulo p of ξ0 ∈ W (Ak ). We have isomorphisms (p) ∼ qk : B + (W (k))/(F p (B + (W (k)))+pB + (W (k))) = W (Ak )/(ξ0p, p) = Ak /(ξ̄0p ) → V (k)/pV (k), where all except the last one are canonical identifications and where the last isomorphism is defined by the epimorphism Ak ։ V (k)/pV (k) that takes (xn )n∈N∪{0} ∈ Ak to x1 (see (p) [Fa1, Sect. 2, p. 30] and [Fa2, Sect. 4, top of p. 126]). The isomorphism qk naturally by a W (k)-epimorphism qk : B + (W (k)) ։ V (k)/pV (k). 8 is defined 1 1 We fix a p-th root p p of p such that qk (ξ0 ) is p p modulo p; thus, if (xn )n∈N∪{0} ∈ Ak 1 defines p, then x1 is p p modulo p. For i ∈ {1, . . . , p − 1}, the ideal qk (F i (B + (W (k)))) i of V (k)/pV (k) is generated by p p modulo p. Let Φ1k : F 1 (B + (W (k))) → B + (W (k)) be the map defined by the rule: if x ∈ F 1 (B + (W (k))), then we have an equality Φk (x) = 2 pΦ1k (x). As Φk (ξ0 ) = (pp , −1p , 0, . . . ), we easily get that Φk (ξ0 ) − pξ0p − p(−1p , 0, . . . ) ∈ p2 W (Ak ). As ξ0p p ∈ F p (B + (W (k))) and as (−1)p and −1 are congruent modulo p, we get 2 that Φ1k (ξ0 ) − (−1p , 0, 0, . . . ) ∈ Ker(qk ). Thus the Frobenius lift Φk of B + (W (k)) and the Φk -linear map Φ1k become under qk the Frobenius endomorphism Φ̄k of V (k)/pV (k) 1 1 and respectively the map Φ̄1k : p p V (k)/pV (k) → V (k)/pV (k) that takes p p x modulo p to (−x)p (equivalently to −xp ) modulo p; here x ∈ V (k). Let D(B + (W (k))/pB + (W (k))) be the reduction of (B + (W (k)), F 1 (B + (W (k))), Φk , Φ1k ) modulo p. Also let 1 D(V (k)/pV (k)) := (V (k)/pV (k), p p V (k)/pV (k), Φ̄k , Φ̄1k ). Let gr0 := V (k) = F 0 (B + (W (k)))/F 1 (B + (W (k))). The V (k)-module gr1 := F 1 (B + (W (k)))/F 2 (B + (W (k))) is generated by the image ξ1 of ξ0 in gr1 . As gr1 is torsion free, we get that gr1 is free of rank 1. The image β1 of β0 in gr1 is λ1 ξ1 , where λ1 ∈ V (k) is a Gm (V (k))-multiple of a p − 1-th root of p (cf. [Fo2, Subsubsects. 5.1.2 and 5.2.4] where the pair (β0 , ξ0 ) is denoted as (t, ξ)). This implies that qk (β0 ) is qk (ξ0 ) times a Gm (V (k)/pV (k))-multiple of 1 1 the reduction modulo p of a p(p − 1)-th root of p. As p1 + p(p−1) = p−1 , we get that qk (β0 ) is the image in V (k)/pV (k) of a Gm (V (k))-multiple of a p − 1-th root of p. 2.2.2. Applications to D. We use the notations of Subsection 1.1. Let (F i (H 1 (D)))i∈{0,1} be the filtration of H 1 (D) defined by F 1 (H 1 (D)) := 0 and F 0 (H 1 (D)) := H 1 (D). We endow M with the trivial Gal(B(k))-action. The Fontaine comparison theory provides us with a B + (W (k))-linear monomorphism (1) iD : M ⊗W (k) B + (W (k)) ֒→ H 1 (D) ⊗Zp B + (W (k)) that respects the tensor product filtrations and the Galois actions. Following [Fa2, Sect. 6], we recall how iD is constructed. Let j : Qp /Zp → DV (k) be a homomorphism of p-divisible groups over Spec(V (k)). For n ∈ N let δncan be the natural divided power structure of the ideal Im(F 1 (B + (W (k)) + pB + (W (k)) → B + (W (k))/pn B + (W (k))) of B + (W (k))/pn B + (W (k)). By evaluating D(j) at the thickening (Spec(V (k)/pV (k)) ֒→ Spec(B + (W (k))/pn B + (W (k))), δncan ), we get a B + (W (k))/pn B + (W (k))-linear map M/pn M ⊗Wn (k) B + (W (k))/pn B + (W (k)) → B + (W (k))/pn B + (W (k)). Passing to limit n → ∞ we get a B + (W (k))-linear map M ⊗W (k) B + (W (k)) → B + (W (k)) i.e., an element of M ∗ ⊗W (k) B + (W (k)). Varying j we get a B + (W (k))-linear monomorphism i∗D : Tp (DB(k) ) ⊗Zp B + (W (k)) ֒→ M ∗ ⊗W (k) B + (W (k)). The dual of i∗D is iD . 9 We check that β0 annihilates Coker(iD ) and Coker(i∗D ). Considering homomorphisms Qp /Zp → DV (k) → µp∞ , it is enough to handle the case when D = µm p∞ , with m ∈ N. In 1 + this case we have Im(iD ) = H (D)⊗Zp β0 B (W (k)), cf. the definition of βk and [BM, Cor. 2.2.4]; thus β0 annihilates Coker(iD ). By duality, we get that β0 annihilates Coker(i∗D ). The B + (W (k))-linear monomorphism iD is strict with respect to filtrations, cf. [Fa2, Thms. 5 and 7] and [Fa2, Ch. 8] (due to the paragraph before Subsubsection 2.2.1, to check this property it is irrelevant if we use iD or its tensorization over B + (W (k)) with B ++ (W (k))). This strictness property implies that the V (k)-linear map (2) jD : F 0 ⊗W (k) gr1 ⊕ F 1 ⊗W (k) gr0 ֒→ H 1 (D) ⊗Zp gr1 defined by iD at the level of one gradings, is injective. As Coker(iD ) is annihilated by β0 , Coker(jD ) is annihilated by the element λ1 ∈ V (k) of Subsubsection 2.2.1. The category of crystalline representations of Gal(B(k)) over Qp is stable under tensor products and duals, cf. [Fo3, Subsect. 5.5]. This implies that there exists a tensor vα ∈ T(H 1 (D)[ p1 ]) that corresponds to tα via the B + (W (k))[ β10 ]-linear isomorphism ∼ T(M [ p1 ]) ⊗B(k) B + (W (k))[ β10 ] → T(H 1 (D)[ p1 ]) ⊗Qp B + (W (k))[ β10 ] defined by iD [ β10 ]. The tensor vα is fixed by Gal(B(k)). Let µcan (β0−1 ) be the B + (W (k))[ β10 ]-linear automorphism of M ⊗W (k) B + (W (k))[ β10 ] defined by the evaluation of the cocharacter µcan : Gm → G at β0−1 . If i ∈ Z and t ∈ {x ∈ F i (T(M ))[ p1 ]|φ(x) = pi x}, then the B + (W (k))[ β10 ]-linear isomorphism T(M ) ⊗W (k) ∼ B + (W (k))[ β10 ] → T(H 1 (D)) ⊗Zp B + (W (k))[ β10 ] defined by iD [ β10 ] ◦ µcan (β0−1 ) takes t to a tensor v ∈ T(H 1 (D))[ p1 ] whose Zp -span < v > is a Gal(B(k))-module isomorphic to Zp (−i). We say that v corresponds (or is associated) to t via Fontaine comparison theory for D (and via the choice of the generator β0 of the free Zp -module Im(βk )). Let W be a direct summand of the T(M ) which has finite rank and P W (k)-module 1 i for which we have an identity W = i∈Z pi φ(W ∩ F (T(M ))). Then Fontaine com´ parison theory show that there exists a unique direct summand W et of T(H 1 (D)) such ∼ that the B + (W (k))[ β10 ]-linear isomorphism T(M ) ⊗W (k) B + (W (k))[ β10 ] → T(H 1 (D)) ⊗Zp ´ B + (W (k))[ β10 ] defined by iD [ β10 ] maps W ⊗W (k) B + (W (k))[ β10 ] to W et ⊗Zp B + (W (k))[ β10 ]. 2.2.3. Lemma. We recall that D is a p-divisible group over Spec(W (k)). If p = 2, we assume that either Dk or Dkt is connected. Then D is uniquely determined up to isomorphism by its filtered Dieudonné module (M, F 1 , φ) and the natural homomorphism of Zp -algebras End(D)opp = End(Dt ) = End(H 1 (D)) → End((M, F 1 , φ)) is an isomorphism. Proof: The classical Dieudonné theory says that Dk is uniquely determined up to isomorphism by (M, φ) and that the natural homomorphism End(Dk )opp = End(Dkt ) → End((M, φ)) is an isomorphism (see [Dem, Ch. III, Sect. 8, Thm.], [Fo1, Ch. III, p. 128 or p. 153], etc.). Thus for p > 2 the Lemma follows from Grothendieck–Messing deformation theory (see also [Fa2, Thm. 5] or Theorem 2.3.4 below). For p ≥ 2 the Lemma (as well as Proposition 2.2.4 below) can be deduced from [Fo1, Ch. IV, 1.6, p. 186]. As loc. cit. is stated in terms of Honda triples of the form (M, φ( p1 F 1 ), φ), we include here a proof of the Lemma for p = 2 that appeals to Subsubsection 2.2.2. 10 Suppose that p = 2; thus p − 1 = 1. As λ1 ∈ 2Gm (V (k)), 2 annihilates Coker(jD ). It suffices to prove the Lemma under the assumption that Dkt is connected. Let D1 be another 2-divisible group over Spec(W (k)) that has (M, F 1 , φ) as its filtered Dieudonné module. As 2 annihilates both Coker(jD ) and Coker(jD1 ), via either jD1 or iD1 we can identify H 1 (D1 ) with a Z2 -submodule of 12 H 1 (D) that contains 2H 1 (D). We show that the assumption H 1 (D) 6= H 1 (D1 ) leads to a contradiction. We can ´ ´ assume k = k̄. Let Det and D1et be the maximal étale 2-divisible groups that are quotients of D and D1 (respectively). Their filtered Dieudonné modules are (M0 , 0, φ), where M0 = ∩n∈N φn (M ) is the maximal direct summand of M that is W (k)-generated by elements fixed by φ. Let r0 ∈ N ∪ {0} be the rank of M0 . Let {a1 , . . . , ar } be a W (k)-basis for M formed by elements of F 1 ∪ F 0 and such that we have φ(ai ) = ai if i ≤ r0 ; this makes sense as the canonical split cocharacter µcan : Gm → G of (M, F 1 , φ) fixes all elements of M fixed by φ (cf. the functorial aspects of [Wi, p. 513]). As iDet´ and iDet´ are 1 ´ ´ isomorphisms, H 1 (Det ) = H 1 (D1et ) is a direct summand of both H 1 (D) and H 1 (D1 ). Let ´ x ∈ H 1 (D) ∩ H 1 (D1 ) be such that Z2 x ⊕ H 1 (Det ) is a direct summand of H 1 (D) ∩ H 1 (D1 ) x and Z2 2 is a direct summand of precisely one of the Z2 -modules H 1 (D) and H 1 (D1 ). To fix the ideas we assumeP that x2 ∈ H 1 (D) \ H 1 (D1 ). As x2 β0 ∈ iD (M ⊗W (k) B + (W (k))), r x we can write 2 β0 = iD ( i=1 ai ⊗ αi ), where αi ∈ B + (W (k)). Let α0i ∈ gr0 be αi modulo F 1 (B + (W (k))). As iD is strict with respect to filtrations, we have αi ∈ F 1 (B + (W (k))) for each i ∈ {1, . . . , r} such that ai ∈ F 0 . One can check that in fact we have β0 ∈ 2B + (W (k)), cf. [Fo1, top of p. 79]. Either from this or from the fact that λ1 ∈ 2Gm (V (k)) we get that the image of x2 β0 = x β20 ∈ H 1 (D1 ) ⊗Z2 B + (W (k))[ 21 ] in H 1 (D1 ) ⊗Z2 gr1 [ 12 ] belongs to H 1 (D1 ) ⊗Z2 gr1 and generates a direct summand which when tensored with k does not ´ belong to H 1 (Det ) ⊗Z2 gr1 ⊗V (k) k. Thus there exists i0 ∈ {r0 + 1, . . . , r} such that: (i) either ai0 ∈ F 1 and αi0 ∈ Gm (B + (W (k))) (ii) or ai0 ∈ F 0 and jD1 (ai0 ⊗ α0i0 ) generates a direct summand of H 1 (D1 ) ⊗Z2 gr1 ´ ) ⊗Z2 gr1 ⊗V (k) k. which when tensored with k does not belong to H 1 (D1et Pr We first assume that (i) holds. As Φk (β0 ) = 2β0 , for n ∈ N we have i=1 φn (ai ) ⊗ nx n + n + Φnk (αi ) = i−1 D (2 2 β0 ) ∈ 2 M ⊗W (k) B (W (k)). As Φk (αi0 ) ∈ Gm (B (W (k))) and as {φn (a1 ), . . . , φn (ar )} is a B(k)-basis for M [ 21 ] formed by elements of M \ 2n+1 M , we get that φn (ai0 ) is divisible by 2n . Taking n >> 0, this implies that (M, φ) has Newton polygon slope 1 with positive multiplicity. Thus Dkt is not connected. Contradiction. We assume that (ii) holds. Due to the property enjoyed by jD1 (ai0 ⊗ α0i0 ), the image ´ of iD1 (ai0 ⊗ 1) in H 1 (D1 ) ⊗Z2 k does not belong to H 1 (D1et ) ⊗Z2 k. This statement holds n also if ai0 is replaced by φ (ai0 ). Taking n >> 0, we get that (M, φ) has Newton polygon slope 0 with multiplicity at least r0 + 1. This contradicts the definition of r0 . Thus H 1 (D) = H 1 (D1 ). Therefore D = D1 , cf. a classical theorem of Tate. The standard trick of replacing D by D ⊕D shows that to prove that End(H 1 (D)) surjects onto End((M, F 1 , φ)), it suffices to show that each automorphism a of (M, F 1 , φ) is the image ´ : of an automorphism of H 1 (D). We identify a via iD [ β10 ] with a Galois automorphism aet 1 ∼ 1 1 1 H (D)[ 2 ] → H (D)[ 2 ]. An argument similar to the one above shows that the assumption ´ ´ H 1 (D) 6= aet (H 1 (D)) leads to a contradiction. Thus H 1 (D) = aet (H 1 (D)) and therefore 11 ´ a is the image of the automorphism aet of H 1 (D). 2.2.4. Proposition. Suppose that p = 2 and either Dk or Dkt is connected. Then each direct summand F̃ 1 of M that lifts F 1 /2F 1 , is the Hodge filtration of a 2-divisible group D̃ over Spec(W (k)) that lifts Dk . Moreover, D̃ is unique (up to a unique isomorphism). Proof: The divided power structure of the ideal (4) of W (k) is nilpotent modulo (2n ) for all n ≥ 2. Thus based on Grothendieck–Messing deformation theory, it suffices to show that there exists a unique lift D̃W2 (k) of Dk to Spec(W2 (k)) whose Hodge filtration is F̃ 1 /4F̃ 1 . Let d be the product of the dimensions of Dk and Dkt . Let Duniv be the universal 2-divisible group over the deformation space D of Dk , cf. [Il, Cor. 4.8 (i)]. We view D as a scheme naturally identified with Spec(W (k)[[x1 , . . . , xd ]]) and not as a formal scheme of the form Spf(W (k)[[x1 , . . . , xd ]]), cf. Subsection 2.1. Let R(d) be the 2-adic completion of (d) the divided power hull of the maximal ideal of W (k)[[x1 , . . . , xd ]]. Let δ2 be the natural divided power structure of the maximal ideal of R(d) . Let ΦR(d) be the Frobenius lift of R(d) that is compatible with σ and that takes xi to x2i for all i ∈ {1, . . . , d}. The evaluation (d) univ (d) of the filtered Dieudonné crystal of DR ), δ2 ) is (d) at the thickening (Spec(k) ֒→ Spec(R 1 defined as in Subsection 2.1 and it is (M ⊗W (k) R(d) , Funiv , φ⊗ΦR(d) , δ0 ), where δ0 is the flat (d) 1 connection on M ⊗W (k) R that annihilates M ⊗ 1 and where Funiv is a direct summand (d) 1 1 d d of M ⊗W (k) R that lifts F /2F . Let Adef (resp. Alift ) be the d dimensional affine space over Spec(k) that parametrizes lifts of Dk to Spec(W2 (k)) (resp. lifts of F 1 /2F 1 to direct summands of M/4M ), the origin corresponding to DW2 (k) (resp. to F 1 /4F 1 ). 1 The existence of Funiv implies that there exists a morphism mDk : Addef → Adlift which on k-valued points takes a lift of Dk to Spec(W2 (k)) to the direct summand of M/4M that is the Hodge filtration of this lift. We know that mDk (k) is injective, cf. Lemma 2.2.3 and the beginning of this proof. As mDk̄ is the pull back of mDk to a morphism of Spec(k̄)schemes, mDk (k̄) is also injective. Thus the morphism mDk is quasi-finite, (due to reasons of dimensions) dominant, and generically purely inseparable. In particular, Addef is an open subscheme of the normalization Bddef of Adlift in the field of fractions of Addef , cf. Zariski Main theorem (see [Ra1, Ch. IV]). We show that the assumption Addef 6= Bddef leads to a contradiction. From [Ma, Subsect. (17.H), Thm. 38] we easily get that Bddef \ Addef is purely of codimension 1 in Bddef and thus contains an irreducible divisor of Bddef . As mDk is generically purely inseparable, such a divisor is the reduced scheme of the pull back of an irreducible divisor C of Adlift . As a polynomial ring over k is a unique factorization domain, C is the principal divisor of a global function f of Adlift that does not belong to k. We get that the polynomial ring over k of global functions of Addef has an invertible element f that does not belong to k. Contradiction. Therefore Addef = Bddef . Thus the morphism mDk is finite. As mDk is also generically purely inseparable and d as Alift is normal, mDk is in fact a finite, purely inseparable morphism. As k perfect, this implies that the map mDk (k) is surjective. As mDk (k) is also injective, the map mDk (k) is a bijection. Thus D̃W2 (k) exists and it is unique. 2.3. Smooth Faltings–Fontaine theory. Let R00 be a smooth W (k)-algebra. Let R0 be a regular, formally smooth R00 -algebra. The main examples of interest are ind-étale algebras over R00 or different completions of them. We recall that an ind-étale algebra 12 over R00 is the direct limit of a functor from a filtered category (see [Mi1, App. A]) to the category of étale R00 -algebras. Let R be either R0 or R0∧ . Let ΦR be a Frobenius lift of R∧ = R0∧ that is compatible with σ. Until Subsubsection 2.3.2 we will assume that R0 and R0 /pR0 are integral domains; thus R is also an integral domain. Let KR̄ be the maximal field extension of KR such that the normalization R̄ of R in KR̄ has the property that Spec(R̄[ p1 ]) is a pro-étale cover of Spec(R[ p1 ]). The field extension KR ֒→ KR̄ is a pro-finite Galois extension (see [Gr, Exp. V, Prop. 8.2]) and thus the notations match (i.e., KR̄ is the field of fractions of R̄). Let AR/pR := proj.lim.n∈N R̄/pR̄, the transition homomorphisms being Frobenius endomorphisms. Using AR/pR as a substitute of Ak , one constructs R-homomorphisms sR/pR : W (AR/pR ) → R̄∧ and sR/pR : B + (R) → R̄∧ which are the analogous to sk : W (Ak ) → V (k) and sk : B + (W (k)) → V (k) of Subsection 2.2 (if R is a smooth W (k)-algebra, then this construction was first performed in [Fa1, Sect. 2]; see [Fa3] for other general forms of it). The W (k)-algebra B + (R) is equipped with a Frobenius lift ΦR/pR , a decreasing and separated filtration F n (B + (R))n∈N∪{0} , and a Gal(KR̄ /KR )-action that respects the filtration. Though AR/pR , sR/pR , and ΦR/pR depend only on R̄/pR̄, they are not always uniquely determined but R/pR. However, to ease and uniformize notations, we will use the lower right index R/pR and not R̄/pR̄. We identify W (k) with the normalization of W (k) in R̄. We have a canonical W (k)monomorphism B + (W (k)) ֒→ B + (R) that respects the Frobenius lifts and the filtrations. In particular, both β0 and ξ0 are naturally identified with elements of F 1 (B + (R)). 2.3.1. Lemma. (a) The R-homomorphism sR/pR : W (AR/pR ) → R̄∧ is onto. (b) The kernel of the R-epimorphism sR/pR : W (AR/pR ) ։ R̄∧ is generated by ξ0 . Proof: Part (a) is due in essence to Faltings. Here is a slight modification of his arguments. It is enough to show that sR/pR modulo p is onto and thus that the Frobenius endomor1 phism of R̄/pR̄ is onto. We fix a 2p-th root p 2p of p in V (k). Let y ∈ R̄. The R̄-algebra 1 R̄y := R̄[z]/(z p − p 2 z − y) is finite and after inverting p becomes étale, cf. proof of [Fa3, p. 219, Lem. 5]. The R̄-algebra R̄y has a section R̄y ։ R̄, cf. the very definition of R̄. 1 Thus there exists s(y) ∈ R̄ such that s(y)p − p 2 s(y) − y = 0. A simple calculation shows 1 that s(y)p − [p 2p s(s(y))]p − y ∈ pR̄. Therefore y modulo p has a p-th root in R̄/pR̄. Thus sR/pR is onto. The proof of (b) is the same as for the case R = W (k). ∇ 2.3.2. Categories MF[0,1] (R) and MF[0,1] (R). In this Subsubsection we do not assume that R0 and R0 /pR0 are integral domains. Let dΦR /p : ΩR∧ /W (k) → ΩR∧ /W (k) be the ∇ differential of ΦR divided by p. Let MF[0,1] (R) be the Faltings–Fontaine Zp -linear category defined as follows (see [FL, Sect. 1] and [Wi, Sect. 1] for R = W (k), see [Fa1, Sect. 2] for R smooth, and [Va2, Sect. 2] in general). Its objects are quintuples (N, F, Φ0 , Φ1 , ∇), where N is a finitely generated torsion R-module, F is a direct summand of N , Φ0 : N → N and Φ1 : F → N are ΦR -linear maps, and ∇ : N → N ⊗R ΩR/W (k) = N ⊗R ΩR∧ /W (k) is an integrable, nilpotent modulo p connection on N , such that the following five axioms hold: 13 (a) we have Φ0 (x) = pΦ1 (x) for all x ∈ F ; (b) the R-module N is R-generated by Φ0 (N ) + Φ1 (F ); (c) we have ∇ ◦ Φ0 (x) = p(Φ0 ⊗ dΦR /p) ◦ ∇(x) for all x ∈ N ; (d) we have ∇ ◦ Φ1 (x) = (Φ0 ⊗ dΦR /p) ◦ ∇(x) for all x ∈ F ; (e) locally in the Zariski topology of Spec(R/pR), N is a finite direct sum of Rmodules of the form R/ps R, where s ∈ N ∪ {0}. A morphism f : (N, F, Φ0 , Φ1 , ∇) → (N ′ , F ′ , Φ′0 , Φ′1 , ∇′ ) between two such quintuples is an R-linear map f0 : N → N ′ such that we have an inclusion f0 (F ) ⊆ F ′ and identities Φ′0 ◦ f0 = f0 ◦ Φ0 , Φ′1 ◦ f0 = f0 ◦ Φ1 , and ∇′ ◦ f0 = (f0 ⊗R 1ΩR/W (k) ) ◦ ∇. Additions of f0 ’s ∇ and multiplications of f0 ’s by elements of Zp define the Zp -linear structure of MF[0,1] (R). ∇ ∇ Due to (e) we have a natural identification MF[0,1] (R∧ ) = MF[0,1] (R). Disregarding the connections and thus axioms (c) and (d), we get the Zp -linear category MF[0,1](R). 2.3.3. The functor D. Let p − F F (Spec(R)) be the Zp -linear category of finite, flat, commutative group schemes over Spec(R) of p power order. We recall from [Va2, Sect. 2] that there exists a contravariant Zp -linear functor ∇ ∇ D : p − F F (Spec(R)) → MF[0,1] (R) = MF[0,1] (R0 ). Let H be an object of p − F F (Spec(R)). We review from loc. cit. the construction of D(H) = (N, F, Φ0 , Φ1 , ∇). Let n ∈ N be such that pn annihilates H. The triple (N, Φ0 , ∇) is part of the evaluation (N, Φ0 , ϑ0 , ∇) of the Dieudonné crystal D(HR/pR ) at the thickening (Spec(R/pR) ֒→ Spec(R/pn R), δ(p)) and this motivates the notation D(H). The direct summand F 1 of N is the Hodge filtration of N defined by the lift HR/pn R . To define Φ1 we can work locally in the Zariski topology of Spec(R/pR) and thus we can assume that R = R∧ = R0∧ and (cf. Raynaud theorem of [BBM, Thm. 3.1.1]) that H is a closed subgroup scheme of an abelian scheme A over Spec(R). Let 1 1 (A/R) = Hcrys (A/R). Let FA be the direct summand of NA that is the NA := HdR Hodge filtration defined by A. Let ∇A be the Gauss–Manin connection on NA . Let Φ0A be the ΦR -linear endomorphism of NA . The embedding H ֒→ A defines a canonical Repimorphism (NA , FA , Φ0A , ∇A ) ։ (N, F, Φ0 , ∇), cf. [BBM, Thm. 3.1.2 and pp. 132–133]. Thus as Φ1 : F → N we can take the natural quotient of the restriction of p1 Φ0A to FA . As the Zp -linear categories of p-divisible groups over Spec(R0∧ ) and over Spf(R0∧ ) are canonically isomorphic, from [Fa1, Thm. 7.1] we get that: 2.3.4. Theorem. Suppose that p > 2, that the W (k)-algebra R0 is smooth, and that R = ∇ R0∧ . Then the Zp -linear functor D : p − F F (Spec(R)) → MF[0,1] (R) is an antiequivalence of Zp -linear categories. One can use [Fa1, Sect. 2, pp. 31–33] and [Ra2, Thm. 3.3.3] to check that the functor D of Theorem 2.3.4 is in fact an antiequivalence of abelian categories. 2.3.5. A more general form of (1). In this Subsubsection we assume that R0 and R0 /pR0 are integral domains, that R = R0∧ , and that there exist elements z1 , . . . , zd ∈ R such that R is a formally étale W (k)[z1 , . . . , zn ]-algebra and the equality ΦR (zi ) = zip holds 14 for all i ∈ {1, . . . , d}. For n ∈ N, each element of Ker(B + (R)/pn B + (R) → R̄∧ /pn R̄∧ ) is nilpotent. Thus the formally étale assumption and the fact that R is p-adically complete imply that there exists a unique W (k)-monomorphism iR : R ֒→ B + (R) that lifts the inclusion R ֒→ R̄∧ and that takes zi to (zi , 0, 0, . . . ) ∈ W (AR/pR ) ֒→ B + (R). Let D be a p-divisible group over Spec(R). Let (N, FN1 , ΦN , ∇N ) be as in Subsection 2.1. As the R-homomorphism sR/pR : B + (R) → R̄∧ is onto (cf. Lemma 2.3.1 (a)), there exists a B + (R)-linear monomorphism (3) iD : N ⊗R B + (R) ֒→ H 1 (DKR ) ⊗Zp B + (R) that is constructed and has the same properties as the B + (W (k))-linear monomorphism iD of Subsubsection 2.2.2. Here (as for H 1 (D)) H 1 (DKR ) is the dual of the Tate-module of DKR . As the W (k)-monomorphism iR : R ֒→ W + (R) respects Frobenius lifts, the Frobenius endomorphism of N ⊗R B + (R) is ΦN ⊗ ΦR/pR . 2.4. Artin–Schreier systems of equations. For a matrix z, let z [p] be the matrix obtained by raising the entries of z to their p-powers. Let n ∈ N. Let x1 , . . . , xn be variables. Let x := (x1 , . . . , xn )t . Let Spec(Z0 ) be an affine Spec(k)-scheme. An Artin– Schreier system of equations in n variables over Z0 is a system (4) x = Bx[p] + C0 , where B ∈ Mn×n (Z0 ) and C0 ∈ Mn×1 (Z0 ). Let Z00 be the finitely generated k-subalgebra of Z0 generated by the entries of B. 2.4.1. Theorem. (a) The system (4) defines an étale, affine Spec(Z0 )-scheme Spec(Z1 ). (b) Each geometric fibre of the morphism q1 : Spec(Z1 ) → Spec(Z0 ) has a number of geometric points equal to pm , where m ∈ {0, . . . , n} depends on the fibre. (1) (c) There exists an open, Zariski dense subscheme U00 of Spec(Z00 ) that depends (1) only on B but not on either the Z00 -algebra Z0 or C0 and such that if U0 := Spec(Z0 )×Spec(Z00 ) (1) (1) U00 , then the morphism U1 (1) (1) := q1∗ (U0 ) → U0 is an étale cover. (d) Suppose that Z0 is noetherian. Let Z∞ be the inductive limit of Fp -algebras Zq , q ∈ N, q ≥ 2, where each Zq is the Zq−1 -algebra defined by an Artin–Schreier system of equations of the form (4) but with C0 replaced by some other matrix Cq−1 ∈ Mn×1 (Zq−1 ). Then the image of each connected component C∞ of Spec(Z∞ ) in Spec(Z0 ) is an open, Zariski dense subscheme of a connected component of Spec(Z0 ). (e) Suppose that Z0 is a finitely generated k-algebra. Then each point p∞ of Spec(Z∞ ) specializes to a point of Spec(Z∞ ) whose residue field is algebraic over k. Proof: We prove (a). The Z0 -algebra Z1 is of finite presentation. We have ΩZ1 /Z0 = 0, cf. the shape of the system (4). We check that the criterion of formal étaleness holds for q1 . Let Z be a commutative Z0 -algebra and let I be an ideal of Z of such that I 2 = 0. Let w̄ be a solution of (4) in Z/I. If w ∈ Mn×1 (Z) lifts w̄, then Bw[p] + C0 does not depend on the choice of w and it is the unique solution of (4) in Z that lifts w̄. We conclude that the morphism q1 is étale, cf. [BLR, Ch. 2, Sect. 2.2, Props. 2 and 6]. 15 To prove (b) we can assume that Z0 = k = k̄. Let B1 ∈ Mn×n (k) be such that = B. We use induction on n ∈ N. The case n = 1 is trivial. For n ≥ 2 the passage from n − 1 to n goes as follows. Let x0 be a new variable. We introduce the homogeneous system of equations in the variables x0 , x1 , . . . , xn [p] B1 (5) x0p−1 x = Bx[p] + xp0 C0 , the multiplications with powers of x0 being scalar multiplications. The system (5) defines a closed subscheme Y of Pnk . By repeatedly using the projective dimension theorem (cf. [Ha, Ch. I, Thm. 7.2]), we get that the k-scheme Y is non-empty. Let H be the hypersurface of Pnk defined by x0 = 0. We first consider the case when the set (Y ∩ H)(k) is empty. In this case we have Spec(Z1 ) = Y and therefore from (a) we get that Y is an étale, closed subscheme of Pnk . Thus the Spec(k)-scheme Y is finite. The degree of Y is pn , cf. [Ha, Ch. n I, Thm. 7.7]. Thus Spec(Z1 ) = Spec(k p ); therefore (b) holds in this case. We consider the case when the set (Y ∩ H)(k) is non-empty. This means that the system of linear equations B1 x = 0 has a non-trivial solution in k. Thus the matrix B1 is not invertible. Up to a renumbering of xi ’s we can assume that we have an identity Pn−1 (n) (i) (i) B1 = i=1 di B1 , where di ∈ k and where B1 is the i-th row of B1 . The system (4) is equivalent to the system X of equations, where we keep the first n − 1 equations of (4) which involve x1 , . . . , xn−1 on the left hand side and where we replace the last equation of (4) which involves xn on the left hand side by the following linear equation (6) xn + n−1 X dpi xi = cn + n−1 X ci dpi , i=1 i=1 where (c1 , . . . , cn ) := C0t . Equation (6) allows us to eliminate the variable xn in X (and thus also in (4)). By performing this elimination we come across an Artin–Schreier system of equations in n − 1 variables over k of the form x̃ = B̃ x̃[p] + C̃0 , where x̃ := (x1 , . . . , xn−1 )t . The entries of B̃ ∈ Mn−1×n−1 (k) depend on B but not on C0 . By induction we have m Z1 = k p for some m ∈ {0, . . . , n − 1}. This proves (b). To prove (c), let Spec(W00 ) be the étale group scheme over Spec(Z00 ) defined by the addition of solutions of the Artin–Schreier system of equations y = By [p] , where (1) y := (y1 , . . . , yn )t . Let U00 be the maximal open subscheme of Spec(Z00 ) such that the (1) (1) morphism Spec(W00 ) ×Spec(Z00 ) U00 → U00 is an étale cover. As the fibres of q1 are non-empty (cf. (b)), the product rule (y, x) → y + x makes Spec(Z1 ) to be a left torsor (1) of the affine group scheme Spec(W00 ) ×Spec(Z00 ) Spec(Z0 ) over Spec(Z0 ). Thus U1 is an (1) étale cover of U0 i.e., (c) holds. To prove (d) and (e) we can assume that Spec(Z0 ) is reduced and connected. We (1) (1) (1) prove (d). For q ∈ N let Uq := Spec(Zq ) ×Spec(Z00 ) U00 . As Uq is an étale cover of (1) (1) (1) (1) Uq−1 (cf. (c)), the pull back U∞ of U00 to Spec(Z∞ ) is a pro-étale cover of U0 . By induction on dim(Spec(Z00 )) we define a stratification of Spec(Z00 ) in reduced, locally (1) (s) closed subschemes U00 , . . . , U00 such that the following property holds: 16 (i) (*) for each i ∈ {2, . . . , s} the scheme U00 is the maximal open subscheme of the (j) reduced scheme of Spec(Z00 ) \ (∪i−1 j=1 U00 ) which has the property that the morphism (i) (i) Spec(W00 ) ×Spec(Z00 ) U00 → U00 is an étale cover. As above (for the case i = 1) we argue that for i ∈ {1, . . . , s} the following scheme (i) (i) (i) := Spec(Z∞ ) ×Spec(Z00 ) U00 is a pro-étale cover of U0 := Spec(Z0 ) ×Spec(Z00 ) U00 . The going up (resp. down) property holds for pro-étale covers (resp. pro-étale morphisms), cf. [Ma, Subsect. (5.E), Thm. 5] (resp. [Ma, Subsect. (5.D), Thm. 4]). We apply this (i) (i) to the pro-étale covers U∞ → U0 and to the pro-étale morphism Spec(Z∞ ) → Spec(Z0 ). We get that the image C0 of C∞ in Spec(Z0 ) is the complement in Spec(Z0 ) of the union (i) of the schematic closures in Spec(Z0 ) of those connected components of some U0 with i ∈ {2, . . . , s} that do not intersect C0 . As Spec(Z0 ) is noetherian, this union is finite and thus C0 is an open, Zariski dense subscheme of Spec(Z0 ). We prove (e). Let I∞ be the prime ideal of Spec(Z∞ ) that defines p∞ . To prove (e), for ∗ ∈ N ∪ {0, ∞} we can replace Z∗ with Z∗ /(I∞ ∩ Z0 )Z∗ . Thus we can assume that I∞ ∩ Z0 = 0 and that Z0 is an integral domain. By localizing we can also assume that (1) U00 is Spec(Z00 ). The morphism Spec(Z∞ /I∞ ) → Spec(Z0 ) is an étale cover and thus surjective. As Spec(Z0 ) has points whose residue fields are algebraic over k and due to the going down property, we get that p∞ specializes to points of Spec(Z∞ ) whose residue fields are algebraic over k. (i) U∞ 2.4.2. Remark. The case C0 = 0 (i.e., the Z00 -algebra W00 ) is studied systematically for the first time in [Ly], using a language of F -modules. We came across Artin–Schreier systems of equations independently in 1998 in connection to formula (12) below. 2.5. Simple group properties. Until the end we use the notations of Subsection 1.1. Let S ∈ {W (k), B +(W (k))}. Let xS ∈ S be p if S = W (k) and be β0 if S = B + (W (k)). Let (tα )α∈J̃ be another family of tensors of F 0 (T(M ))[ p1 ] fixed by φ and such that GB(k) is also the subgroup of GL M [ 1 ] that fixes tα for all α ∈ J̃. For α ∈ J̃ let vα ∈ T(H 1 (D)[ p1 ]) p be such that iD [ β10 ] maps tα to vα , cf. Subsubsection 2.2.2. ∼ 2.5.1. Fact. (a) There exists an isomorphism ρ̃S : (M ⊗W (k) S, (tα )α∈J̃ ) → (H 1 (D) ⊗Zp ∼ S, (tα )α∈J̃ ) if and only if there exists an isomorphism ρS : (M ⊗W (k) S, (tα )α∈J ) → (H 1 (D)⊗Zp S, (tα )α∈J ). ∼ (H 1 (D)⊗Zp (b) Suppose that there exists an isomorphism iS : (M ⊗W (k) S[ x1S ], (tα )α∈J ) → ∼ S[ x1S ], (tα )α∈J ). Then there exists an isomorphism ρS : (M ⊗W (k) S, (tα )α∈J ) → (H 1 (D)⊗Zp S, (tα )α∈J ) if and only if there exists an element h ∈ G(S[ x1S ]) that takes the S-submodule 1 1 i−1 S (H (D) ⊗Zp S) of M ⊗W (k) S[ xS ] onto M ⊗W (k) S. Proof: We prove (a). If ρ̃S exists, then its tensorization with B + (W (k))[ β10 ] is of the form iD [ β10 ] ◦ h, where h ∈ GL M (B + (W (k))[ β10 ]) fixes tα for all α ∈ J̃. Thus h ∈ G(B + (W (k))[ β10 ]) and therefore ρ̃S takes tα to vα for all α ∈ J. Thus we can take ρS to be defined by ρ̃S . Similarly, if ρS exists, then we can take ρ̃S to be defined by ρS . 17 Part (b) is an elementary exercise: one regains each element of the set {ρS , h} from the other element via the identity ρS [ x1S ] = iS ◦ h−1 . Let σφ := φµ(p). It is a σ-linear automorphism of M that normalizes Lie(GB(k) ) ∩ EndW (k) (M ). Until Subsection 2.6 we assume k = k̄. Thus MZp := {x ∈ M |σφ (x) = x} is a Zp -structure of M . As µ(p) and φ fix tα , we have σφ (tα ) = tα . Thus we have tα ∈ T(MZp [ p1 ]) for all α ∈ J. Thus G is the pull back to Spec(W (k)) of a flat, closed subgroup scheme GZp of GL MZp . If G is smooth, then Lie(GZp ) = {x ∈ Lie(G)|σφ (x) = x}. 2.5.2. Lemma. (a) Suppose that G is smooth and Gk is connected. Then there exists ∼ an isomorphism ρZp : (MZp , (tα )α∈J ) → (H 1 (D), (vα )α∈J ) if and only if there exists an ∼ 1 isomorphism ρ : (M, (tα )α∈J ) → (H (D) ⊗Zp W (k), (vα )α∈J ). (b) Suppose that G is smooth. Then there exists an isomorphism ρ as in (a) if and ∼ only if there exists an isomorphism ρB+ (W (k)) : (M ⊗W (k) B + (W (k)), (tα )α∈J ) → (H 1 (D)⊗Zp B + (W (k)), (vα )α∈J ). Proof: To prove the Lemma we can assume that we have tα ∈ T(M ) and vα ∈ T(H 1 (D)) for all α ∈ J. Let Y0 be the affine Spec(Zp )-scheme that parametrizes isomorphisms between (MZp , (tα )α∈J ) and (H 1 (D), (vα )α∈J ). Let Y be the schematic closure of Y0Qp in Y0 . The group scheme GZp acts on Y from the left. The only if part of (a) is trivial. We prove the if part of (a). As ρ exists, YW (k) is the trivial torsor of G. Thus Y is a torsor of GZp in the flat topology and therefore it is a smooth Spec(Zp )-scheme of finite type. Thus Y has points with values in finite fields. As GFp is connected, Y has Fp -valued points (cf. Lang’s theorem). Thus Y has Zp -valued points i.e., ρZp exists. This proves (a). The proof of (b) is similar, starting from the fact that each torsor of G in the flat topology of Spec(W (k)) has k-valued points and therefore (as G is smooth) it also has W (k)-valued points. 2.5.3. Lemma. We recall that k = k̄. Let (MZp , GZp ) and σφ = φµ( 1p ) be as above. Let G̃ be a flat, closed subgroup scheme of G such that the following three properties hold: (i) the cocharacter µ : Gm → G factors through G̃; (ii) we have φ(Lie(G̃B(k) )) = Lie(G̃B(k) ); (iii) the generic fibre G̃B(k) is connected. Then there exists a unique closed, flat subgroup scheme G̃Zp of GZp whose pull back to Spec(W (k)) is G̃. If G̃B(k) is a reductive group, then the generic fibre G̃Qp of G̃Zp is also a reductive group and therefore there exists a set J̃ that contains J and such that there exist a family of tensors (tα )α∈J̃ of T(MZp [ p1 ]) with the property that G̃Qp is the subgroup of GLMZp [ 1 ] which fixes tα for all α ∈ J̃. p Proof: We know that σφ normalizes Lie(G̃B(k) ), cf. properties (i) and (ii). The following Lie algebra g̃Qp := {x ∈ Lie(G̃B(k) )|σφ (x) = x} over Qp is the unique Lie subalgebra of Lie(GQp ) such that we have Lie(G̃B(k) ) = g̃Qp ⊗Qp B(k). We check that there exists a unique connected subgroup G̃Qp of GQp whose Lie algebra is g̃Qp . The uniqueness part is implied by [Bo, Ch. II, Subsect. 7.1]. To check the existence part, we consider commutative Qp algebras Ã such that there exists a closed, flat subgroup scheme G̃Ã of GÃ whose Lie algebra 18 is g̃Qp ⊗Qp Ã. For instance, Ã can be B(k) itself and thus we can assume Ã is a finitely generated Qp -subalgebra of B(k). By replacing Ã with its quotient through a maximal ideal, we can assume Ã is a finite field extension of Qp . Even more, we can assume that Ã is a finite Galois extension of Qp and that G̃Ã is connected. As Lie(G̃Ã ) = g̃Qp ⊗Qp Ã, from [Bo, Ch. II, Subsect. 7.1] we get that the natural action of the Galois group Gal(Ã/Qp ) on g̃Qp ⊗Qp Ã is defined naturally by a natural action of Gal(Ã/Qp ) on G̃Ã . This last action is free. As G̃Ã is an affine scheme, the quotient G̃Qp of G̃Ã by Gal(Ã/Qp ) exists (cf. [BLR, Ch. 6, Sect. 6.1, Thm. 5]) and it has all the desired properties. From [Bo, Ch. II, Subsect. 7.1] and the property (iii) we get that our notations match i.e., the pull back of the connected group G̃Qp to Spec(B(k)) has the same Lie algebra as G̃B(k) and therefore it is G̃B(k) . Let G̃Zp be the schematic closure of G̃Qp in GZp ; it is the unique closed, flat subgroup scheme of GZp whose pull back to Spec(W (k)) is G̃. Obviously G̃Qp is reductive if and only if G̃B(k) is so. The last part of the Lemma that pertains to the set J˜ follows from [De2, Prop. 3.1 c)]. 2.6. The smoothening G′ of G. We recall from [BLR, Ch. 7, Sect. 7.1, Thm. 5] that there exists a unique smooth group scheme G′ over Spec(W (k)) which is equipped with a homomorphism iG : G′ → G and for which the following two properties hold: (i) the generic fibre of iG is an isomorphism, and (ii) if Y is an arbitrary smooth Spec(W (k))-scheme, then each morphism Y → G of Spec(W (k))-schemes factors uniquely through iG . See [BLR, Ch. 3, Sect. 3.2] for dilatations. As iG is obtained using a sequence of dilatations centered on special fibres (see [BLR, Ch. 7, Sect. 7.1, pp. 174–175]), the scheme G′ is affine. We denote also by µ : Gm → G′ the factorization through G′ of the cocharacter µ : Gm → G of the end of Subsection 2.1, cf. (ii). From (ii) we get that: (iii) each smooth, closed subgroup scheme U of G is naturally a closed subgroup scheme of G′ . We identify Lie(G′ ) with a W (k)-lattice of Lie(GB(k) ) = Lie(G′B(k) ) contained in 0 ′ EndW (k) (M ). Let G′0 k be the identity component of Gk . Let G(W (k̄)) be the subgroup of G(W (k̄)) = G′ (W (k̄)) formed by W (k̄)-valued points of G′ whose special fibres factor 0 0 through G′0 k . Let G(W (k)) := G(W (k̄)) ∩ G(W (k)). 2.7. Two unipotent group schemes. Let µ : Gm → G and M = F 1 ⊕ F 0 be as in Subsection 2.1. Let T(M ) = ⊕i∈Z F̃ i (T(M )) be the direct sum decomposition such that Gm acts via µ on F̃ i (T(M )) as the −i-th power of the identity character of Gm . Let Ubig be the smooth, unipotent, commutative, closed subgroup scheme of GL M whose Lie algebra is F̃ −1 (EndW (k) (M )) := F̃ −1 (T(M )) ∩ EndW (k) (M ). Under the direct sum decomposition EndW (k) (M ) = HomW (k) (F 1 , F 0 ) ⊕ EndW (k) (F 1 ) ⊕ EndW (k) (F 0 ) ⊕ HomW (k) (F 0 , F 1 ) of W (k)-modules, we have F̃ −1 (EndW (k) (M )) = HomW (k) (F 1 , F 0 ). If Spec(R) is an affine Spec(W (k))-scheme, then we have Ubig (R) = 1M ⊗W (k) R + F̃ −1 (EndW (k) (M ))⊗W (k) R. The 19 intersection Lie(GB(k) ) ∩ F̃ −1 (EndW (k) (M )) is a direct summand of F̃ −1 (EndW (k) (M )). Let U be the smooth, closed subgroup scheme of Ubig (and thus also of GL M ) defined by the following rule on valued points: U (R) = 1M ⊗W (k) R + Lie(GB(k) ) ∩ F̃ −1 (EndW (k) (M )) ⊗W (k) R. We have Lie(U ) = Lie(GB(k) ) ∩ F̃ −1 (EndW (k) (M )). As UB(k) is connected and as we have Lie(UB(k) ) ⊆ Lie(GB(k) ), the group UB(k) is a subgroup of GB(k) (cf. [Bo, Ch. II, Subsect. 7.1]). As U and G are schematic closures in GLM of their generic fibres, U is a smooth, closed subgroup scheme of G and thus also of G′ (cf. property 2.6 (iii)). 3. Global deformations In this Section we use the notations of Subsections 2.1 and 2.6. We first construct global deformations of (M, F 1 , φ, G, (tα )α∈J ) over p-adic completions of ind-étale algebras over smooth W (k)-algebras whose reductions modulo p are regular, formally smooth over k, geometrically connected, and define spectra which have Zariski dense sets of k̄valued points. Then we use such deformations to “connect” (M, F 1 , φ, G, (tα )α∈J ) with (M, F 1 , gφ, G, (tα)α∈J ), where g is an arbitrary element of G(W (k))0 . If k = k̄, then we truly connect both (M, F 1 , φ, G, (tα )α∈J ) and (M, F 1 , gφ, G, (tα )α∈J ) with the same quadruple (M, F 1 , hφ, G, (tα )α∈J ) for some h ∈ G(W (k))0 that has certain properties (see Subsection 3.5). Until Subsection 3.4 we do not use the fact that (even if p = 2) the triple (M, F 1 , φ) is the filtered Dieudonné module of a p-divisible group D over Spec(W (k)). In Subsection 3.1 we develop the language needed to state the Theorem 3.2 which pertains to the existence of some connections and implicitly of some global deformations of (M, F 1 , φ, G, (tα )α∈J ). In Subsection 3.3 we prove the basic results Lemma 3.1.3 and Theorem 3.2. In Subsection 3.4 we translate Theorem 3.2 in terms of p-divisible groups as allowed by Theorem 2.3.4 and by a variant of it for p = 2. In Subsection 3.5 we apply Subsection 3.4 to show that to prove the Main Theorem we can assume that G is a torus. 3.1. Notations and a language. Let dM := rkW (k) (M ), d := dim(GB(k) ) = dim(Gk ), S(M ) := {1, . . . , dM }, and S(G) := {1, . . . , d}. Let G′ and G(W (k̄))0 be as in Subsection 2.6. Let Zk be the origin of G′k ; as a scheme it is Spec(k). Let z1 , . . . , zd be free variables. Let O := Spec(W (k)[z1 , . . . , zd ]). Let Y = Spec(R) be an open, affine subscheme of G′ through which the identity section a : Spec(W (k)) → G′ factors and which has a geometrically connected special fibre as well as the following property: (i) There exists an étale, affine morphism b : Y → O of Spec(W (k))-schemes, such that the morphism b ◦ a : Spec(W (k)) → O at the level of rings maps each zl to 0. We also view Zk as a closed subscheme of Y , Yk , or Y ∧ . We will choose Y such that there exists a section h : Spec(W (k)) ֒→ Y = Spec(R) with the property that the W (k)-epimorphism W (k)[z1 , . . . , zd ] ։ W (k) associated to b ◦ h sends each zl to an element ul ∈ Gm (W (k)). If k is infinite, then the existence of h 20 is implied by the fact that the set of k-valued points of Yk is Zariski dense in Yk (cf. [Bo, Ch. V, Cor. 18.3]). Composing b with an automorphism of O which takes zl to u−1 l zl for all l ∈ S(G), we can assume that we have ul = 1 for all l ∈ S(G). The W (k)-monomorphism W (k)[z1 , . . . , zd ] ֒→ R associated to b allows us to identify zl with an element of R. Let ΦR be the Frobenius lift of either R∧ or Y ∧ that is compatible with σ and such that we have ΦR (zl ) = zlp for all l ∈ S(G). Let r ∈ G′ (Y ∧ ) = G′∧ (Y ∧ ) be the universal element of G′∧ defined by the p-adic completion of the inclusion Y ֒→ G′ . Here and it what follows we identify points of G′∧ (or of G′ ) with values in a flat, affine Spec(W (k))-scheme Spec(∗) with ∗-linear automorphisms of M ⊗W (k) ∗. See the end of Subsection 2.1 for the direct sum decomposition M = F 1 ⊕ F 0 . 3.1.1. Definition. A morphism Ỹ0 → Y ∧ will be called formally quasi-étale if either it is formally étale or there exists a formally étale morphism Ỹ → Y ∧ such that Ỹ0 is the spectrum of the completion of a local ring of Ỹ whose residue field has characteristic p and Ỹ0 → Y ∧ is the natural morphism. Thus Ỹ0 is regular and a formally smooth Y ∧ -scheme. 3.1.2. Definition and notations. (a) Let c : Spec(Q̃∧ ) → Y ∧ be a formally quasi-étale, affine morphism of Spec(W (k))-schemes, with Q̃ a regular W (k)-algebra. Let ΦQ̃ be the only Frobenius lift of Spec(Q̃∧ ) (or Q̃∧ ) that satisfies the identity c ◦ ΦQ̃ = ΦR ◦ c. Let MQ̃ := (M ⊗W (k) Q̃∧ , F 1 ⊗W (k) Q̃∧ , Φ(Q̃)0 , Φ(Q̃)1 ), where Φ(Q̃)0 := r ◦ c(φ ⊗ ΦQ̃ ) and where Φ(Q̃)1 : F 1 ⊗W (k) Q̃∧ → M ⊗W (k) Q̃∧ is the ΦQ̃ -linear map such that for x ∈ F 1 ⊗W (k) Q̃∧ we have pΦ(Q̃)1 (x) = Φ(Q̃)0 (x). For n ∈ N we also denote by Φ(Q̃)i its reduction modulo pn , i ∈ {0, 1}. (b) Let δ0 be the flat connection on either M ⊗W (k) Q̃∧ or M ⊗W (k) Q̃/pn Q̃ that annihilates M ⊗ 1. (c) Let MQ̃ /pn MQ̃ be the reduction modulo pn of MQ̃ ; it is an object of the category ∇ MF[0,1] (Q̃). We say MQ̃ /pn MQ̃ potentially can be viewed as an object of MF[0,1] (Q̃) if there exists a connection ∇ : M ⊗W (k) Q̃/pn Q̃ → M ⊗W (k) Q̃/pn Q̃⊗Q̃/pn Q̃ Ω(Q̃/pn Q̃)/Wn (k) = M ⊗W (k) Ω(Q̃/pn Q̃)/Wn (k) such that we have: (7) (8) ∇ ◦ Φ(Q̃)0 (x) = p(Φ(Q̃)0 ⊗ dΦQ̃ /p) ◦ ∇(x) ∇ ◦ Φ(Q̃)1 (x) = (Φ(Q̃)0 ⊗ dΦQ̃ /p) ◦ ∇(x) ∀x ∈ F 0 ⊗W (k) Q̃/pn Q̃ and ∀x ∈ F 1 ⊗W (k) Q̃/pn Q̃. Here dΦQ̃ /p is the differential of ΦQ̃ divided by p and then taken modulo pn . About ∇ we say it is a connection on MQ̃ /pn MQ̃ . We emphasize that the system of equations obtained by putting (7) and (8) together, does not depend on the choice of the direct supplement F 0 of F 1 in M ; moreover, as we chose Φ(Q̃)0 and Φ(Q̃)1 to be ΦQ̃ -linear maps and not Q̃∧ -linear maps, we got x ∈ F i ⊗W (k) Q̃/pn Q̃ and not in F i ⊗W (k) σ Q̃/pn Q̃ (here i ∈ {0, 1}). 21 (d) We say the connection ∇ of (c) respects the G-action, if for each l ∈ S(G), the Q̃/pn Q̃-linear endomorphism of M ⊗W (k) Q̃/pn Q̃ which for x ∈ M maps x ⊗ 1 to ∂ ∇( ∂z )(x⊗1) ∈ M ⊗W (k) Q̃/pn Q̃, is an element of (Lie(GB(k) )∩EndW (k) (M ))⊗W (k) Q̃/pn Q̃. l 3.1.3. Lemma. Let n ∈ N. We consider a formally quasi-étale, affine morphism c : Spec(Q̃∧ ) → Y ∧ of Spec(W (k))-schemes, with Q̃ a regular W (k)-algebra. If all connected components of Spec(Q̃/pQ̃) have non-empty intersections with c−1 (Zk ), then there exists at most one connection on MQ̃ /pn MQ̃ . 3.2. Basic Theorem. The following four things hold: (a) For each n ∈ N there exists an étale R-algebra Qn such that the natural formally ∧ étale, affine morphism ℓn : Spec(Q∧ has the following three properties: n) → Y (i) ℓ−1 n (Zk ) = Spec(k) and Spec(Qn /pQn ) is a geometrically connected Spec(k)-scheme; (ii) there exists a unique connection ∇n on MQn /pn MQn and moreover ∇n is integrable, nilpotent modulo p, and respects the G-action; (iii) for each formally quasi-étale, affine morphism c : Spec(Q̃∧ ) → Y ∧ of Spec(W (k))schemes (with Q̃ a regular W (k)-algebra) such that all connected components of Spec(Q̃/pQ̃) have a non-empty intersection with c−1 (Zk ) and MQ̃ /pn MQ̃ poten∇ tially can be viewed as an object of MF[0,1] (Q̃), there exists a unique morphism cn : Spec(Q̃∧ ) → Spec(Q∧ n ) such that c = ℓn ◦ cn and the unique connection on MQ̃ /pn MQ̃ is the extension of ∇n via cn modulo pn . ∧ (b) Let n ∈ N. We consider the unique morphism ℓ(n) : Spec(Q∧ n+1 ) → Spec(Qn ) such that ℓn+1 = ℓn ◦ ℓ(n) and the extension of ∇n via ℓ(n) modulo pn is ∇n+1 modulo pn , cf. property (iii) of (a). Then ℓ(n) modulo p is étale, quasi-finite, and generically an étale 2 cover of degree at most pd . Similarly, ℓ1 modulo p has all these properties. (c) Let Spec(Q∞ ) be the projective limit of Spec(Q∧ n ) under the transition morphisms (n) ∧ ∧ ℓ , n ∈ N. Let Q := Q∞ . Let ℓ : Spec(Q) → Y be the resulting morphism. Then the image Yk0 of ℓ modulo p is an open subscheme of Yk and moreover each point of Spec(Q/pQ) specializes to a point of Spec(Q/pQ) whose residue field is an algebraic extension of k. (d) There exists a reduced, closed subscheme B(φ) of G′k of dimension at most d − 1, that depends on φ and on the reduction modulo p of dΦR /p but not on the choice of h : Spec(W (k)) ֒→ Y in Subsection 3.1, and such that Yk \ B(φ) ⊆ Yk0 . 3.3. The proofs of 3.1.3 and 3.2. We prove Lemma 3.1.3 and Theorem 3.2 in the following nine steps. 1) The complete local case. Let R̂ be the completion of R with respect to the ideal of R that defines the factorization Spec(W (k)) ֒→ Y of the identity section a : Spec(W (k)) ֒→ G′ . See Subsubsection 3.1.2 (a) for ΦR̂ and Φ(R̂)0 . There exists a unique connection ∇R̂ on M ⊗W (k) R̂ such that Φ(R̂)0 is horizontal i.e., we have ∇R̂ ◦ Φ(R̂)0 = (Φ(R̂)0 ⊗dΦR̂ ) ◦ ∇R̂ . The connection ∇R̂ is integrable and nilpotent modulo p. As we have ΦR (zl ) = zlp for all l ∈ S(G), the last two sentences follow from [Fa2, Thm. 10] applied 22 to the triple (M ⊗W (k) R̂, F ⊗W (k) R̂, Φ(R̂)0 ) (the R̂-linear map (M + p1 F 1 ) ⊗W (k) σ R̂ → M ⊗W (k) R̂ defined naturally by Φ(R̂)0 is an R̂-linear isomorphism). We recall the argument ˜ be another connection on M ⊗W (k) R̂ such that Φ(R̂)0 is for the uniqueness part. Let ∇ R̂ horizontal. As we have ΦR (zl ) = zlp for all l ∈ S(G) and • ◦ Φ(R̂)0 = (Φ(R̂)0 ⊗ dΦR̂ ) ◦ • ˜ }, by induction on q ∈ N we get that ∇ ˜ − ∇ ∈ EndW (k) (M ) ⊗W (k) for • ∈ {∇R̂ , ∇ R̂ R̂ R̂ 1 ∧ q ∧ (z1 , . . . , zd ) ΩR̂/W (k) [ p ], where ΩR̂/W (k) = ΩR/W (k) ⊗R R̂ is the p-adic completion of ˜ =∇ . Ω . As R̂ is complete in the (z1 , . . . , zd )-topology, we get that ∇ R̂/W (k) R̂ R̂ See Subsubsection 3.1.2 (b) for δ0 . We check the following relation (9) γR̂ := ∇R̂ − δ0 ∈ (Lie(GB(k) ) ∩ EndW (k) (M )) ⊗W (k) ΩR/W (k) ⊗R R̂. If G is smooth, then (9) is implied by [Fa2, Sect. 7, Rm. ii)]. In general, we view T(M ) as a module over the Lie algebra (associated to) EndW (k) (M ) and we denote also by ∇R̂ the connection on T(M ⊗W (k) R̂[ p1 ]) which extends naturally the connection ∇R̂ on M ⊗W (k) R̂. The ΦR̂ -linear action of Φ(R̂)0 on M ⊗W (k) R̂ extends to a horizontal ΦR̂ -linear action of Φ(R̂)0 on T(M ⊗W (k) R̂[ p1 ]). For instance, if f ∈ M ∗ ⊗W (k) R̂ = (M ⊗W (k) R̂)∗ and if x ∈ M ⊗W (k) R̂, then Φ(R̂)0 (f ) ∈ M ∗ ⊗W (k) R̂[ p1 ] maps Φ(R̂)0 (x) to ΦR̂ (f (x)). As φ and G fix tα , the tensor tα ∈ T(M ⊗W (k) R̂[ p1 ]) is also fixed by Φ(R̂)0 . Thus we have ∇R̂ (tα ) = (Φ(R̂)0 ⊗ dΦR̂ )(∇R̂ (tα )). As we have dΦR̂ /p(zl ) = zlp−1 dzl for all l ∈ S(G), by induction on q ∈ N we get that ∇R̂ (tα ) = γR̂ (tα ) ∈ T(M ) ⊗W (k) (z1 , . . . , zd )q Ω∧ [ 1 ]. R̂/W (k) p As R̂ is complete with respect to the (z1 , . . . , zd )-topology, we get ∇R̂ (tα ) = γR̂ (tα ) = 0. But Lie(GB(k) ) ∩ EndW (k) (M ) is the Lie subalgebra of EndW (k) (M ) that centralizes tα for all α ∈ J. From the last two sentences we get that (9) holds. This implies that ∇R̂ respects the G-action. 2) General connections. Let {ei |i ∈ S(M )} be a W (k)-basis for M formed by elements of F 1 ∪ F 0 . For i ∈ S(M ) let εi ∈ {0, 1} be such that ei ∈ F εi . Let {eij |i, j ∈ S(M )} be the W (k)-basis for EndW (k) (M ) such that eij (es ) = δjs ei . The R-module ΩR/W (k) is free: {dzl |l ∈ S(G)} is an R-basis for it, cf. property 3.1 (i). Let c : Spec(Q̃∧ ) → Y ∧ be as in Lemma 3.1.3. Any connection ∇ on M ⊗W (k) Q̃∧ /pQ̃ is of the form X (10) ∇ = δ0 + xijl eij ⊗ dzl , (i,j,l)∈S(M )×S(M )×S(G) with xijl ∈ Q̃∧ /pQ̃ for all (i, j, l) ∈ S(M ) × S(M ) × S(G). 3) The equations. We start proving Lemma 3.1.3, Theorem 3.2 (a), and the part of Theorem 3.2 (b) that pertains to ℓ1 . The condition that ∇ of (10) is a connection on MQ̃ /pMQ̃ (i.e., that (7) and (8) hold for n = 1) gets translated into a system of equations in the variables xijl ’s as follows. Let j ∈ S(M ). As φ(F 0 ⊕ p1 F 1 ) = M , M ⊗W (k) R∧ is R∧ -generated by the Φ(R)εi (ei )’s with i ∈ S(M ). Thus we can write X ej = aj,i Φ(R)εi (ei ), i∈S(M ) 23 where aj,i ∈ R∧ . Let φ̄, ēj , and āj,i be the reductions modulo p of φ, ej , and aj,i (respectively). We have ∇(āj,i Φ(R)εi (ēi )) = āj,i ∇(Φ(R)εi (ēi )) + Φ(R)εi (ēi )dāj,i. Thus by plugging ēi ’s in the formula (7 + εi ), by multiplying the result with āj,i , and by summing up with i ∈ S(M ) we get that if ∇ is a connection on MQ̃ /pMQ̃ , then ∇(ēj ) = P (i,l)∈S(M )×S(G) xijl ēi ⊗ dzl is equal to (11) X i∈S(M ), εi =1 = āj,i (Φ(R)0 ⊗ dΦR /p)∇(ēi ) + X (i,i′ ,l)∈S(M )×S(M )×S(G), (εi′ ,εi )=(0,1) X Φ(R)εi (ēi )dāj,i i∈S(M ) āj,i xpi′ il φ̄(ēi′ )zlp−1 ⊗ dzl + X Φ(R)εi (ēi )dāj,i . i∈S(M ) By identifying the coefficients of the two expressions of ∇(ēj ) with respect to the R/pRbasis {ēi ⊗ dzl |(i, l) ∈ S(M ) × S(G)} for M ⊗W (k) ΩR/pR/k = M/pM ⊗k ΩR/pR/k , we come across an Artin–Schreier system of equations in d2m d variables of the form (12) xijl = Lijl (xp11l , xp12l , . . . , xpdM dM l ) + aijl (0), (i, j, l) ∈ S(M ) × S(M ) × S(G), where the form Lijl is homogeneous and linear and has coefficients in R/pR and where aijl (0) ∈ R/pR. The form Lijl involves only the variables xi′ il ’s, with (εi′ , εi ) = (0, 1). By varying j ∈ S(M ), we obtain all possible equations in the xijl ’s i.e., any other equation in the xijl ’s produced by (7) and (8) is a linear combination of the equations of (12). We consider the affine morphism ℓ(1) : S1 → S0 := Yk = Spec(R/pR) defined by the system of equations (12) (with the xijl ’s viewed as variables over R/pR). Thus ∇ is a connection on MQ̃ /pMQ̃ if and only if (12) holds i.e., if and only if c modulo p factors naturally through the morphism ℓ(1). 4) The uniqueness part. Let I0 be the ideal of R/pR generated by zl ’s. It suffices to prove Lemma 3.1.3 for n = 1 under the hypotheses that ∇ is a connection on MQ̃ /pMQ̃ and that Q̃/pQ̃ is a complete, local ring whose residue field is k̄ and whose maximal ideal is generated by I0 . The existence of another (i.e., different from ∇) connection on MQ̃ /pMQ̃ corresponds to a non-trivial solution in Q̃/pQ̃ of the system of equations (13) xijl = Lijl (xp11l , xp12l , . . . , xpdM dM l ), (i, j, l) ∈ S(M ) × S(M ) × S(G). The coefficients of the linear forms Lijl belong to I0p−1 , cf. (11). By induction on q ∈ N, we get that for each solution of (13) in Q̃/pQ̃ we have xijl ∈ I0p−1+pq (Q̃/pQ̃) for all (i, j, l) ∈ S(M ) × S(M ) × S(G). Thus the only solution of (13) in Q̃/pQ̃ is given by xijl = 0 for all (i, j, l) ∈ S(M ) × S(M ) × S(G). Thus Lemma 3.1.3 holds for n = 1. 5) Construction of Q1 . We know that the morphism ℓ(1) is étale, cf. Theorem 2.4.1 (a). Let S01 be the maximal open closed subscheme of S1 with the property that each connected component of it has a non-empty intersection with the closed subscheme 24 ℓ(1)−1 (Zk ) of S1 . From 4) we get that ℓ(1)−1 (Zk ) is either empty or Spec(k). Thus S01 is a connected, affine Spec(k)-scheme. From Theorem 2.4.1 (b) we get that ℓ(1)−1 (Zk ) is non-empty. Thus the scheme S01 is non-empty and we have ℓ(1)−1 (Zk ) = Spec(k). Standard arguments that involve a lift of (12) to a system over R, the Jacobi criterion for étaleness, and localizations, show the existence of an étale R-algebra Q1 such that the ∧ special fibre of the morphism ℓ1 : Spec(Q∧ is the morphism S01 → S0 defined 1) → Y naturally by ℓ(1). Let ∇1 be the unique connection on MQ1 /pMQ1 , cf. end of 3) and 4). 6) Proofs of 3.2 (a) for n = 1 and of 3.2 (b) for ℓ1 modulo p. As ℓ−1 1 (Zk ) = −1 0 ℓ(1) (Zk ) = Spec(k) and as the scheme S1 = Spec(Q1 /pQ1 ) is connected, property (i) of Theorem 3.2 (a) holds for n = 1. If ∇ is a connection on MQ̃ /pMQ̃ and if each connected component of Spec(Q̃/pQ̃) has a non-empty intersection with c−1 (Zk ), then from the construction of ℓ1 and the end of 3) we get that c modulo p factors uniquely through ℓ1 modulo p. Thus c factors uniquely as a morphism c1 : Spec(Q̃∧ ) → Spec(Q∧ 1) of Spec(W (k))-schemes in such a way that c = ℓ1 ◦ c1 and ∇ is the extension of ∇1 via c1 modulo p. Thus property (iii) of Theorem 3.2 (a) holds for n = 1. Applying this with c as the natural formally quasi-étale, affine morphism Spec(R̂) → Y ∧ , we get that the extension of ∇1 to a connection on M ⊗W (k) R̂/pR̂ (via the natural morphism Spec(R̂) → Spec(Q∧ 1 ) of Y -schemes), is the reduction modulo p of ∇R̂ and thus it is a connection that is integrable, nilpotent modulo p, and respects the G-action. This implies that the connection ∇1 itself is integrable, nilpotent modulo p, and respects the G-action. Thus property (ii) of Theorem 3.2 (a) also holds for n = 1. Therefore Theorem 3.2 (a) holds for n = 1. Related to the part of Theorem 3.2 (b) that refers to ℓ1 modulo p, we are left to check 2 that the fibres of ℓ1 modulo p over geometric points have at most pd points. We consider a k-linear map lG : EndW (k) (M )/(pEndW (k) (M ) + Lie(GB(k) ) ∩ EndW (k) (M )) → k. As ∇1 respects the G-action, for each map lG the variables xijl ’s satisfy the following relation (14) X (i,j)∈S(M )×S(M ) xijl lG (ēij ) = 0, where l ∈ S(G). As Lie(GB(k) ) ∩ EndW (k) (M ) is a direct summand of EndW (k) (M ) of rank d, the k-linear span of the lG ’s maps has dimension d2M − d. Thus for a fixed l our variables xijl ’s also satisfy a system of d2M − d-linear equations with coefficients in k whose rank is precisely d2M − d and which is naturally attached to (14). By varying l ∈ S(G), as in the proof of Theorem 2.4.1 (b) we can use the equations (14) to eliminate d2M d − d2 of our present xijl ’s variables at once; in other words S01 is a connected component of the S0 -scheme S1 that is also definable using an Artin–Schreier system of equations in d2 = d2M − (d2M − d2 ) variables and with coefficients in R/pR. Thus from Theorem 2.4.1 (b) and (c) we get that ℓ1 modulo p (i.e., the morphism S01 → S0 ) is generically an étale cover of degree at most 2 pd . This proves Theorem 3.2 (b) for ℓ1 modulo p. 7) The inductive statement. By induction on n ∈ N we prove Lemma 3.1.3 and Theorem 3.2 (a). We assume that Lemma 3.1.3 folds for n and that we managed to construct Qm for m ∈ {1, . . . , n}. We will show first that Lemma 3.1.3 holds for n + 1 and then we will construct Qn+1 . We work with a formally quasi-étale morphism 25 c : Spec(Q̃∧ ) → Y ∧ that factors through a formally quasi-étale morphism cn : Spec(Q̃∧ ) → lift n+1 Spec(Q∧ Q̃ which modulo pn is the extension n ). We fix a connection ∇n on M ⊗W (k) Q̃/p of ∇n via cn modulo pn and which respects the G-action. The general form of a connection on M ⊗W (k) Q̃/pn+1 Q̃ that lifts the extension of ∇n via cn modulo pn , is of the form (15) ∇ = ∇lift n + X (i,j,l)∈S(M )×S(M )×S(G) pn xijl eij ⊗ dzl , with all xijl ∈ Q̃/pQ̃ and with pn xijl identified naturally with an element of pn Q̃/pn+1 Q̃. 8) The key point. The condition that ∇ of (15) is a connection on MQ̃ /pn+1 MQ̃ is expressed by an Artin–Schreier system of equations in the variables xijl ’s of (15) over Qn /pQn . The key point is that the same computations as in (11) show that this system has the form (16) xijl = Lijl (xp11l , xp12l , . . . , xpdM dM l ) + aijl (n), (i, j, l) ∈ S(M ) × S(M ) × S(G), where the homogeneous linear forms Lijl are as in (12) and where aijl (n) ∈ Qn /pQn . As the coefficients aijl (0) played no role in steps 4) to 6), we can repeat the arguments of these three steps. First we get that the system (16) defines an étale, affine scheme Sn+1 over S0n := Spec(Qn /pQn ). Second we get that the maximal open closed subscheme S0n+1 of Sn+1 such that each connected component of it intersects the pull back of Zk through the natural morphism ℓ(n) : Sn+1 → S0 , is a connected scheme. Third we get that ℓ(n)−1 (Zk ) = Spec(k) and that Lemma 3.1.3 holds for n + 1. Fourth we get the existence ∧ of an étale Qn -algebra Qn+1 such that the special fibre of ℓ(n) : Spec(Q∧ n+1 ) → Spec(Qn ) is the étale, affine morphism S0n+1 → S0n . Fifth, as in 6) we use ∇R̂ to get that the unique connection ∇n+1 on MQn+1 /pn+1 MQn+1 is integrable, nilpotent modulo p, and respects the G-action. This repetition takes care of Lemma 3.1.3 and Theorem 3.2 (a) for n + 1 and thus ends our induction. This ends the proofs of Lemma 3.1.3 and Theorem 3.2 (a). 9) Proofs of 3.2 (b) to (d). We know that ℓ(n) modulo p is étale. As S0n+1 is a closed subscheme of a scheme of finite type over S0n , ℓ(n) modulo p is of finite type. The 2 fact that the geometric fibres of ℓ(n) modulo p have at most pd geometric points is argued as in 6) (as the connection ∇n+1 respects the G-action). Thus Theorem 3.2 (b) holds. Based on 8), Theorem 3.2 (c) is implied by Theorem 2.4.1 (d) and (e) applied to the sequence of étale morphisms · · · → S0n → S0n−1 → · · · → S01 → S0 that are obtained by taking certain connected components of Artin–Schreier systems of equations that involve the same linear forms Lijl ’s. From Theorem 2.4.1 (c) (see also the proof of Theorem 2.4.1 (d)), we get that there exists an open, Zariski dense subscheme Wk of Yk over which all special fibres of ℓn ’s are étale covers and which depends only on the coefficients of Lijl ’s and thus only on dΦR /p modulo p. Obviously, Wk is contained in the image Yk0 of Spec(Q/pQ) in Yk . The choice of another morphism h : Spec(W (k)) → Y corresponds to −1 p a replacement of ΦR by another Frobenius lift Φ′R of R that takes u−1 l zl to (ul zl ) , where σ(u )u−p −1 l l ∈ W (k). Thus dΦR /p ul ∈ G(W (k)). Thus Φ′R (zl ) = zlp + pvl zlp , where vl := p ′ ′ and dΦR /p coincide modulo p. Therefore B(φ) := Gk \ Wk does not depend on the choice 26 of h, has dimension at most d − 1 = dim(Y ) − 1, and its complement in Yk is contained in Wk and thus also in Yk0 . Thus B(φ) has all the desired properties i.e., Theorem 3.2 (d) holds. This ends the proofs of Lemma 3.1.3 and Theorem 3.2. 3.4. Geometric translation of 3.2. We translate Theorem 3.2 in terms of p-divisible groups. The Spec(k)-scheme Spec(Q/pQ) is geometrically connected and there exists a unique section a0 : Spec(W (k)) ֒→ Spec(Q) such that the composite of ℓ ◦ a0 with the natural morphism Y ∧ → G′ is the identity section a : Spec(W (k)) → G′ , cf. the étaleness and the (i) parts of Theorem 3.2 (a). Let ∇∞ be the connection on M ⊗W (k) Q which modulo pn is the natural extension of ∇n ; this makes sense due to (iii) of Theorem 3.2 (a). The reduction modulo pn of ∇∞ respects the G-action. This implies that ∇∞ annihilates tα ∈ T(M ⊗W (k) Q[ p1 ]) for all α ∈ J. As Q/pQ is an ind-étale k[z1 , . . . , zd ]-algebra, the set of the reductions modulo p of zl ’s with l ∈ S(G), is a finite p-basis for Q/pQ in the sense of [BM, Def. 1.1.1]. 3.4.1. Theorem. If p = 2, we assume that the property (C) of Definition 1.1.1 holds. (a) There exists a unique p-divisible group DQ/pQ over Spec(Q/pQ) such that the evaluation of D(DQ/pQ ) at the thickening Q := (Spec(Q/pQ) ֒→ Spec(Q), δ(p)) is the triple (M ⊗W (k) Q, Φ(Q)0 , ∇∞ ). (b) There exists a unique p-divisible group D over Spec(Q) such that the evaluation of the filtered Dieudonné crystal of D at the thickening Q is the quadruple (M ⊗W (k) Q, F 1 ⊗W (k) Q, Φ(Q)0 , ∇∞ ). Proof: We first prove (a) and (b) for p > 2. For p > 2 there exists a unique finite, flat, commutative group scheme Dn over Spec(Q∧ n ) of order a power of p and such that ∇ the object D(Dn ) of MF[0,1] (Qn ) is defined naturally by the pair (MQn /pn MQn , ∇n ), cf. Theorem 2.3.4. Due to the uniqueness of ∇n (cf. (ii) of Theorem 3.2 (a)), we can identify D((ℓ(n) )∗ (Dn )) = D(Dn+1 [pn ]). Thus we can also identify (ℓ(n) )∗ (Dn ) = Dn+1 [pn ], cf. Theorem 2.3.4. Therefore there exists a unique p-divisible group D over Spec(Q) such that for all n ∈ N we have D[pn ] = Dn Q = Dn+1 [pn ]Q . The evaluation at the thickening Q of the filtered Dieudonné crystal of D is (M ⊗W (k) Q, F 1 ⊗W (k) Q, Φ(Q)0 , ∇∞ ) (we recall from Section 2 that we disregard the Verschiebung maps of such evaluations). As Q/pQ has a finite p-basis, from [BM, Prop. 1.3.3] we get that each (filtered) F -crystal on CRIS(Spec(Q/pQ)/Spec(W (k))) is uniquely determined by its evaluation at the thickening Q. Thus DQ/pQ is uniquely determined by (M ⊗W (k) Q, Φ(Q)0 , ∇∞ ), cf. [BM, Thm. 4.1.1]. Thus (a) holds for p > 2 and moreover from Grothendieck–Messing deformation theory we also get that (b) holds for p > 2. Next we include two extra ways of proving (a) that work for all primes p ≥ 2. The first way does not assume that the property (C) holds for p = 2 and it goes as follows. We work with an arbitrary prime p ≥ 2. The existence and the uniqueness of DQ/pQ can be deduced from [dJ, Main Thm. 1]. Strictly speaking, loc. cit. is stated in a way that applies only to smooth k-algebras. But as the field KQ/pQ has a finite p-basis, loc. cit. applies to show that DKQ/pQ exists and is unique. Descent and extension arguments as in [dJ, Subsect. 4.4] show that DQ/pQ itself exists and it is unique. 27 We describe with full details the second way (as it is the simplest). Again we work with an arbitrary prime p ≥ 2 but if p = 2 we assume that the property (C) holds. Let A be the set of points of Spec(Q/pQ) whose residue fields are algebraic extensions of k. Let Q0 be the localization of Q with respect to a point p0 ∈ A. The residue field k0 of p0 is an algebraic extension of k and thus it is a perfect field. The ring S0 := Q0 /pQ0 has a finite p-basis as Q/pQ does. Let Qh0 and Q̂0 be the henselization and the completion of Q0 (respectively). Thus S0h := Qh0 /pQh0 is the henselization of S0 (this follows easily from [BLR, Ch. 2, Sect. 2.3, Prop. 4]). Let Ŝ0 := Q̂0 /pQ̂0 . Let x1 , . . . , xd ∈ Q̂0 be such that we can identify Q̂0 = W (k0 )[[x1 , . . . , xd ]]. Let Φ̃Q̂0 be the Frobenius lift of Q̂0 that is compatible with σk0 and that takes xl to xpl for all l ∈ S(G). Let (M ⊗W (k) Q̂0 , F 1 ⊗W (k) Q̂0 , Φ0 , ∇0 ) be the extension of (M ⊗W (k) Q, F 1 ⊗W (k) Q, Φ(Q)0 , ∇∞ ) via Q ֒→ Q̂0 but with Φ0 as a Φ̃Q̂0 -linear map. As in Subsection 3.3 1) we argue that the connection ∇0 on M ⊗W (k) Q̂0 is uniquely determined by the equality ∇0 ◦ Φ0 = (Φ0 ⊗ dΦ0 ) ◦ ∇0 . We recall what Φ0 is. ∼ The Q̂0 -linear isomorphism (M + p1 F 1 ) ⊗W (k) σ Q̂0 → M ⊗W (k) Q̂0 defined naturally by Φ0 is the composite of a correction Q̂0 -linear automorphism 1 1 ∼ K : (M + F 1 ) ⊗W (k) σ Q̂0 → (M + F 1 ) ⊗W (k) σ Q̂0 p p ∼ M ⊗W (k) Q̂0 defined by Φ(Q̂0 )0 and of the Q̂0 -linear isomorphism (M + p1 F 1 ) ⊗W (k) σ Q̂0 → (see Subsubsection 3.1.2 (a) for Φ(Q̂0 )0 ). For l ∈ S(G) let sl := Φ̃Q̂0 (zl ) − zlp ∈ pQ̂0 . For x ∈ M + p1 F 1 we have (cf. [De1, Formula (1.1.3.4)]) K(x ⊗ 1) := σ Q̂ X ( d Y i1 ,i2 ,... ,id ∈N∪{0} l=1 d ∇∞ ( Y sil d il l ) )(x ⊗ 1) . dzl il ! l=1 As ∇∞ respects the G-action, K fixes each tα viewed as a tensor of T(M + p1 F 1 )⊗W (k) 0 1 [ p ]. Thus the reduction of (M ⊗W (k) Q̂0 , F 1 ⊗W (k) Q̂0 , Φ0 , (tα )α∈J ) modulo the ideal (x1 , . . . , xd ) of Q̂0 , is a quadruple of the form (M ⊗W (k) W (k0 ), F 1 ⊗W (k) W (k0 ), h̃(φ ⊗ σk0 ), (tα )α∈J ) for some element h̃ ∈ GL M (W (k0 )) that fixes tα for all α ∈ J. We have h̃ ∈ G(W (k0 )). Thus (even for p = 2) the triple (M ⊗W (k) W (k0 ), F 1 ⊗W (k) W (k0 ), h̃(φ⊗σk0 )) is the filtered Dieudonné module of a uniquely determined p-divisible group D(h̃) over Spec(W (k0 )) (for p = 2, cf. Proposition 2.2.4 and the fact that the property (C) holds). There exists a unique p-divisible group DQ̂0 over Spec(Q̂0 ) that lifts D(h̃) and such that the evaluation of its filtered Dieudonné crystal at the thickening (Spec(Q̂0 /pQ̂0 ) ֒→ Spec(Q̂0 ), δ(p)) is (M ⊗W (k) Q̂0 , F 1 ⊗W (k) Q̂0 , Φ0 , ∇0 ). The argument for this goes as follows. The existence of DQ̂0 is implied by [Fa2, Thm. 10]. The uniqueness of DQ̂0 ×Spec(Q̂0 ) 28 Spec(Q̂0 /pQ̂0 ) is implied by [BM, Prop. 1.3.3 and Thm. 4.1.1]. Thus the uniqueness of DQ̂0 itself follows from Grothendieck–Messing deformation theory and the fact that the natural divided power structure of the ideal p(x1 , . . . , xd ) of Q̂0 is nilpotent modulo p(x1 , . . . , xd )q for all q ∈ N. We recall (cf. beginning of Section 1) that DŜ0 := DQ̂0 ×Spec(Q̂0 ) Spec(Ŝ0 ). In the next two paragraphs we will use descent in order to show that DŜ0 is defined over Spec(S0h ). Q Q (p) (p) The Ŝ0 ⊗S0h Ŝ0 -module Ŝ0 ⊗S h (p) Ŝ0 is free and has l∈S(G) zlnl ⊗ l∈S(G) zlml as a 0 Ŝ0 ⊗S0h Ŝ0 -basis, where nl , ml ∈ {0, . . . , p − 1} for all l ∈ S(G). Thus Ŝ0 ⊗S0h Ŝ0 has a finite p-basis with d2 elements. The normalization of S0h in Ŝ0 is S0h itself and thus Ŝ0 ⊗S0h Ŝ0 is an integral domain. We check that Ŝ0 ⊗S0h Ŝ0 is normal. We write Ŝ0 as an inductive limit Ŝ0 = lim.ind.δ∈I S0δ of normal Ŝ0h -algebras of finite type indexed by the set of objects I of a filtered category. Thus Ŝ0 ⊗S0h Ŝ0 = lim.ind.δ∈I S0δ ⊗S0h Ŝ0 . As R/pR is an excellent ring (see [Ma, Ch. 13, Sect. 34]), the homomorphism S0δ ⊗S0h Ŝ0 → S0δ is regular. Thus the scheme S0δ ⊗S0h Ŝ0 is normal, cf. [Ma, Ch. 13, Sect. 33, Lemmas 2 and 4]. Therefore the scheme Ŝ0 ⊗S0h Ŝ0 is normal. Let s1 , s2 : Spec(Ŝ0 ⊗S0h Ŝ0 ) → Spec(Ŝ0 ) be the two natural projection morphisms. Both D(s∗1 (DŜ0 )) and D(s∗2 (DŜ0 )) are defined naturally by the triple (M ⊗W (k) Q, Φ(Q)0 , ∇∞ ) and therefore we have a canonical identification D(s∗1 (DŜ0 )) = D(s∗2 (DŜ0 )). ∼ ∗ Let θ : s∗1 (DŜ0 ) → s2 (DŜ0 ) be the unique isomorphism such that D(θ) is this identification ∗ D(s1 (DŜ0 )) = D(s∗2 (DŜ0 )), cf. [BM, Thm. 4.1.1]. The local rings of Ŝ0 ⊗S0h Ŝ0 ⊗S0h Ŝ0 are normal and have finite p-bases (this is argued as for Ŝ0 ⊗S0h Ŝ0 ). Thus based on loc. cit. we get naturally a descent datum on DŜ0 with respect to the faithfully flat morphism Spec(Ŝ0 ) → Spec(S0h ) (i.e., θ satisfies the cocycle condition s∗23 (θ) ◦ s∗12 (θ) = s∗13 (θ)). Thus standard descent of coherent sheaves of modules (see [BLR, Ch. 6, Sect. 6.1, Thm. 4]) applied to the finite Spec(Ŝ0 )-scheme DŜ0 [pn ] and to the evaluation of D(DŜ0 [pn ]) at the thickening (Spec(Ŝ0 ) ֒→ Spec(Q̂0Wn (k) ), δ(p)), shows that DŜ0 is the pull back of a pdivisible group DS0h over Spec(S0h ) whose Dieudonné crystal is uniquely determined by its h∧ h 0 h evaluation (M ⊗W (k) Qh∧ 0 , Φ(Q0 ) , ∇∞ ) at the thickening (Spec(S0 ) ֒→ Spec(Q0 ), δ(p)); here and below we denote also by ∇∞ its natural extensions. Repeating the arguments but this time using descent in the context of the pro-étale morphism Spec(S0h ) → Spec(S0 ), we get that DS0h is also the pull back of a p-divisible group DS0 over Spec(S0 ) whose Dieudonné crystal is uniquely determined by its evaluation 0 ∧ (M ⊗W (k) Q∧ 0 , Φ(Q0 ) , ∇∞ ) at the thickening (Spec(S0 ) ֒→ Spec(Q0 ), δ(p)). Each point of Spec(Q/pQ) specializes to such a point p0 ∈ A, cf. Theorem 3.2 (c). Based on [BM, Thm. 4.1.1] we get that the DQ0 /pQ0 ’s glue together to define a p-divisible group DQ/pQ over Spec(Q/pQ) whose Dieudonné crystal is uniquely determined by its evaluation (M ⊗W (k) Q, Φ(Q)0 , ∇∞ ) at the thickening Q. Thus (a) holds. We are left to prove (b) in the case when p = 2 and the property (C) holds. It suffices to prove the existence and the uniqueness of the lift DQ/p2 Q of DQ/pQ to Spec(Q/p2 Q). We fix a lift D′Q/p2 Q of DQ/pQ to Spec(Q/p2 Q), cf. [Il, Thm. 4.4 a) and f)]. Let δ(p)tr 29 be the trivial divided power structure of the ideal (p) of Q/p2 Q defined by the identities (p)[s] = 0, s ∈ N \ {1}. We recall F̃ −1 (EndW (k) (M )) is the maximal direct summand of EndW (k) (M ) on which Gm acts via µ as the identity character of Gm , cf. Subsection 2.7. Let Lcrys-lift (resp. Llift ) be the free Q/pQ-module of lifts of F 1 ⊗W (k) Q/pQ to direct summands of M ⊗W (k) Q/p2 Q, the zero element corresponding to the Hodge filtration defined by D′Q/p2 Q and by the divided power structure δ(p) (resp. δ(p)tr ) of the ideal (p) of Q/p2 Q. The Q/pQ-module structure of Lcrys-lift (resp. of Llift ) is defined naturally by identifying Lcrys-lift (resp. Llift ) with the set of images of the lift of M ⊗W (k) Q/p2 Q that defines the zero element of Lcrys-lift (resp. of Llift ) through elements of the form 1M ⊗W (k) Q/p2 Q + pu ∈ GL M (Q/p2 Q), where u ∈ F̃ −1 (EndW (k) (M )) ⊗W (k) Q/p2 Q. Let L ∈ Lcrys-lift be such that it corresponds to F 1 ⊗W (k) Q/p2 Q. We define a natural map of sets MQ/pQ : Llift → Lcrys-lift as follows. Let x ∈ Llift and let DxQ/p2 Q be the lift of DQ/pQ defined by x and Grothendieck– Messing deformation theory (the divided power structure of the ideal (p) of Q/p2 Q being δ(p)tr ). We define MQ/pQ (x) to be the Hodge filtration of M ⊗W (k) Q/p2 Q defined by DxQ/p2 Q using the divided power structure δ(p) of the ideal (p) of Q/p2 Q. The map MQ/pQ has a functorial aspect with respect to pulls back of DQ/pQ . Let T : Spec(W (k̃)) → Spec(Q̂0 ) be a Teichmüller lift whose special fibre is dominant. Here k̃ is a big enough perfect field that contains the field of fractions KŜ0 . The map Mk̃ which is the analogue of MQ/pQ but obtained using lifts of Dk̃ to W2 (k̃), is injective (see proof of Proposition 2.2.4). Thus MQ/pQ is injective. Thus, if DQ/p2 Q exists, then it is unique. To DQ̂0 ×Spec(Q̂0 ) Spec(Q̂0 /p2 Q̂0 ) corresponds an element L0 ∈ Llift ⊗Q/pQ Ŝ0 . The images of L0 in Llift ⊗Q/pQ S0 ⊗S0 Ŝ0 ⊗S0h Ŝ0 via s1 and s2 are equal. Thus we have L0 ∈ Llift ⊗Q/pQ S0h , cf. [BLR, Ch. 6, Sect. 6.1, Lemma 2]. Repeating the arguments in the context of the pro-étale morphism Spec(S0h ) → Spec(S0 ), we get that L0 ∈ Llift ⊗Q/pQ S0 . From the injectivity of MQ/pQ and of its analogue “localization” MS0 (thought to be MQ/pQ ⊗ 1S0 ) and from the fact that MS0 (L0 ) = L, we get that L0 ∈ Llift ⊗Q/pQ ∩p0 ∈A S0 ; here we use quotations for “localization” as we will not stop to check that the map MQ/pQ is indeed Q/pQ-linear. But Theorem 3.2 (d) implies that Q/pQ = ∩p0 ∈A S0 . Thus L0 ∈ Llift . The element MQ/pQ (L0 ) − L ∈ Lcrys-lift is 0 as this is so after tensorization with Ŝ0 over Q/pQ. Thus MQ/pQ (L0 ) = L i.e., DQ/p2 Q exists. This ends the proof of (b). 3.4.2. Étale Tate-cycles. If p = 2 we continue to assume that the property (C) holds. Let the p-divisible group D be as in Theorem 3.4.1 (b). We consider the B + (Q)-linear monomorphism iD : M ⊗W (k) B + (Q) = M ⊗W (k) Q ⊗Q iQ B + (Q) → H 1 (DKQ ) ⊗Zp B + (Q) 30 constructed as (3), the Frobenius endomorphism of M ⊗W (k) B + (Q) = M ⊗W (k) Q ⊗Q + 0 + iQ B (Q) being Φ(Q) ⊗ ΦQ/pQ (here iQ : Q ֒→ B (Q) and ΦQ/pQ are as in Subsection 2.3 and Subsubsection 2.3.5 but for Q; thus the W (k)-monomorphism iQ is compatible with the Frobenius lifts ΦQ and ΦQ/pQ ). Let Vα ∈ T(H 1 (DKQ )[ p1 ]) ⊗Qp B + (Q)[ β10 ] correspond to tα via the B + (Q)[ β10 ]-linear isomorphism iD [ β10 ]. We check that Vα ∈ T(H 1 (DKQ )[ p1 ]). Let T : Spec(W (k̃)) → Spec(Q̂0 ) be as in the last paragraph of the proof of Theorem 3.4.1. We denote by T1 the W (k)-homomorphism B + (Q) → B + (W (k̃)) defined naturally by T and the choice of a W (k)-homomorphism T0 : Q̄ ⊗Q W (k̃) → W (k̃) (see Subsection 2.3 and the end of Subsection 2.1 for Q̄ and W (k̃)). As T is dominant, the following restriction T00 : Q̄ → W (k̃) of T0 is injective. Based on Lemma 2.3.1 (a) and (b) we easily get that the W (k)-homomorphism W (AQ/pQ ) → W (Ak̃ ) defined naturally by T00 is also injective. This implies that the W (k)-homomorphism T1 [ β10 ] : B + (Q)[ β10 ] → B + (W (k̃))[ β10 ] is also injective. The image of Vα via T1 [ β10 ] is a tensor of T(H 1 (DB(k̃) )[ p1 ]), cf. end of Subsubsection 2.2.2. As we have a canonical identification H 1 (DKQ )[ p1 ] = H 1 (DB(k̃) )[ p1 ] of Zp -modules, the relation Vα ∈ T(H 1 (DKQ )[ p1 ]) is implied by the injectivity of T1 [ β10 ]. The tensor Vα ∈ T(H 1 (DKQ )[ p1 ]) is fixed by Gal(KQ̄ /KQ ) (as tα is so) and thus it is an étale Tate-cycle of D (more precisely, of DQ[ p1 ] ). The pull back of (D, (tα )α∈J ) (resp. of (D, (Vα )α∈J )) via a0 is (D, (tα )α∈J ) (resp. is (D, (vα )α∈J )); this is so as a0 : Spec(W (k)) → Spec(Q) is a Teichmüller lift with respect to ΦQ . 3.4.3. Lemma. If p = 2 we assume that the property (C) holds. To prove the Main Theorem for (D, (tα )α∈J ) it is enough to prove the Main Theorem for an arbitrary pull back of (D, (tα )α∈J ) via a W (k)-valued point of Spec(Q). Proof: All pulls back of (D, (Vα )α∈J ) via B(k)-valued points of Spec(Q) are isomorphic to (H 1 (D), (vα )α∈J ). Moreover each pull back of (D, (tα )α∈J ) via a W (k)-valued point of Spec(Q) is of the form (D(h̃), (tα )α∈J ), where the p-divisible group D(h̃) over Spec(W (k)) has a filtered Dieudonné module of the form (M, F 1 , h̃φ) for some element h̃ ∈ G(W (k)) (see the paragraph of the proof of Theorem 3.4.1 that pertains to D(h̃), applied with k0 = k). The Lemma follows from the last two sentences. 3.4.4. Remarks. (a) Suppose that p = 2 and G is smooth. One can use Artin approximation theory to show that D always exists, provided modulo p we work in the étale topology of Spec(Q/pQ) and we allow changes in the filtration F 1 ⊗W (k) Q of M ⊗W (k) Q (i.e., we replace the cocharacter µQ : Gm → GQ by a Ker(G(Q) → G(Q/pQ))-conjugate of it). In general, D is not unique and (unfortunately) there exists nothing to guarantee that we can work with a single connected étale cover of Spec(Q/pQ). (b) Suppose that p = 2 and that there exists no element h ∈ G(W (k̄))0 such that (M ⊗W (k) W (k̄), h(φ⊗σk̄ )) has both Newton polygon slopes 0 and 1 with positive multiplicities. We refer to the element h̃ ∈ G(W (k0 )) of the proof of Theorem 3.4.1. Considering a Teichmüller lift Spec(W (k0 )) → Spec(Q) that factors through Spec(Q̂0 ), we get that (M ⊗W (k) W (k0 ), h̃(φ ⊗ σk0 )) is isomorphic to (M ⊗W (k) W (k0 ), h(φ ⊗ σk0 )), for some h ∈ G(W (k̄))0 . Thus Subsubsections 3.4.1 to 3.4.3 continue to hold in this case (the same proof applies). 31 (c) If p is arbitrary and the property (C) holds, then one can check that Q = R∧ . 3.5. Application. Until Section 4 we assume that k = k̄. We recall that G′0 k is the identity ′ 0 component of Gk (see Subsection 2.6). Let g ∈ G(W (k)) ; thus g modulo p belongs to ∧ G′0 → Y ∧ , Qg , MQg , Dg , and B(gφ) ⊆ Yk be the analogues k (k). Let bg : Y → O, ΦgR : Y of b : Y → O, ΦR : Y ∧ → Y ∧ , Q, MQ , D, and B(φ) ⊆ Yk but obtained working with gφ instead of with φ. We take h ∈ Y (W (k)) ∩ Y (W (k))g such that modulo p does not define a k-valued point of B(gφ) ∪ B(φ) and the composites of h ◦ b and hg −1 ◦ bg are defined at the level of rings by homomorphisms W (k)[z1 , . . . , zd ] → W (k) and W (k)[z1 , . . . , zd ] → W (k) that map each zl to Gm (W (k)); here l ∈ S(G). This makes sense as g modulo p belongs to G′0 k (k) and as B(gφ) ∪ B(φ) does not depend on h (cf. Theorem 3.2 (d)). As in Subsection 3.1 we can assume that in fact both these two homomorphisms W (k)[z1 , . . . , zd ] → W (k) are such that they map each zl to 1 (cf. also Theorem 3.2 (d)). Let ah : Spec(W (k)) → Spec(Q) (resp. ahg −1 : Spec(W (k)) → Spec(Qg )) be a morphism that lifts the W (k)-valued point h (resp. hg −1 ) of Y ; it is a Teichmüller lift with respect to the Frobenius lift ΦQ (resp. ΦQg ) due to the fact that ΦQ (resp. ΦQg ) preserves the ideal of Q (resp. of Qg ) generated by (z1 − 1, . . . , zd − 1). Thus the pull back of (MQ , (tα )α∈J ) (resp. (MQg , (tα )α∈J )) via ah (resp. ahg −1 ) is (M, F 1 , hφ, hφ1 , (tα )α∈J ), . This achieves the connecwhere the σ-linear map φ1 : F 1 → M takes x ∈ F 1 to φ(x) p tion of each (M, F 1 , φ, G, (tα )α∈J ) and (M, F 1 , gφ, G, (tα)α∈J ) with the same quintuple (M, F 1 , hφ, G, (tα )α∈J ). ′0 Let Tk be a maximal torus of G′0 k through which µk : Gm → Gk factors (see Subsec′ tion 2.6). Let T be a maximal torus of G that lifts Tk , cf. [DG, Exp. IX, Thms. 3.6 and 7.1]. Up to G′ (W (k))-conjugation, we can assume that µ : Gm → G′ factors through T (cf. loc. cit). As the T -module M is a direct sum of T -modules of rank 1, the group scheme Ker(T → GL M ) is of multiplicative type and thus flat over Spec(W (k)). The generic fibre of Ker(T → GL M ) is trivial. From the last two sentences we get that Ker(T → GL M ) is a trivial group scheme over Spec(W (k)). Thus we can also identify T with a torus of GL M that is a maximal torus of G. Let σφ := φµ(p); it is a σ-linear automorphism of M . The Lie algebra φ(Lie(T )) = σφ (Lie(T )) is the Lie algebra of a maximal torus σφ (T ) of G and thus also of G′ , cf. −1 property 2.6 (iii). Let ḡ0 ∈ G′0 k (k) be such that ḡ0 σφ (T )k ḡ0 = Tk , cf. [Bo, Ch. V, Thm. ′ 15.14]. Let g0 ∈ G (W (k)) be such that it lifts ḡ0 and we have g0 σφ (T )g0−1 = T , cf. [DG, Exp. IX, Thm. 3.6]. Until the end of this Section, we will take g := g0 . The triple (M, gφ, T ) is such that µ factors through T and gφ(Lie(T )) = Lie(T ). We consider a set JT that contains J and a family of tensors (tα )α∈JT of F 0 (T(M ))[ p1 ] that extends (tα )α∈J and that has the following two properties (cf. Lemma 2.5.3): (i) gφ fixes each tα with α ∈ JT , and (ii) TB(k) is the subgroup of GL M [ p1 ] that fixes tα for all α ∈ JT . 3.5.1. Theorem. It suffices to prove the Main Theorem under the extra hypothesis that G is a torus. Proof: We first prove this Theorem in the case when either p > 2 or p = 2 and the property (C) holds. For h̃ ∈ G′ (W (k)) let D(h̃) be the p-divisible group over Spec(W (k)) whose filtered Dieudonné module is (M, F 1 , h̃φ), cf. Proposition 2.2.4. If p = 2, then the property 32 (C) also holds for either (M, gφ, T ) or (M, hφ, G). The pull back of (D, (tα )α∈J ) (resp. of its analogue (Dg , (tα )α∈J ) over Spec(Qg )) through ah (resp. ahg −1 ) is (D(h), (tα )α∈J ). Thus to prove the Main Theorem for (D, (tα )α∈J ) (resp. (D(g), (tα)α∈J )) is equivalent to proving the Main Theorem for (D(h), (tα )α∈J ), cf. Lemma 3.4.3. Thus the Main Theorem holds for (D, (tα )α∈J ) if and only if the Main Theorem holds for (D(g), (tα)α∈J ). If the Main Theorem holds for (D(g), (tα)α∈JT ), then the Main Theorem also holds for (D(g), (tα)α∈J ). As TB(k) is the subgroup of GL M [ p1 ] that fixes tα for all α ∈ JT , to prove the Main Theorem in this case we can assume that G = T . ´ We are left to prove the Theorem in the case when p = 2 and Get Zp is a torus. ´ be the semisimple Zp -subalgebra of It is enough to show that G is a torus. Let Eet ´ ´ 1 et EndZp (H (D)) formed by all endomorphisms fixed by Get is a semisimple Zp . Thus E ´ Zp -subalgebra of End(Dt ) and by functoriality we can identify Eet with a semisimple Zp ´ 1 et algebra E of endomorphisms of (M, F , φ). The centralizer of E in GL H 1 (D) is a torus ´ ´ ´ 1 et et Get 1Zp that contains GZp . As iD [ β0 ] is an isomorphism, the group schemes G and GZp are isomorphic over Spec(B + (W (k))[ β10 ]); thus GB(k) is a torus. The centralizer C of the semisimple W (k)-algebra E ⊗Zp W (k) in GL M is a product of general linear group schemes ´ whose extension to Spec(B + (W (k))[ β10 ]) is isomorphic to Get . Thus C is a split 1B + (W (k))[ 1 ] β0 torus. As GB(k) is a subtorus of CB(k) , G is a subtorus of C. Thus G is a torus. 4. Proof of the Main Theorem In Subsections 4.1 and 4.2 we prove the Main Theorem. See Subsection 4.3 for an example that illustrates the computations of Subsections 4.1 to 4.2. See Subsection 4.4 for the proof of Corollary 1.4. Subsection 4.5 contains two remarks. We will use the notations of Subsubsections 2.1, 2.2.1, and 2.2.2. Until Section 5 we assume that k = k̄. 4.1. Notations and simple properties. We start the proof of the Main Theorem. To prove the Main Theorem we can assume that G is a torus T , cf. Theorem 3.5.1. Let µ : Gm → T be as in Subsection 2.1 (it is easy to see that µ = µ−1 can ). Let σφ , MZp , and ´ ´ et et GZp be as before Lemma 2.5.2. Let TZp := GZp and let TZp := GZp . The hypothesis of the ´ Main Theorem for p = 2 says: if p = 2, then either the group scheme TZet over Spec(Zp ) is p t a torus or at least one of the two p-divisible groups Dk and Dk is connected. Let T0Zp be the smallest subtorus of TZp such that µ : Gm → T factors through T0 := T0W (k) . Thus T0 is the subtorus of T generated by the images of the conjugates of µ under integral powers ´ of φ. Let E (resp. Eet ) be the Zp -subalgebra of endomorphisms of MZp (resp. of H 1 (D) ´ and thus also of Dt ) fixed by TZp (resp. by TZetp ). As TZp is a torus, E is a semisimple Zp -algebra. Let T1Zp be the double centralizer of TZp in GL MZp (i.e., the centralizer of E in GL MZp ). Thus T1 := T1W (k) is the double centralizer of T in GL M . Both T1Zp and T1 are tori. Let T2Zp be a maximal torus of GL MZp that contains T1Zp . Thus T2 := T2W (k) is a maximal torus of GL M that contains T1 and such that we have φ(Lie(T2 )) = Lie(T2 ). Let n and m be the ranks of F 0 and F 1 (respectively). 33 Let B := {a1 , . . . , an+m } be a W (k)-basis for M = F 1 ⊕ F 0 such that we have inclusions {a1 , . . . , an } ⊆ F 0 and {an+1 , . . . , an+m } ⊆ F 1 and moreover T2 normalizes W (k)x for all x ∈ B. For i ∈ {1, . . . , n} let εi := 0. For i ∈ {n + 1, . . . , n + m} let εi := 1. As φ(Lie(T2 )) = Lie(T2 ), there exists a permutation π of the set {1, . . . , n + m} such that we have φ(ai ) ∈ Gm (W (k))pεi aπ(i) for all i ∈ {1, . . . , n + m}. By replacing each ai with a suitable Gm (W (k))-multiple of it, we can assume that we have φ(ai ) = pεi aπ(i) ∀i ∈ {1, . . . , n + m}. Let φ1 : F 1 → M be the σ-linear map such that for all i ∈ {n + 1, . . . , n + m} we have φ1 (ai ) := aπ(i) . The reduction of (M, F 1 , φ, φ1 ) modulo p is D(D[p]) of Subsubsection 2.3.3. ´ ´ 4.1.1. Lemma. We have Eet = E and TZet is a torus. p ´ Proof: Each element of E (resp. of Eet ) when viewed as a tensor of T(M [ p1 ]) (resp. of T(H 1 (D))) is fixed by both φ and µ (resp. by Gal(B(k))). The functorial aspect of ´ 1 ´ Fontaine comparison theory allows us to identify Eet [ p ] = E[ p1 ] in such a way that Eet is a Zp -subalgebra of E. If p > 2 or if p = 2 and either Dk or Dkt is connected, then ´ ´ 1 we have Eet = Eet [ p ] ∩ End(H 1 (D)) = E[ p1 ] ∩ End((M, F 1 , φ)) = E (cf. Lemma 2.2.3). ´ ´ If TZet is a torus, then Eet is a semisimple Zp -algebra and therefore a maximal order p ´ 1 ´ ´ ´ = E. Therefore we always have Eet = E and thus Eet is a of Eet [ p ] = E[ p1 ]; thus Eet semisimple Zp -algebra. An argument similar to the one of the proof of Theorem 3.5.1 ´ shows that TZet ×Spec(Zp ) Spec(W (k)) is a subtorus of the split torus that is the centralizer p ´ ´ of Eet ⊗Zp W (k) in GL H 1 (D)⊗Zp W (k) . Thus TZet is a torus. p Q 4.1.2. Decomposing D. Let π = l∈C(π) πl be the decomposition of π into cycles. If l ∈ C(π) and if πl = (i1 , . . . , iq ) with q ∈ N, let Ml := ⊕qs=1 W (k)ais . We get a direct sum decomposition (M, F 1 , φ) = ⊕l∈C(π) (Ml , Ml ∩ F 1 , φ). The projection of M on ´ . Thus the direct sum decomposition Ml along ⊕l′ ∈C(π)\{l} Ml′ is an element of E = Eet Q 1 1 (M, F , φ) = ⊕l∈C(π) (Ml , Ml ∩ F , φ) defines a product decomposition D = l∈C(π) Dl into p-divisible groups over Spec(W (k)) whose special fibres have a unique Newton polygon slope. Below we will often use this fact in order to reduce our computations to the simpler case when π is a cycle. 4.1.3. Lubin–Tate quadruples. We say that (M, F 1 , φ, T, (tα )α∈J ) is a Lubin–Tate quadruple if m = 1 and π is a cycle (equivalently, if the W (k)-module F 1 has rank 1 and 1 ). the F -isocrystal (M, φ) over k is simple of Newton polygon slope n+1 1 We check that if (M, F , φ, T, (tα )α∈J ) is a Lubin–Tate quadruple, then we have T0 = T1 = T = G = T2 . The cocharacter µ : Gm → GL M of (M, F 1 , φ) acts trivially on ai for i ∈ {2, . . . , n + m} and non-trivially on a1 . Thus T0 contains the rank 1 subtorus of T2 that fixes a2 , a3 , . . . , and an+m . But φ normalizes Lie(T0,B(k) ) (cf. the very definition of T0 ) and thus T0 contains the subgroup scheme of T2 generated by the images of the conjugates of µ under powers of φ. Therefore by induction on i ∈ {1, . . . , n + m} we get 34 that T0 contains the rank 1 subtorus of T2 that fixes a1+i , . . . , an+m , a1 , . . . , and ai−1 . Thus T2 6 T0 . Therefore we have T0 = T1 = T = G = T2 . 4.1.4. Theorem. There exists t ∈ T (B + (W (k))[ β10 ]) that takes the B + (W (k))-submodule (iD [ β10 ])−1 (H 1 (D) ⊗Zp B + (W (k))) of M ⊗W (k) B + (W (k))[ β10 ] onto M ⊗W (k) B + (W (k)). 4.2. Proofs of 1.2 and 4.1.4. To prove the Main Theorem is equivalent to proving Theorem 4.1.4, cf. Lemma 2.5.2 (b) and Fact 2.5.1 (b). We will prove the Main Theorem and Theorem 4.1.4 in the next eleven Subsubsections. 4.2.1. Lemma. Suppose that T = T1 (for instance, this holds if (M, F 1 , φ, T, (tα )α∈J ) is a Lubin–Tate quadruple). Then the Main Theorem and Theorem 4.1.4 hold. Proof: As T = T1 , T is the centralizer of E in GL M . Thus to prove the Main Theorem and ´ Theorem 4.1.4 we can assume that E = {tα |α ∈ J} and Eet = {vα |α ∈ J}, cf. Fact 2.5.1 ´ (a) and (b). Let A :=QEet ⊗Zp W (k) = E ⊗Zp W (k), cf. Lemma 4.1.1. The W (k)-algebra A is a finite product i∈I Mri (W (k)) of matrix W (k)-algebras (here each ri ∈ N). Each Q representation of i∈I Mri (W (k)) on a free W (k)-module of finite rank is a direct sum indexed by i ∈ I of a finite number fi of copies of the standard representation of Mri (W (k)) of rank ri . The representations of A on M and H 1 (D) ⊗Zp W (k) involve the same numbers fi , as the tensorizations of these representations with B + (W (k))[ β10 ] are isomorphic (cf. Subsubsection 2.2.2). Thus the representations of A on M and H 1 (D) ⊗Zp W (k) are ∼ isomorphic i.e., there exists an isomorphism ρ : (M, (tα )α∈J ) → (H 1 (D)⊗Zp W (k), (vα )α∈J ). Thus the Main Theorem and Theorem 4.1.4 hold. 4.2.2. The wi elements. Let t ∈ T1 (B + (W (k)) β10 ) be an element that takes the B + (W (k))-module (iD [ β10 ])−1 (H 1 (D)⊗Zp B + (W (k))) onto M ⊗W (k) B + (W (k)), cf. Lemma 4.2.1 . As iD is a B + (W (k))-monomorphism and as β0 annihilates Coker(iD ), there exist elements wi ∈ B + (W (k)) ∩ Gm (B + (W (k)) β10 ) such that we have t(ai ) = wi ai , ∀i ∈ {1, . . . , n + m}. Let qk : B + (W (k)) ։ V (k)/pV (k) be as in Subsubsection 2.2.2. To compute the qk (wi )’s (see Proposition 4.2.5 below) we need few extra preliminaries. In all that follows we assume that the different roots of p which will show up are powers of a fixed high order 1 root (like the (pm+n )!-th root) of p and that qk (ξ0 ) ∈ V (k)/pV (k) is p p modulo p. We consider the system S of equations Xip = (−p) επ(i) p Xπ(i) , i ∈ {1, . . . , n + m} in n + m-variables X1 , . . . , Xn+m over V (k). We fix a non-zero solution (Z1 , . . . , Zn+m ) of S. Let v : K(k) \ {0} → Q be the valuation of K(k) normalized by v(p) = 1. 4.2.3. Lemma. (a) If π is a cycle, then any other non-zero solution of S is of the form (17) ηn+m η2 (γZ1 , γ p Z2 , . . . , γ p 35 Zn+m ), where γ ∈ µpn+m −1 (W (k)) ⊆ V (k) and where for i ∈ {2, . . . , n + m} the number ηi ∈ {1, . . . , n + m − 1} is such that π ηi (1) = i. (b) For i ∈ {1, . . . , n+m} there exists a rational function Qi (x) ∈ Q(x) that depends 1 on π and n but not on p and such that we have v(Zi ) = Qi (p) ∈ [0, p(p−1) ] ∩ Q. (c) If (i1 , . . . , iq ) is a cycle of π, then we have Qi1 (p) = · · · = εiq = 1. 1 p(p−1) if and only if εi1 = Proof: Part (a) is trivial. To prove (b) and (c), let (i1 , . . . , iq ) be a cycle of π of length q and let iq+1 := i1 . Let P q−j−1 j∈{1,... ,q}, εij+1 =1 x . (18) Qi1 (x) := xq − 1 P q−j−1 q p ij+1 From the shape of S we get Zip1 = (−p) Zi1 . As Zi1 ∈ V (k) \ {0}, Pq pq−j−1 1 1 we get that v(Zi1 ) = Qi1 (p) ≤ j=1 pq −1 = p(p−1) . Thus Qi1 (p) ∈ [0, p(p−1) ] ∩ Q and P 1 therefore (b) holds. We have Qi1 (p) = p(p−1) if and only if j∈{1,... ,q}, εi =1 xq−j = j+1 Pq q−j i.e., if and only if εi1 = · · · = εiq = 1. Thus (c) also holds. j=1 x j∈{1,... ,q}, ε =1 4.2.4. Proposition. We recall that i∗D : Tp (DB(k) )⊗Zp B + (W (k)) ֒→ M ∗ ⊗W (k) B + (W (k)) is the dual of the B + (W (k))-linear monomorphism iD : M ⊗W (k) B + (W (k)) ֒→ H 1 (D)⊗Zp ∗ B + (W (k)) of (1). Let rD : M ֒→ H 1 (D) ⊗Zp B + (W (k)) and rD : Tp (DB(k) ) ֒→ M ∗ ⊗W (k) B + (W (k)) be the natural restrictions of iD and i∗D (respectively). Then the Fp -linear maps ∗ rD modulo p and rD modulo p are injective. 1 Proof: We recall that β0 annihilates Coker(iD ) and Coker(i∗D ) and that qk (β0 ) is p p−1 modulo p times a unit of V (k)/pV (k) (see Subsubsections 2.2.1 and 2.2.2). Thus if p > 2, ∗ then the Fp -linear maps rD modulo p and rD modulo p are injective. We check that rD modulo p is injective even for p = 2. To check this, we can assume π is a cycle (cf. Subsection 4.1.2). We show that the assumption that rD modulo p is not injective, leads to a contradiction. This assumption implies that there exists i ∈ {1, . . . , n + m} such that wi ∈ pB + (W (k)). By induction on j ∈ {0, . . . , n + m − 1} we check that wi+j ∈ pB + (W (k)) (here wn+m+s := ws for s ∈ {1, . . . , i − 1}). The case j = 0 is obvious and the passage from j to j + 1 goes as follows. If i + j ≤ n, then iD (ai+j+1 ⊗ 1) = (1H 1 (D) ⊗ Φk )(iD (ai+j ⊗ 1)) ∈ (1H 1 (D) ⊗ Φk )(H 1 (D) ⊗Zp pB + (W (k))) ⊆ H 1 (D) ⊗Zp pB + (W (k)). If i + j > n, then due to the fact that iD respects filtrations we have iD (ai+j+1 ⊗1) = (1H 1 (D) ⊗Φ1k )(iD (ai+j ⊗1)) ∈ (1H 1 (D) ⊗ Φ1k )(H 1 (D) ⊗Zp pF 1 (B + (W (k))) ⊆ H 1 (D) ⊗Zp pB + (W (k)). Thus regardless of what i + j is, we have wi+j+1 ∈ pB + (W (k)). This ends the induction. Thus rD modulo p is the 0 map. Referring to (2), we get that Im(jD ) ⊆ pH 1 (D) ⊗Zp 1 gr . As we also have pH 1 (D) ⊗Zp gr1 ⊆ Im(jD ) (see proof of Lemma 2.2.3) we get that Im(jD ) = pH 1 (D) ⊗Zp gr1 . Thus for each j ∈ {1, . . . , n}, the element bj ∈ H 1 (D) ⊗Zp B + (W (k)) defined by the equality iD (aj ⊗ 1) = pbj is such that its image in H 1 (D) ⊗Fp k is non-zero. Thus (1H 1 (D) ⊗ Φk )s (bj ) modulo p is non-zero for all s ∈ N. Taking s >> 0 m of (M, φ) is 0. Therefore D is étale. Thus iD we get that the Newton polygon slope m+n 36 is an isomorphism and therefore rD modulo p is injective. Contradiction. Thus rD modulo ∗ p is injective even if p = 2. This implies that rD modulo p is injective even if p = 2. 4.2.5. Proposition. Let (x1 , . . . , xn+m ) be the reduction modulo p of the solution (Z1 , . . . , Zn+m ) we fixed in Subsubsection 4.2.2. Then for each i ∈ {1, . . . , n + m} there exists vi ∈ Gm (V (k)/pV (k)) such that we have (19) qk (wi ) = qk (ξ0 )εi xi vi . Proof: To check Formula (19) we can assume that π is a cycle (i1 , . . . , im+n ), cf. Subsection m 4.1.2. Thus (M, φ) has only one Newton polygon slope m+n . We first check (19) in the 1 m (cf. Lemma case when n = 0. As n = 0, we have D = µp∞ and thus each Qi (p) is p(p−1) 1 + 4.2.3 (c)). We also have Im(iD ) = H (D) ⊗Zp β0 B (W (k)) (see Subsubsection 2.2.2) and thus each w1 , . . . , wm is a Gm (B + (W (k)))-multiple of β0 . But up to Gm (V (k)/pV (k))1 1 multiples, xi is p p(p−1) modulo p (cf. Lemma 4.2.3 (c)) and qk (β0 ) is p p−1 modulo p (see 1 1 1 Subsubsection 2.2.1). As p p−1 = p p p p(p−1) , we easily get that Formula (19) holds if n = 0. We prove Formula (19) in the case when n > 0. As n > 0 and as π is a cycle, we 1 have Qi (p) ∈ [0, p(p−1) ) for all i ∈ {1, . . . , n + m} (cf. Lemma 4.2.3 (b) and (c)). The rD modulo p and the extension <, > to B + (W (k))/pB +(W (k)) of the perfect pairing <, >: Tp (DB(k) )/pTp (DB(k) ) × H 1 (D)/pH 1 (D) → Fp define an Fp -linear map j1 : Tp (DB(k) )/pTp (DB(k) ) → Hom(D(D[p]), D(B +(W (k))/pB + (W (k)))) via the formula (j1 (x))(y) =< x, z >∈ B + (W (k))/pB + (W (k)), where y ∈ M/pM and x ∈ Tp (DB(k) )/pTp (DB(k) ) and where z ∈ H 1 (D)/pH 1 (D) ⊗Fp B + (W (k))/pB +(W (k)) is the image of y through rD modulo p. The epimorphism D(B + (W (k))/pB + (W (k))) ։ D(V (k)/pV (k)) defined by qk , defines an Fp -linear map j2 : Hom(D(D[p]), D(B +(W (k))/pB + (W (k)))) → Hom(D(D[p]), D(V (k)/pV (k))). As rD modulo p is injective, j1 is an Fp -linear monomorphism. We check that j2 is also an Fp -linear monomorphism. Let x̃ : M/pM → B + (W (k))/pB +(W (k)) be a k-linear map that defines an element of Ker(j2 ). The kernel of the k-epimorphism B + (W (k))/pB +(W (k)) ։ V (k)/pV (k) defined by qk is annihilated by the Frobenius endomorphism of B + (W (k))/pB +(W (k)). Thus x̃ annihilates Im(φ) modulo p. As n > 0, we get that there exists i ∈ {1, . . . , n + m} such that x̃ annihilates ai modulo p. As Im(φ) modulo p is contained in Ker(x̃) and as φ1 modulo p maps Ker(x̃) ∩ F 1 /pF 1 to Ker(x̃), by induction on s ∈ {0, . . . , n + m − 1} we get that x̃ annihilates aπ s (i) modulo p. Thus, as π is a cycle and as Ker(x̃) is a k-vector space, we have x̃ = 0. Therefore j2 is an Fp -linear monomorphism. Let x : M/pM → V (k)/pV (k) be a k-linear map that defines an element of Hom(D(D[p]), D(V (k)/pV (k))). As x(F 1 /pF 1 ) belongs to qk (ξ0 )V (k)/pV (k), the image through x of ai modulo p is of the form qk (ξ0 )εi yi , where yi ∈ V (k)/pV (k). But x takes φεi (ai ) modulo p to qk (ξ0 )επ(i) yπ(i) as well as to (−yi )p (cf. the definition of Φ̄k and Φ̄1k in Subsubsection 2.2.1). Thus (y1 , . . . , yn+m ) is a solution of the reduction of S modulo 37 p. Conversely, if (y1 , . . . , yn+m ) is a solution of the reduction of S modulo p, then the map x : M/pM → V (k)/pV (k) that takes ai modulo p to qk (ξ0 )εi yi does define an element of Hom(D(D[p]), D(V (k)/pV (k))). Thus, as a set, Hom(D(D[p]), D(V (k)/pV (k))) is in bijection with the set of solutions of the reduction of S modulo p. Thus, as V (k) is strictly henselian and S defines a finite, flat V (k)-algebra of degree pn+m , the number n0 of elements of Hom(D(D[p]), D(V (k)/pV (k))) is at most pn+m and therefore at most equal to the number of elements of Tp (DB(k) )/pTp (DB(k) ). As j1 and j2 are Fp -linear monomorphisms and as Tp (DB(k) )/pTp (DB(k) ) has pn+m elements, by reasons of orders of finite abelian groups we get that n0 = pn+m and that both j1 and j2 are isomorphisms. Let {a∗1 , . . . , a∗n+m } be the W (k)-basis for M ∗ which is the dual of the W (k)-basis B = {a1 , . . . , am } of M . Let sD : Tp (DB(k) )/pTp (DB(k) ) → M ∗ ⊗W (k) V (k)/pV (k) be the Fp -linear map defined naturally by i∗D ⊗1V (k)/p(V (k) (the tensorization being with respect to qk : B + (W (k)) ։ V (k)/pV (k)). As j2 ◦j1 is an isomorphism, from the description of Hom(D(D[p]), D(V (k)/pV (k))) we get that [V (k)/pV (k)]Im(sD ) is the VP (k)/pV (k)n+m ∗ ∗ submodule of M ⊗W (k) V (k)/pV (k) generated by elements of the form i=1 ai ⊗ qk (ξ0 )εi yi , where (y1 , . . . , yn+m ) runs through all reductions modulo p of solutions of S (for p ≥ 3 this result is a particular case of [Fa2, Sect. 4, p. 128]). The Moore determiηn+m η2 nant of the square matrix of Mn+m (W (k)) whose rows are (γ, γ p , . . . , γ p ), with γ running through µpn+m −1 (W (k)), is invertible (cf. [Go, Def. 1.3.2 and Lem. 1.3.3]). From this and (17) we get that V (k)/pV (k)Im(sD ) is generated by all a∗i ⊗ qk (ξ0 )εi xi ’s with 1 1 ε = p−1 ≤ 1, each element qk (ξ0 ) i xi ∈ i ∈ {1, . . . , m + n}. As p1 + Qi (p) < p1 + p(p−1) V (k)/pV (k) is non-zero; therefore sD is injective (even if p = 2). On the other hand, as i∗D is the dual of iD and as we have t(ai i) = wi ai (see Subsubsection 4.2.2), we easily get that V (k)/pV (k)Im(sD ) is generated by all a∗i ⊗ qk (wi )’s with i ∈ {1, . . . , m + n}. Thus for i ∈ {1, . . . , n + m}, we get that qk (wi ) is a Gm (V (k)/pV (k))-multiple of qk (ξ0 )εi xi . Therefore (19) holds even if n > 0. 4.2.6. A Lubin–Tate quadruple. To the quintuple (M, F 1 , φ, T, (tα )α∈J ) we will associate a Lubin–Tate quintuple (M̃ , F̃ 1 , φ̃, T̃ , (t̃α )α∈J̃ ) to which we can apply Subsubsections 4.2.1 to 4.2.5. Let o(π) be the order of π. Let n(π) ∈ N be the smallest number with the property that for each cycle (i1 , . . . , iq ) of π, there exist at most n(π) distinct cycles of π of the form (i′1 , . . . , i′q ) and such that up to a cyclic rearrangement we have εi′j = εij for all j ∈ {1, . . . , q}. Let r ∈ o(π)n(π)N. Let {ã1 , . . . , ãr } be a W (k)-basis for M̃ := W (k)r . For s ∈ N and i ∈ {1, . . . , r} let ãsr+i := ãi . Let D̃ be the p-divisible group over Spec(W (k)) whose filtered Dieudonné module is (M̃ , F̃ 1 , φ̃), where F̃ 1 := W (k)ã1 , φ̃(ã1 ) = pã2 , and φ̃(ãi ) = ãi+1 for i ∈ {2, . . . , r} (cf. Lemma 2.2.3 and Proposition 2.2.4). Let (F i (T(M̃ )))i∈Z be the filtration of T(M̃ ) defined by F̃ 1 . Let (M̃ , F̃ 1 , φ̃)⊗s := (M̃ ⊗s , (F i (T(M̃ )) ∩ M̃ ⊗s )i∈Z , φ̃). Let (t̃α )α∈J̃ be the set of all tensors of F 0 (T(M̃ ))[ p1 ] fixed by φ̃. Let ṽα ∈ T(H 1 (D̃)[ p1 ]) be the element that corresponds to t̃α via the following B + (W (k))[ β10 ]-linear isomorphism ∼ iD̃ [ β10 ] : M̃ ⊗W (k) B + (W (k))[ β10 ] → H 1 (D̃) ⊗Zp B + (W (k))[ β10 ] obtained as iD of (1) was. 38 Let G̃ be the schematic closure in GLM̃ of the subgroup G̃B(k) of GL M̃ [ 1 ] that fixes p t̃α for all α ∈ J̃. Let T̃ be the maximal torus of GL M̃ that normalizes W (k)ãi for all i ∈ {1, . . . , r}. Let µ̃ : Gm → G̃ be the inverse of the canonical split cocharacter of (M̃ , F̃ 1 , φ̃), cf. end of Subsection 2.1. As G̃ contains the conjugates of µ̃ under φ̃, it also contains T̃ . As Lie(T̃ ) is generated by elements of F 0 (T(M̃ )) fixed by φ̃, G̃ fixes Lie(T̃ ) and thus G̃ is contained in T̃ . Therefore G̃ = T̃ (this also follows from Subsection 4.1.3). 4.2.7. The w̃i elements. The double centralizer of T̃ in GL M̃ is T̃ itself. Thus there exists t̃ ∈ T̃ (B + (W (k))[ β10 ]) such that it takes (iD̃ [ β10 ])−1 (H 1 (D̃) ⊗Zp B + (W (k))) onto M̃ ⊗W (k) B + (W (k)), cf. Lemma 4.2.1 applied to (D̃, (t̃α )α∈J̃ ). As in Subsubsection 4.2.2, for i ∈ {1, . . . , r} let w̃i ∈ B + (W (k)) ∩ Gm (B + (W (k))[ β10 ]) be such that t̃(ãi ) = w̃i ãi . Let w̃i+r := w̃i . Thus qk (w̃i ) ∈ V (k)/pV (k) is well defined for i ∈ {1, . . . , 2r}. We apply Lemma 4.2.3 and Proposition 4.2.5 to (M̃ , F̃ 1 , φ̃) (thus (n, m) and π get replaced by the 1 pair (r − 1, 1) and by the cycle (1 2 . . . r)). As qk (ξ0 ) is p p modulo p, from (18) and p−1 1 (19) we get that qk (w̃1 ) is a Gm (V (k)/pV (k))-multiple of p p + pr −1 modulo p and that for i ∈ {2, . . . , r} the element qk (w̃i ) is a Gm (V (k)/pV (k))-multiple of p p−2+i pr −1 modulo p. 4.2.8. Proposition. The triple (M, F 1 , φ) is a direct summand of ⊕rs=1 (M̃ , F̃ 1 , φ̃)⊗s . Pq Proof: We fix a cycle π1 = (i1 , . . . , iq ) = (i1,1 , . . . , iq,1 ) of π. Let ε(π1 ) := s=1 εis . Let π1 , π2 , . . . , πu be all distinct cycles of π of length q that are of the form πj = (i1,j , . . . , iq,j ) with (εi1,j , . . . , εiq,j ) = (εi1 , . . . , εiq ) (here j ∈ {1, . . . , u} and u ∈ N). We have qr ∈ N and u ≤ qr , cf. the definitions of o(π) and n(π) in Subsubsection 4.2.6. For j ∈ {1, . . . , qr } let ζj ∈ Gm (W (k)) be such that σ r (ζj ) = ζj and the reductions modulo p of ζ1 , . . . , ζ qr are linearly independent over Fpq . For d ∈ {1, . . . , ε(π1 )} let sd ∈ {1, . . . , q} be such that we have s1 < s2 < · · · < sε(π1 ) and {sd |d ∈ {1, . . . , ε(π1 )}} = {s ∈ {1, . . . , q}|εis = 1}. For (d, i) ∈ {1, . . . , ε(π1 )} × {0, . . . , r−q q } let ld+iε(π1 ) := r + 2 − sd + (r − q)i. We note that the numbers l1 modulo r, . . . , l qr ε(π1 ) modulo r are all distinct. r For j ∈ {1, . . . , u} let cj : (⊕qs=1 W (k)ais ,j , F 1 ∩⊕qs=1 W (k)ais ,j , φ) → (M̃ , F̃ 1 , φ̃)⊗ q ε(π1 ) be the unique morphism that maps ai1 ,j to the sum r−q q X v=0 σ vq (ζj )ãvq+l1 ⊗ ãvq+l2 ⊗ · · · ⊗ ãvq+l r ε(π1 ) . q For s ∈ {1, . . . , q}, we have φ̃(ãs−1+vq+l1 ⊗ · · · ⊗ ãs−1+vq+l r ε(π1 ) ) = pu(s) ãs+vq+l1 ⊗ · · · ⊗ q ãs+vq+l r ε(π1 ) , where u(s) ∈ {0, 1}. We have u(s) = 1 if and only if there exists (d, i) ∈ q {1, . . . , ε(π1 )}×{0, . . . , r−q q } such that s−1+vq +ld+iε(π1 ) = s−1+vq +r +2−sd +(r −q)i is congruent modulo r to 1 i.e., if and only if s ∈ {s1 , . . . , sε(π1 ) }. This implies that cj exists; the uniqueness of cj follows from the fact that πj is a cycle. 39 For j ∈ {u + 1, . . . , qr }, let Im(cj ) be the natural analogue of Im(c1 ), . . . , Im(cu ). As the reductions modulo p of ζj ’s are linearly independent, the Moore determinant of the square matrix of M qr (k) whose j-th row is the reduction modulo p of (ζj , σ q (ζj ), . . . , σ r−q (ζj )) Pu is invertible (cf. [Go, Def. 1.3.2 and Lem. 1.3.3]). Thus the morphism j=1 cj is injective r q Im(cj )⊕(S(π1 ), (F i (T(M̃ ))∩S(π1 ))i∈Z , φ̃) as a direct supplement and its image has ⊕s=u+1 r in (M̃ , F̃ 1 , φ̃)⊗ q ε(π1 ) . Here S(π1 ) is the W (k)-span of those elements ãj1 ⊗ · · · ⊗ ãj r ε(π1 ) q r of M̃ ⊗ q ε(π1 ) that are not of the form ãw+l1 ⊗ · · · ⊗ ãw+l r ε(π1 ) for some w ∈ {1, . . . , r}. q We note that rq ε(π1′ ) ≤ r. We consider a different sequence π1′ , . . . , πu′ ′ of cycles of π that is analogue to the sequence π1 , . . . , πu . The numbers ε(π1 ), l1 , . . . , l qr ε(π1 ) are canoniPu′ Pu cally associated to π1 . Thus the morphism j=1 c′j analogue to j=1 cj but obtained using the cycles π1′ , . . . , πu′ ′ , is such that its image is contained either in (S(π1 ), (F i(T(M̃ )) ∩ ⊗ qr ε(π1′ ) 1 S(π1 ))i∈Z , φ̃) or in some ( M̃ , F̃ , φ̃) with ε(π1′ ) 6= ε(π1 ). Thus by adding such morPu phisms j=1 cj together we get that (M, F 1 , φ) is a direct summand of ⊕rs=1 (M̃ , F̃ 1 , φ̃)⊗s . 4.2.9. The projector t̃α0 . We fix a direct sum decomposition ⊕s∈N (M̃ , F̃ 1 , φ̃)⊗s = (M, F 1 , φ) ⊕ (M ⊥ , (F i (T(M̃ )) ∩ M ⊥ )i∈Z , φ̃) such that M is a direct summand of ⊕rs=1 M̃ ⊗s (cf. Proposition 4.2.8). Thus M is also naturally a direct summand of T(M̃ ). Therefore there exists α0 ∈ J̃ such that t̃α0 is a projector of T(M̃ ) on M . As M is a direct summand of ⊕rs=1 M̃ ⊗s , we get that M ∗ is a direct summand of ⊕rs=1 M̃ ∗⊗s . Thus we can identify T(M ) with a direct summand of T(M̃ ) and under this identification each tα is identified with a t̃α̃ for some α̃ ∈ J̃. Let L1 (D) be the direct summand of ⊕rs=1 H 1 (D̃)⊗s that corresponds to (M, F 1 , φ) via iD̃ [ β10 ]. The element −1 (L1 (D) ⊗Zp B + (W (k))) onto M ⊗W (k) B + (W (k)). We t̃ (of Subsubsection 4.2.7) takes iD̃ have L1 (D) = ṽα0 (T(H 1 (D̃))) and therefore Gal(B(k)) normalizes L1 (D). As we have w̃i ∈ B + (W (k)) for all i ∈ {1, . . . , n + m} and as M ⊆ ⊕rs=1 M̃ ⊗s , the B + (W (k))[ β10 ]linear isomorphism ĩD [ 1 1 ∼ 1 1 ] : M ⊗W (k) B + (W (k))[ ] → L (D) ⊗Zp B + (W (k))[ ] β0 β0 β0 induced naturally by iD̃ [ β10 ], induces via restriction a B + (W (k))-linear monomorphism ĩD : M ⊗W (k) B + (W (k)) ֒→ L1 (D) ⊗Zp B + (W (k)). 4.2.10. Proposition. The B + (W (k))-linear monomorphism ĩD : M ⊗W (k) B + (W (k)) ֒→ L1 (D) ⊗Zp B + (W (k)) defined by iD̃ is iD . In particular, we have L1 (D) = H 1 (D). Proof: As the crystalline representations of Gal(B(k)) over Qp are stable under subobjects that are direct summands (see [Fo3, Subsect. 5.5]), we can identify naturally the triples (iD [ β10 ], H 1 (D)[ p1 ], (vα )α∈J ) = (ĩD [ β10 ], L1 (D)[ p1 ], (ṽα̃ )α∈J ) and therefore we only need to check that H 1 (D) = L1 (D). To check this it suffices to consider the case when π is 40 a cycle π1 = (i1 , . . . , iq ) (cf. Subsection 4.1.2); thus q = n + m and ε(π1 ) = m. If w ∈ {1, . . . , r}, then the element t̃ (of Subsubsection 4.2.7) takes ãw+l1 ⊗ ãw+l2 ⊗ · · · ⊗ ãw+l r ε(π1 ) ∈ B + (W (k)) to itself times q r q ε(π1 ) w̃(w) := Y j=1 We have qk (w̃(w)) = εl w,j p Q qr ε(π1 ) j=1 ηl w,j + ppr −1 w̃w+lj ∈ B + (W (k)). qk (w̃w+lj ). But qk (w̃w+lj ) up to a Gm (V (k)/pV (k))-multiple modulo p, where εlw,j ∈ {0, 1} and ηlw,j ∈ {−1, . . . , r − 2} (see is of the form p the end of Subsubsection 4.2.7). At most one of εlw,j ’s is 1. The ηlw,j ’s are distinct (as lj modulo r is uniquely determined by j ∈ {1, . . . , qr ε(π1 )}) and their number is exactly rq ε(π1 ) and thus it is at most r. Thus qk (w̃(w)) is the image in V (k)/pV (k) of an Pr−2 η 1 1 = p−1 . element of V (k) whose p-adic valuation is at most p1 + η=−1 prp−1 = p1 + p(p−1) 1 This last p-adic valuation is p−1 if and only if rq ε(π1 ) = r i.e., if and only if we have n + m = q = ε(π1 ) = m. These last equalities are equivalent to n = 0. In particular we get that if either n > 0 or p > 2, then qk (w̃(w)) 6= 0. We check that H 1 (D) = L1 (D) in the case when (n, p) = (0, 2). In this case we can assume (n, m) = (0, 1). As H 1 (D) has rank 1, there exists s ∈ Z such that H 1 (D) =Qps L1 (D). r As iD̃ is strict with filtrations, we have w̃1 ∈ F 1 (B + (W (k))). Therefore w̃(w) = i=1 w̃i ∈ F 1 (B + (W (k)) is congruent modulo F p (B + (W (k))) to ξ0 times an element of B + (W (k)) 1 whose image through qk is p p(p−1) modulo p, cf. the last part of the previous paragraph. Thus w̃(w) is congruent modulo F p (B + (W (k))) to a Gm (B + (W (k))/F p (B + (W (k))))multiple of β0 modulo F p (B + (W (k))) (cf. end of Subsubsection 2.2.1). On the other hand we have Im(iD ) = ps L1 (D) ⊗Zp β0 B + (B(k)) and therefore w̃(w) is a Gm (B + (W (k)))multiple of ps β0 . As the V (k)-module gr1 = F 1 (B + (W (k)))/F p (B + (W (k))) is torsion free, we have s = 0. Therefore we have H 1 (D) = L1 (D) if n = 0 and p = 2. We check that H 1 (D) = L1 (D) in the case when either n > 0 or p > 2. We first show that we have an inclusion H 1 (D) ⊆ L1 (D). If p = 2 and n > 0, then L1 (D1 ) := H 1 (D) ∩ L1 (D) is a Zp -submodule of H 1 (D) with the property that iD factors through the inclusion L1 (D1 )⊗Zp B + (W (k)) ֒→ H 1 (D)⊗Zp B + (W (k)) (cf. the fact that ĩD is a B + (W (k))-linear monomorphism); as in the proof of Lemma 2.2.3 (applied with L1 (D1 ) instead of H 1 (D1 )) we argue first that p annihilates H 1 (D)/L1 (D1 ) and second that H 1 (D) = L1 (D1 ) ⊆ L1 (D). If p > 2, then the equality H 1 (D) ∩ L1 (D) = H 1 (D) is implied by the fact that Coker(jD ) is annihilated by λ1 ∈ V (k) \ pV (k) (see the end of Subsubsection 2.2.2). Thus we always have H 1 (D) ⊆ L1 (D) and therefore in the dual context we have an inclusion jD,D̃ : L1 (D)∗ ֒→ Tp (D) = H 1 (D)∗ . As either n > 0 or p > 2, we have qk (w̃(w)) 6= 0 for all w ∈ {1, . . . , r}. This implies that the Fp -linear map s1D : L1 (D)∗ /pL1 (D)∗ → M ∗ ⊗W (k) V (k)/pV (k) that is the natural analogue of the map sD (of the proof of Proposition 4.2.5) is injective. But s1D is the composite of jD,D̃ modulo p with sD . Thus jD,D̃ modulo p is injective and therefore jD,D̃ is an isomorphism. Thus we have H 1 (D) = L1 (D) if either n > 0 or p > 2. If we have qr ε(πl ) ≤ p − 2 for all cycles πl of π (with l ∈ C(π)), then the identity H 1 (D) = L1 (D) is also implied by [Fa2, Thm. 5]. 41 4.2.11. End of the proof of 1.2 and 4.1.4. As T̃ = G̃ is generated by conjugates of µ̃ via powers of φ̃ (see Subsection 4.2.6), the image of T̃ in GL M is the subtorus T0 of T . Thus the automorphism of M ⊗W (k) B + (W (k))[ β10 ] defined by the element t̃ of Subsubsection 4.2.7 is an element t ∈ T0 (B + (W (k))[ β10 ]) that takes the B + (W (k))submodule (iD [ β10 ])−1 (H 1 (D) ⊗Zp B + (W (k))) of M ⊗W (k) B + (W (k))[ β10 ] onto M ⊗W (k) B + (W (k)); here we used the fact (see Proposition 4.2.10) that we have H 1 (D) = L1 (D) regardless of what π is. As T0 is a subtorus of T , we have t ∈ T (B + (W (k))[ β10 ]) and thus Theorem 4.1.4 holds. This ends the proofs of Theorem 4.1.4 and the Main Theorem. 4.3. Example. We consider the case when T = T0 has rank 2. There exist two subcases: (a) all Newton polygon slopes of (M, φ) are 12 , or (b) the Newton polygon slopes of (M, φ) are elements of the set {0, 12 , 1}, the multiplicity of the Newton polygon slope 12 is positive, and either the multiplicity of the Newton polygon slope 0 or of the Newton polygon slope 1 is positive. In the subcase (a) we have n = m and T1 has rank 2. This is so as we can choose B such that we have π(i) = i + n for all i ∈ {1, . . . , n}. As T0 = T1 , Lemma 4.2.1 applies. We consider the subcase (b). Let M = M0 ⊕ M 12 ⊕ M1 be the Newton polygon slope decomposition of (M, φ); thus for v ∈ {0, 21 , 1} all Newton polygon slopes of (Mv , φ) are v. For s ∈ {0, 1} let M s1 := M 21 ∩ F s . We have a direct sum decomposition M 21 = 2 M 11 ⊕ M 01 . We have no a priori relation between n and m. The torus T1 is the center 2 2 of GL M0 ×Spec(W (k)) GL M 11 ×Spec(W (k)) GL M 01 ×Spec(W (k)) GLM1 . Thus T1 has rank 4 if 2 2 M1 and M0 are non-zero and has rank 3 otherwise. To fix the ideas we will assume that there exist numbers q0 ∈ {0, . . . , n} and q1 ∈ {0, . . . , m} such that {1, . . . , q0 , n + m + 1 − q1 , . . . , n + m} is the set of elements fixed by π and we have π(q0 + s) = n + m + 1 − q1 − s for all s ∈ {1, . . . , n − q0 }. We have q0 + q1 > 0 and n − q0 = m − q1 > 0. We can choose qk (wi ) ∈ V (k)/pV (k) to be a Gm (V (k)/pV (k))-multiple of the reduction modulo p of the following elements of V (k) (cf. Proposition 4.2.5): (i) 1 for i ∈ {1, . . . , q0 }; 1 (ii) (−p) p(p−1) for i ∈ {n + m + 1 − q1 , . . . , n + m}; 1 (iii) (−p) p(p2 −1) for i ∈ {q0 + 1, . . . , n}; 1 1 (iv) (−p) p2 −1 p p for i ∈ {n + 1, . . . , n + m − q1 }. The value of (ii) is the product of the values of (iii) and (iv). We have o(π) = 2 and n(π) = max{q0 , n − q0 , q1 }. If q1 > 0, then T0 is the closed subgroup scheme of GL M that fixes the following three types of tensors fixed by φ: – all endomorphisms of (M, φ), – the elements ai ∈ M for i ∈ {1, . . . , q0 }, and – the elements aq0 +s ⊗an+m+1−q1 −s ⊗a∗i +an+m+1−q1 −s ⊗aq0 +s ⊗a∗i ∈ M ⊗2 ⊗W (k) M ∗ , with s ∈ {1, . . . , n − q0 } and with i ∈ {n + m + 1 − q1 , . . . , n + m}. 42 4.4. Proof of Corollary 1.4. We continue to assume that k = k̄ but we come back to the general situation of Subsection 1.1. Thus G is not any more assumed to be a torus. We take µ := µ−1 can , cf. Subsection 2.1. Let σφ , MZp , and GZp be as in Subsection 2.5.1. Let Gmin,B(k) be the smallest subgroup of GL M [ p1 ] that contains the images of conjugates of µB(k) under powers of φ; it is connected and we have φ(Lie(Gmin,B(k) )) = Lie(Gmin,B(k) ). Let Gmin be the schematic closure of Gmin,B(k) in GLM . As G is normalized by σφ and as µ factors through G, GB(k) contains the images of conjugates of µB(k) under powers of φ. Thus Gmin is a closed subgroup scheme of G. From Lemma 2.5.3 we get that Gmin is the pull back to Spec(W (k)) of a flat, closed subgroup scheme Gmin,Zp of GZp . Let Gmin,Qp be the generic fibre of Gmin,Zp . Groups like Gmin,Qp were first considered in [Wi, Prop. 4.2.3]. Let u ∈ N ∪ {0} be the largest number such that Gum is a quotient of Gmin,Qp . The image of µB(k) in Gum (extended to B(k)) is defined over Qp and it is normalized by both σφ and φ. From the very definition of Gmin,B(k) we get that this image is Gum (extended to B(k)) and therefore we have u ∈ {0, 1}. The relation u ∈ {0, 1} also follows from the fact that each crystalline representation of Gal(B(k)) of rank 1 is isomorphic to a tensor power of the cyclotomic character of Gal(B(k)). In this paragraph we assume that u = 1. Let G0min,Qp be the normal, connected subgroup of Gmin,Qp such that Gmin,Qp /G0min,Qp is isomorphic to Gm . The image of µ is not contained in G0min,Qp and thus µ is non-trivial. This implies that Gmin,Qp is not contained in SL MZp [ p1 ] := GL der . Therefore Gmin,Qp /[Gmin,Qp ∩ SL MZp [ p1 ] ] is isomorphic to Gm . MZ [ 1 ] p p This implies that G0min,Qp is the identity component of Gmin,Qp ∩ SL MZp [ p1 ] . Therefore the GLMZp [ 1 ] , Gm ) → Hom(Gmin,Qp , Gm ) has finite cokernel. restriction homomorphism Hom(GL p GLMZp [ 1 ] , Gm ) → Hom(Gmin,Qp , Gm ) has trivial If u = 0, the homomorphism Hom(GL p codomain and therefore it is onto. GLMZp [ 1 ] , Gm ) → Hom(Gmin,Qp , Gm ) has finite Thus regardless of what u is, Hom(GL p cokernel. Therefore Gmin,Qp is the subgroup of GL MZp [ p1 ] that fixes a family of tensors (tα )α∈Jmin of T(MZp [ p1 ]), cf. [De2, Prop. 3.1 c)]. As for each α ∈ Jmin the tensor tα is fixed by both µB(k) and σφ , we have tα ∈ {x ∈ F 0 (T(M ))[ p1 ]|φ(x) = x}. Let (vα )α∈Jmin be the family of tensors of T(H 1 (D))[ p1 ] that corresponds to (tα )α∈Jmin via Fontaine comparison 1 theory for D. Let Gét ] min be the schematic closure in GL H 1 (D) of the subgroup of GL H 1 (D)[ p ét ét that fixes each vα with α ∈ Jmin . The group Gét is a min,Qp is a subgroup of G ; thus if G ét ét torus, then Gmin is a subtorus of G . If p = 2 and the condition (C) holds for (M, φ, G), then as Gmin ⊆ G we get that the condition (C) holds as well for (M, φ, Gmin ). Based on the last paragraph and the fact that we have an inclusion Gmin ⊆ G, we get that to prove the Corollary 1.4 we can assume that J = Jmin is a subset of Jtwist , that G = Gmin , and that (tα )α∈J is the family of all tensors of {x ∈ F 0 (T(M ))[ p1 ]|φ(x) = x}. Let W ∈ W, cf. the notations of Corollary. From the functorial aspects of the canonical split cocharacters we get that W is normalized by the image of µcan . From this and the fact that φ(W [ p1 ]) = W , we get that W [ p1 ] is normalized by all conjugates of µB(k) under powers of φ. This implies that each W is normalized by G = Gmin . 43 ∼ Let ρ : (M, (tα )α∈J ) → (H 1 (D)⊗Zp W (k), (vα )α∈J ) be an isomorphism as in the Main ∼ Theorem. Let ρ : T(M ) → T(H 1 (D) ⊗Zp W (k)) be the isomorphism induced by ρ. ´ We check that for each W ∈ W we have ρ(W ) = W et ⊗Zp W (k). As iD [ β10 ] : ∼ M ⊗W (k) B + (W (k))[ β10 ] → H 1 (D) ⊗Zp B + (W (k))[ β10 ] maps W ⊗W (k) B + (W (k))[ β10 ] to ´ ´ W et ⊗Zp B + (W (k))[ β10 ], to check that we have ρ(W ) = W et ⊗Zp W (k) it suffices to show 1 −1 1 + that the B (W (k))[ β0 ]-linear automorphism nD := iD [ β0 ] ◦ ρB+ (W (k))[ β1 ] of M ⊗W (k) 0 B + (W (k))[ β10 ] normalizes W ⊗W (k) B + (W (k))[ β10 ]. As nD fixes each element tα with α ∈ J, we have nD ∈ G(B + (W (k))[ β10 ]). As W is normalized by G, we get that nD normalizes ´ W ⊗W (k) B + (W (k))[ β10 ]. Thus we have ρ(W ) = W et ⊗Zp W (k) for all W ∈ W. To end the proof of Corollary 1.4 we are left to show that we can choose ρ such that it maps tα to vα for all α ∈ Jtwist . To check this let D̃ := D ⊕ µp∞ . The Dieudonné module of D̃ is (M̃ , φ̃) := (M, φ) ⊕ (W (k), pσ). The Hodge filtration of M̃ defined by D̃ is F̃ 1 := F 1 ⊕ W (k). Let (F i (T(M̃ )) be the filtration of T(M̃ ) defined by F̃ 1 . Let µ̃ : Gm → GL M̃ be the inverse of the canonical split cocharacter of (M̃ , F̃ 1 , φ̃); it acts on M as µ does (cf. the functorial properties of canonical split cocharacters). Let (t̃α )α∈J̃ be the family of all tensors of F 0 (T(M̃ ))[ p1 ] fixed by φ. Let (ṽα )α∈J̃ be the family of tensors of T(H 1 (D̃))[ p1 ] that corresponds to (t̃α )α∈J̃ via Fontaine ∼ comparison theory for D̃. Let ρ̃ : (M̃ , (t̃α )α∈J̃ ) → (H 1 (D̃), (ṽα )α∈J̃ ), cf. Main Theorem applied to D̃. By composing ρ̃ with an automorphism of (M̃ , (t̃α )α∈J̃ ) defined by an element of the image of µ̃(W (k)) : Gm (W (k)) → GL M̃ (W (k)) we can assume that ρ̃ takes each element t of the set ∪i∈Z {x ∈ F i (T(W (k))|(pσ)(x) = pi x} to the element v of T(H 1 (µp∞ )) that corresponds to t via Fontaine comparison theory. Let ∼ ρ : (M, (tα )α∈J ) → (H 1 (D) ⊗Zp W (k), (vα )α∈J ) be defined by the restriction of ρ̃ to M . For α ∈ Jtwist , let i ∈ Z be such that tα ∈ {x ∈ F i (T(M ))[ p1 ]|φ(x) = pi x}. Let t−i be a generator of the Zp -module {x ∈ F −i (T(W (k))|(pσ)(x) = p−i x}. We have tα ⊗ t−i ∈ {x ∈ F 0 (T(M̃ ))[ p1 ]|φ̃(x) = x}. Thus ρ̃ maps tα ⊗ t−i to vα ⊗ v−i , where v−i corresponds to t−i via Fontaine comparison theory for µp∞ . As ρ̃ also maps t−i to v−i , we conclude that ρ̃ and thus also ρ maps tα to vα . Therefore we can take ∼ ρtwist : (M, (tα )α∈Jtwist ) → (H 1 (D) ⊗Zp W (k), (vα )α∈Jtwist ) to be defined by ρ. 4.4.1. Example. Suppose that the hypotheses of the Main Theorem hold and we have an isogeny λD : D → Dt . Let λM : M × M → W (k) and λH 1 (D) : H 1 (D) × H 1 (D) → Zp be bilinear forms on M and H 1 (D) defined naturally by λD . We can naturally view λM as a tensor of {x ∈ F −1 (M ∗⊗2 )|φ(x) = p−1 x}. From Corollary 1.4 we get that there exists ∼ an isomorphism ρ : (M, (tα )α∈J ) → (H 1 (D), (vα )α∈J ) such that for all x, y ∈ M we have λM (x, y) = λH 1 (D) (ρ(x), ρ(y)). 4.5. Remarks. (a) Suppose that the hypotheses of the Main Theorem hold. If G is not smooth or if G is smooth but Gk is not connected, then we do not know when there exists ∼ an isomorphism ρZp : (MZp , (tα )α∈J ) → (H 1 (D), (vα )α∈J ) (cf. the limitations of Lemma 2.5.2 (a)). But the proofs of Lemma 3.4.3 and Theorem 3.5.1 can be adapted to show that we can always choose the cocharacter µ : Gm → G so that the isomorphism ρZp does exist. 44 (b) Let G(W (k))0 be as in Subsection 2.6. Suppose that p = 2, that k = k̄, and that there exists no h̃ ∈ G(W (k))0 such that (M, h̃φ) has both Newton polygon slopes 0 and 1 with positive multiplicities. Then the Main Theorem continues to hold. This is so as due to Remark 3.4.4 (b), Subsections 3.5 and 4.2 apply entirely. 5. The ramified context Let k be again an arbitrary perfect field of characteristic p > 0. Let V , K, e, X, ee , Φk , and ie be as in Subsection 2.2. Thus V = W (k)[[X]]/(fe). The p-adic fe , R e , R P∞ completion of ΩRe /W (k) is Re dX. For q ∈ N we consider the ideals Iee (q) := { n=0 an X n ∈ ee |a0 = · · · = aq−1 = 0} and Ie (q) := {P∞ an X n ∈ Re |a0 = · · · = aq−1=0 } of R ee and R n=0 Re (respectively). We have Ie (q) = Re ∩ I˜e (q). In this Section we study ramified analogues of the Main Theorem over Re and R̃e . Some preliminaries are gathered in Subsection 5.1. Theorem 5.2 refines the deformation theory of [Fa2, Ch. 7]. See Subsections 5.3 and 5.4 for the main results of this Section. The counterexample of Subsection 5.5 emphasizes that for p > 2 the hypotheses of Theorem 5.3 are needed in general. 5.1. The setting. Let H be a p-divisible group over Spec(V ). Let (N, φN , ∇N ) be the evaluation of D(HV /pV ) at the thickening associated naturally to the closed embedding Spec(V /pV ) ֒→ Spec(Re ). Thus N is a free Re -module of rank equal to the height of H, φN : N → N is a Φk -linear endomorphism, and ∇N : N → N ⊗Re Re dX is a connection on N with respect to which φN is horizontal i.e., we have ∇N ◦ φN = (φN ⊗ dΦk ) ◦ ∇N . The connection ∇N is integrable and nilpotent modulo p. ˜ N is a connection on N such that we have 5.1.1. Uniqueness of connections. If ∇ ˜ ˜ ∇N ◦ φN = (φN ⊗ dΦk ) ◦ ∇N , then as Φk (X) = X p by induction on q ∈ N we get ˜ N . A similar argument shows that ˜ N ∈ X q EndR (N ) ⊗R dXRe [ 1 ]. Thus ∇N = ∇ ∇N − ∇ e e p ∇N modulo Ie (q) is uniquely determined by ΦN modulo Ie (q) and that the extension of ee (to be denoted also by ∇N ) is the unique connection ∇N to a connection on N ⊗Re R ee is horizontal. such that the Frobenius endomorphism ΦN ⊗ Φk of N ⊗Re R 5.1.2. Tensors. Let (F i (T(N/F 1 (Re )N )))i∈Z be the filtration of T(N/F 1 (Re )N ) defined by the direct summand FV1 of N/F 1 (Re )N that is the Hodge filtration of H. We consider a family (tHα )α∈J of tensors of T(N [ p1 ]) fixed by φN and whose images in T(N/F 1 (Re )N )[ p1 ] t belong to F 0 (T(N/F 1 (Re )N ))[ p1 ]. Let H 1 (H) := Tp (HK )(−1). Let vHα ∈ T(H 1 (H)[ p1 ]) correspond to tHα via the B + (W (k))-linear monomorphism iH : N ⊗Re ie B + (W (k)) ֒→ H 1 (H) ⊗Zp B + (W (k)) obtained as iD of (1) was (this time iH is Gal(K)-invariant; see [Fa2, Sect. 4, p. 127] for the canonical action of Gal(K) on N ⊗Re ie B + (W (k))). Let (M, φ, (tα )α∈J ) := (N, φN , (tHα )α∈J ) ⊗Re Re /Ie (1). Let GB(k) be the subgroup of GL M [ p1 ] that fixes t − α for all α ∈ J. Let G be the schematic closure of GB(k) in GL M . It is well known that there exist isomorphisms 1 1 ∼ (N ⊗Re Re [ ], φN ⊗ Φk ) K : (M ⊗W (k) Re [ ], φ ⊗ Φk ) → p p 45 ee instead of with Re ). (to be compared with [Fa2, Sect. 6, p. 132] which works with R We choose such an isomorphism K that lifts 1M [ p1 ] . There exists no element of T(M )[ p1 ] ⊗ Ie (1)[ p1 ] fixed by φ ⊗ Φk . This implies (i) that K is the unique such isomorphism that lifts 1M [ p1 ] and (ii) that each tHα is the extension of tα via the Re [ p1 ]-linear isomorphism ∼ T(M )[ p1 ] ⊗B(k) Re [ p1 ] → T(N )[ p1 ] induced by K. Thus we can speak about the smooth subgroup scheme GRe [ p1 ] of GL N[ p1 ] that fixes tHα for all α ∈ J; it is isomorphic to G ×Spec(W (k)) Spec(Re [ p1 ]). 5.1.3. Three possible ramified analogues. If k = k̄, then one would like to know if any one of the following three conditions holds: ∼ (i) there exists an isomorphism (N, (tHα )α∈J ) → (H 1 (H) ⊗Zp Re , (vHα )α∈J ); ∼ (ii) there exists an isomorphism (M, (tα )α∈J ) → (H 1 (H) ⊗Zp W (k), (vHα )α∈J ); ∼ (iii) there exists an isomorphism (N/F 1 (Re )N, (tHα )α∈J ) → (H 1 (H)⊗Zp V, (vHα )α∈J ). Obviously condition (i) implies condition (ii). In this paragraph we assume that e ≤ p − 1. It is known that there exists an ∼ isomorphism (M, (tα )α∈J ) → (N/F 1 (Re )N, (tHα )α∈J ), cf. [La, Thm. 2.1]. Thus condition (ii) implies condition (iii). If we know that G is smooth, then as in the proof of Lemma 2.5.2 (b) we argue that conditions (iii) and (iii) are equivalent. 5.1.4. Condition (L). Until Subsection 5.5 we will assume that the following (liftable type of) condition holds: (L) there exists a direct summand FR1 e of N that lifts FV1 and such that we have tHα ∈ F 0 (T(N ))[ p1 ] for all α ∈ J, where (F i (T(N )))i∈Z is the filtration of T(N ) defined by FR1 e . The condition (L) is a necessary condition to be able to lift the pair (H, (tHα )α∈J ) to an analogue pair (HRe , (tHα )α∈J ) over Spec(Re ). The goals of this Section are to show that for p > 2 condition (L) is also sufficient for the existence of the lift (HRe , (tHα )α∈J ) and to obtain a ramified analogue of the Main Theorem (see Subsections 5.3 and 5.4 below). 5.1.5. Extra notations. Let F 1 := FR1 e ⊗Re Re /Ie (1). Let M = F 1 ⊕F 0 and µ : Gm → G be as in Subsection 2.1; these notations make sense even if p = 2 and (M, F 1 , φ) is not the filtered Dieudonné module of a p-divisible group over Spec(W (k)). Let G′ be as in Subsection 2.6. Let Q be as in Theorem 3.2 (c). Let ΦQ and MQ = (M ⊗W (k) Q, F 1 ⊗W (k) Q, Φ(Q)0 , Φ(Q)1 ) be obtained as in Subsubsection 3.1.2 (a) using the morphism ℓ : Spec(Q) → Y ∧ of Theorem 3.2 (c). Let a0 : Spec(W (k)) ֒→ Spec(Q) and ∇∞ be as in the beginning of Subsection 3.4. Let I 0 be the ideal of Q that defines the 0 section a0 . Let MQ := (M ⊗W (k) Q, F 1 ⊗W (k) Q, Φ(Q)0 ). ee be a W (k)-homomorphism that maps I 0 to Iee (1). 5.1.6. Extensions. Let mH : Q → R ee ) → Spec(Q). By the extension of We denote also by mH the resulting morphism Spec(R 0 (MQ , ∇∞ , (tα )α∈J ) through mH we mean the quintuple ee , F 1 ⊗W (k) R ee , Φm , ∇m , (tα )α∈J ), (M ⊗W (k) R H H 46 ee , Φm , ∇m ) is the evaluation at the thickening (Spec(R̃e /pR̃e ) ֒→ where (M ⊗W (k) R H H ee /pR ee )/Spec(W (k))) of the F -crystal Spec(R̃e ), δ(p)) of the pull back to CRIS(Spec(R on CRIS(Spec(Q/pQ)/Spec(W (k))) whose evaluation at the thickening (Spec(Q/pQ) ֒→ Spec(Q), δ(p)) is (M ⊗W (k) Q, Φ(Q)0 , ∇∞ ). Thus ΦmH is defined using a correction automorphism K as in the proof of Theorem 3.4.1 and ∇mH is the unique connection on M ⊗W (k) ee with respect to which Φm is horizontal. Similarly for q ∈ N we define the extension R H ee /Iee (q), F 1 ⊗W (k) R ee /Iee (q), Φm (q) , ∇m (q) , (tα )α∈J ) of (M 0 , ∇∞ , (tα )α∈J ) (M ⊗W (k) R H H Q ee /Iee (q) = Re /Ie (q) that maps I 0 to through a W (k)-homomorphism mH (q) : Q → R Iee (1)/Iee (q) = Ie (1)/Ie (q). 5.2. Theorem. If p = 2 we assume that (M, F 1 , φ) is the filtered Dieudonné module of a p-divisible group D over Spec(W (k)). We recall that the condition 5.1.4 (L) holds. ee such that mH (I 0 ) ⊆ Iee (1) and the Then there exists a W (k)-homomorphism mH : Q → R ee , F 1 ⊗R R ee , φN ⊗ Φk , ∇N , (tHα )α∈J ) is isomorphic to the extension quintuple (N ⊗Re R e Re 0 of (MQ , ∇∞ , (tα )α∈J ) through mH , under an isomorphism which modulo Iee (1) is 1M . Proof: This proof is a group theoretical refinement of [Fa2, Sect. 7, pp. 135–136]. We 0 can replace (MQ , ∇∞ , (tα )α∈J ) by its extension (MR̂0 , ∇R̂ , (tα )α∈J ) to the completion R̂ of Q with respect to the ideal I 0 of Q (see Subsection 3.3 1); here MR̂0 := (M ⊗W (k) R̂, F 1 ⊗W (k) R̂, Φ(R̂)0 ). We denote also by mH and mH (q) their factorizations through ee and R̂ → R ee /Iee (q) (respectively). If p ≥ 3 let D be the W (k)-homomorphisms R̂ → R p-divisible group over Spec(W (k)) whose filtered Dieudonné module is (M, F 1 , φ). Let DR̂ be the unique p-divisible group over Spec(R̂) that lifts D and such that the evaluation of its filtered Dieudonné crystal at the thickening (Spec(R̂/pR̂) ֒→ Spec(R̂), δ(p)) is (defined by) (MR̂0 , ∇R̂ ) (cf. proof of Theorem 3.4.1 applied with Q̂0 = R̂). The natural divided power structure of the ideal (pX) of W (k)[[X]]/(X e) is nilpotent. Thus there exists a unique p-divisible group HW (k)[[X]]/(X e ) over Spec(W (k)[[X]]/(X e)) that lifts both HV /pV and D and such that the evaluation of the filtered Dieudonné crystal of HW (k)[[X]]/(X e ) at the following thickening (Spec(V /pV ) ֒→ Spec(W (k)[[X]]/(X e)), δ(p)) is (N, FR1 e , φN , ∇N ) modulo Ie (e), cf. Grothendieck–Messing deformation theory. ee /Iee (q) By induction on q ∈ N we construct a W (k)-homomorphism mH (q) : R̂ → R 0 0 that maps I to Iee (1)/Iee (q) and such that the extension of (M , ∇ , (tα )α∈J ) through R̂ R̂ ee , F 1 ⊗R R ee , φN ⊗ Φk , ∇N , (tHα )α∈J ) modulo Iee (q), mH (q) is isomorphic to (N ⊗Re R e Re under an isomorphism Iq which modulo Iee (1)/Iee (q) is defined by 1M . Such an isomorphism Iq is unique as we have Φqk (Iee (1)/Iee (q)) = 0. The construction of mH (1) is obvious. ee /Iee (q) to The passage from q to q + 1 goes as follows. We lift mH (q) : R̂ → R ee /Iee (q + 1). We endow the ideal an arbitrary W (k)-homomorphism m̃H (q + 1) : R̂ → R ee /Iee (q + 1) with the trivial divided power structure (thus Jee (q) := Iee (q)/Iee (q + 1) of R ee /Iee (q + Jee (q)[l] = 0 if l ≥ 2). We consider an identification Ĩq+1 between (M ⊗W (k) R 1 ee , F 1 ⊗R R ee , φN ⊗ Φk , ∇N ) modulo Iee (q + 1) 1), Fq+1 , Φ′m̃H (q+1) , ∇′m̃H (q+1) ) and (N ⊗Re R e Re ee /Iee (q + 1) that which modulo Jee (q) is Iq . Here F 1 is a direct summand of M ⊗W (k) R q+1 47 ′ ee /Iee (q), Φ′ lifts F 1 ⊗W (k) R m̃H (q+1) lifts Φm̃H (q) , and ∇m̃H (q+1) lifts ∇m̃H (q) . In the next two paragraphs we show that we can choose Ĩq+1 such that we have (Φ′m̃H (q+1) , ∇′m̃H (q+1) ) = (Φm̃H (q+1) , ∇m̃H (q+1) ). (20) We first consider the case q < e. By a second induction on s ∈ {1, . . . , q} we show that the pulls back of DR̂ and HW (k)[[X]]/(X e ) to Spec(W (k)[[X]]/(X s)) are isomorphic, under a unique isomorphism whose evaluation at the thickening (Spec(k[[X]]/(X s)) ֒→ Spec(W (k)[[X]]/(X s)), δ(p)) is Is ; here the W (k)-homomorphism R̂ → W (k)[[X]]/(X s) used for the pull back of DR̂ is mH (s). The case s = 1 holds by constructions. Due to the existence of Is+1 for s < q − 1, the passage from s to s + 1 follows from the fact that Jee (s) has a nilpotent divided power structure and from the Grothendieck–Messing deformation theory. This ends the second induction. Thus the Dieudonné crystals of the pulls back of DR̂ and HW (k)[[X]]/(X e ) to Spec(W (k)[[X]]/(X q )) are canonically identified. Thus as Jee (q) has a nilpotent divided power structure, we can choose Ĩq+1 such that (20) holds. We next consider the case q ≥ e. We have Φk (Iee (q)) ⊆ pIee (q + 1) and thus for ee ) → GL N (R ee /Iee (q))) the Φk -linear map gq ΦN g −1 is of the form hq ΦN , GLN (R gq ∈ Ker(GL q e e where hq ∈ GL N (Re ) is congruent modulo Ie (q + 1) to gq . When gq varies, Φ′m̃H (q+1) ee /Iee (q + 1)) → GLM (R varies by a left multiple of it by an arbitrary element gqM ∈ Ker(GL ee /Iee (q))). We show that Φ′ GL M (R is of the form m̃H (q+1) ee /Iee (q + 1)) → GL M (R ee /Iee (q))). GLM (R g̃q Φm̃H (q+1) , where g̃q ∈ Ker(GL 1 ee /Iee (q + 1) coincide modulo Jee (q), As Φk (Iee (q)) ⊆ Iee (q + 1) and as Fq+1 and F 1 ⊗W (k) R 1 ee /Iee (q + 1)) = Φ′ e e we have Φ′m̃H (q+1) (F 1 ⊗W (k) R m̃H (q+1) (Fq+1 ) ⊆ pM ⊗W (k) Re /Ie (q + 1). ee /Iee (q + 1)-linear isomorphisms (M + Thus both Φ′ and Φm̃ (q+1) give birth to R m̃H (q+1) 1 1 ee /Iee (q F ) ⊗W (k) σ R p H ∼ ee /Iee (q + 1). Thus indeed Φ′ + 1) → M ⊗W (k) R m̃H (q+1) is of the form ee /Iee (q + 1)). As Φ′ g̃q Φm̃H (q+1) , where g̃q ∈ GL M (R m̃H (q+1) and Φm̃H (q+1) coincide modulo Jee (q), g̃q modulo Jee (q) is the identity element. We choose gq such that gqM = g̃q−1 . By ee , F 1 ⊗R R ee , φN ⊗ Φk , ∇N , (tHα )α∈J ) with its conjugate under gq , replacing (N ⊗R R e Re e we can choose Ĩq+1 such that Φ′m̃H (q+1) = Φm̃H (q+1) . Thus we also have ∇′m̃H (q+1) = ∇m̃H (q+1) (cf. Subsubsection 5.1.1) and therefore (20) holds. From now on we take Ĩq+1 such that (20) holds. As Ĩq+1 lifts Iq and as no element of T(M )[ p1 ]⊗B(k) Jee (q)[ p1 ] is fixed by Φm̃H (q+1) , under the identification Ĩ−1 q+1 the image of tHα ee /Iee (q +1)[ 1 ]) gets identified with the tensor tα ∈ T(M ⊗W (k) R ee /Iee (q +1)[ 1 ]) in T(N ⊗R R e p p (here α ∈ J). Thus we have 0 ee /Iee (q + 1)))[ 1 ], tα ∈ Fq+1 (T(M ⊗W (k) R p i ee /Iee (q + 1))))i∈Z is the filtration of T(M ⊗W (k) R ee /Iee (q + 1)) where (Fq+1 (T(M ⊗W (k) R 1 defined by Fq+1 . Let T(M ) = ⊕i∈Z F̃ i (T(M )) be as in Subsection 2.7. We have tα ∈ 48 F̃ 0 (T(M ))[ p1 ] for all α ∈ J and the filtration of T(M ) defined by F 1 is (F i (T(M )))i∈Z , where s −1 F i (T(M )) := ⊕∞ (EndW (k) (M )), Ubig , and U be as in Subsection 2.7. s=i F̃ (T(M )). Let F̃ −1 e We identify 1M with 1M ⊗ e /Ie (q+1) . Let uq+1 ∈ F̃ (EndW (k) (M )) ⊗W (k) Je (q) R W (k) e e ee /Iee (q + 1)) = F 1 . We have be the unique element such that (1M + uq+1 )(F 1 ⊗W (k) R q+1 1 1 0 e e (1M − uq+1 )(tα ) ∈ F (T(M ))[ p ] ⊗B(k) Re /Ie (q + 1)[ p ]. We view T(M ) as a module over EndW (k) (M ) and thus also over Lie(U ). As uq+1 ∈ F̃ −1 (T(M )) and tα ∈ F̃ 0 (T(M ))[ p1 ], ee /Iee (q + 1)[ 1 ] is −uq+1 (tα ). the component of (1M − uq+1 )(tα ) in F̃ −1 (T(M ))[ 1 ] ⊗B(k) R p p ee /Iee (q + 1)[ 1 ], the mentioned component But as (1M − uq+1 )(tα ) ∈ F ⊗B(k) R p is 0. Thus uq+1 annihilates tα for all α ∈ J. But Lie(GB(k) ) is the Lie subalgebra of EndB(k) (M [ p1 ]) that centralizes tα for all α ∈ J. Thus, as Jee (q) is a free W (k)-module of rank 1, we have 0 (T(M ))[ p1 ] ee /Iee (q+1)[ 1 ]∩F̃ −1 (EndW (k) (M ))⊗W (k) Jee (q) = Lie(U )⊗W (k) Jee (q). uq+1 ∈ Lie(GB(k) )⊗B(k) R p We identify HomW (k) (F 1 , F 0 ) with EndW (k) (M )/EndW (k) (M )∩F 0 (T(M )) and with Lie(Ubig ). As ∇R̂ respects the G-action (see Subsection 3.3 1)), the image of the Kodaira– Spencer map KR̂ of ∇R̂ is contained in the image of (Lie(GB(k) )∩EndW (k) (M ))⊗W (k) R̂ in HomW (k) (F 1 , F 0 )⊗W (k) R̂. Thus Im(KR̂ ) is contained in the direct summand Lie(U )⊗W (k) R̂ of HomW (k) (F 1 , F 0 ) ⊗W (k) R̂. We recall that we can identify R̂ = W (k)[[z1 , . . . , zd ]] in such a way that ΦR̂ (zl ) = zlp (see Subsections 3.1 and 3.3 1)). Also the ΦR̂ -linear map Φ(R̂)0 of M ⊗W (k) R̂ (see Subsubsection 3.1.2 (a)) is guniv (φ ⊗ ΦR̂ ), where guniv is the −1 universal element guniv of G′ (R̂). Thus ∇R̂ modulo (p, (z1 , z2 , . . . , zd )p−1 ) is δ0 −guniv dguniv p−1 modulo (p, (z1 , z2 , . . . , zd ) ) (this can be read out from (11); see Subsubsection 3.1.2 (b) for δ0 ). As U is a closed subgroup scheme of G′ , we get that the R̂-submodule Im(KR̂ ) of Lie(U ) ⊗W (k) R̂ surjects onto Lie(U ) ⊗W (k) k. Therefore Im(KR̂ ) = Lie(U ) ⊗W (k) R̂. Pd ∂ As uq+1 ∈ Lie(U )⊗W (k) Jee (q) we can write uq+1 = l=1 xl KR̂ ( ∂z ), where xl ∈ Jee (q) l and where we denote also by K ( ∂ ) its reduction modulo Iee (q + 1). Let mH (q + 1) : R̂ → R̂ ∂zl ee /Iee (q + 1) be the W (k)-homomorphism that takes zl to m̃H (q + 1)(zl ) − xl . By replacing R 1 1 ee /Iee (q + m̃H (q +1) with mH (q +1), Fq+1 gets replaced by (1M −uq+1 )(Fq+1 ) = F 1 ⊗W (k) R ee /Iee (q +1), F 1 ⊗W (k) R ee /Iee (q + 1). Thus Iq+1 is defined by the identification of (M ⊗W (k) R ee , F 1 ⊗R R ee , φN ⊗ Φk , ∇N , (tHα )α∈J ) 1), ΦmH (q+1) , ∇mH (q+1) , (tα )α∈J ) with (N ⊗Re R e Re modulo Iee (q + 1). This ends the induction. Let mH : R̂ → R̃e be such that it lifts all mH (q)’s. The extension of (MR̂0 , ∇R̂ , (tα )α∈J ) ee , F 1 ⊗R R ee , φN ⊗ Φk , ∇N , (tHα )α∈J ), under an through mH is isomorphic to (N ⊗Re R e Re isomorphism which modulo Iee (q) is Iq for all q ∈ N. 5.3. Theorem. Suppose that k = k̄, that p > 2, and that the condition 5.1.4 (L) holds. ∼ Then there exists an isomorphism ρRe : (N, (tHα )α∈J ) → (H 1 (H) ⊗Zp Re , (vHα )α∈J ). Thus the schematic closure of GRe [ p1 ] in GL N is a flat, closed subgroup scheme of GL N that is isomorphic to G ×Spec(W (k)) Spec(Re ). 49 Proof: Let D be as in the proof of Theorem 5.2. As p > 2, let D be as in Theorem 3.4.1 (b). Let (Vα )α∈J be as in Subsubsection 3.4.2. Let (vα )α∈J be as in the end of ee be as in Theorem 5.2. Let H1 be the p-divisible Subsubsection 2.2.2. Let mH : Q → R group over Spec(V ) that is the pull back of D via the composite morphism Spec(V ) = ee /Iee (1)) ֒→ Spec(R ee ) → Spec(Q) defined naturally by mH . Let the quadruple Spec(R (N1 , ΦN1 , ∇N1 , (tH1 α )α∈J ) be the analogue of the quadruple (N, ΦN , ∇N , (tHα )α∈J ) but for H1 and the extension of (tα )α∈J via the W (k)-homomorphism Q → V defined by mH . We can identify (N1 , (tH1 α )α∈J ) = (M ⊗W (k) Re , (tα )α∈J ), cf. also Subsubsection 5.1.6. ∼ ee , ΦN ⊗Φk , ∇N , (tH α )α∈J ) → ee , ΦN ⊗Φk , ∇N , (tHα )α∈J ) Let f˜ : (N1 ⊗Re R (N ⊗Re R 1 1 1 1 ee defined by m∗ (D) of N1 ⊗Re R be an isomorphism that takes the Hodge filtration F H ee 1R ee , cf. Theorem 5.2. Let ΦN N be the Φk -linear endomorphism of onto F 1 ⊗R R Re e 1 HomRe (N1 , N )[ p1 ] such that for x ∈ HomRe (N1 , N ) and y ∈ N1 we have ΦN1 N (x)(ΦN1 (y)) = ee ) ⊆ Re (see Subsection 2.2) we get that ΦN (x(y)) ∈ N . As Φk (R ee ) ∩ HomR (N1 , N )[ 1 ] = HomR (N1 , N ). f˜ = (ΦN1 N ⊗ Φk )(f˜) ∈ (HomRe (N1 , N ) ⊗Re R e e p ee of an isomorphism f between (N1 , ΦN , ∇N , (tH α )α∈J ) and Thus f˜ is the extension to R 1 1 1 (N, ΦN , ∇N , (tHα )α∈J ). As f respects also the Verschiebung maps (of ΦN1 and ΦN ), f ∼ defines naturally an isomorphism fV /pV : D(H1V /pV ) → D(HV /pV ). e As V /pV = k[X]/(X ), HV /pV is uniquely determined by its Dieudonné crystal D(HV /pV ) (cf. [BM, Rm. 4.3.2 (i)]). Thus there exists a unique isomorphism hV /pV : ∼ ee , H1V /pV → HV /pV such that D(hV /pV ) = fV /pV . As p > 2 and f˜(F 1 ) = FR1 e ⊗Re R ee 1R ∼ the isomorphism hV /pV lifts naturally to an isomorphism hV : H1 → H (cf. Grothendieck– Messing deformation theory). Thus we can identify (N, (tHα )α∈J ) with (N1 , (tH1 α )α∈J ) = (M ⊗W (k) Re , (tα )α∈J ) and (H 1 (H), (vHα )α∈J ) with (H 1 (DKQ ), (Vα )α∈J ) = (H 1 (D), (vα )α∈J ). But we have ∼ (M, (tα )α∈J ) → (H 1 (D)⊗Zp W (k), (vα )α∈J ), cf. the Main Theorem applied to (D, (tα )α∈J ). From the last two sentences we get the first part of the Theorem. The second part of the Theorem is a direct consequence of the first part. 5.4. Remark. We refer to Theorem 5.2 with p > 2. Let nH : Q → Re be a W (k)homomorphism such that following diagram is commutative n Q −−−H−→   mH y Re  e yV ẽV ee −−− R −→ V ; here mH is as in Theorem 5.2 and eV and ẽV are as in Subsection 2.2. Let HRe be the pull back of D (of Theorem 3.4.1 (b)) via the morphism Spec(Re ) → Spec(Q) defined by nH . Let (F0i (T(N )))i∈Z be the filtration of T(N ) defined by the direct summand of N that is the Hodge filtration of HRe . The p-divisible group HRe over Spec(Re ) lifts H, cf. proof of 50 Theorem 5.3 and the commutativity of the above diagram. For α ∈ J, the tensor tHα is the extension of tα ∈ F 0 (T(M )) ⊗W (k) Q[ p1 ] via nH and therefore we have tHα ∈ F00 (T(N ))[ p1 ]. Thus indeed the condition 5.1.4 (L) is for p > 2 a sufficient condition to lift (H, (tHα )α∈J ) to an analogue pair (HRe , (tHα )α∈J ) over Spec(Re ). 5.5. Counterexample. Suppose that e > 1, that k = k̄, and that there exists an embedding V ֒→ C. Let d ∈ N. Let AV be an abelian scheme over AV which has complex multiplication and relative dimension d. Let A be the special fibre of AV . We consider a semisimple, commutative Q–subalgebra B of End(AV ) ⊗Z Q of dimension 2d. There exist √ examples in which d = 1, A is a supersingular elliptic curve, B := Q( −p), and e = 2. As B is a products of number fields, we can speak about the ring of integers OB of B. Let ŌB be the largest subring of OB such that AV has complex multiplication by ŌB . By replacing AV with an abelian scheme isogenous to it, we can assume that ŌB = Z + pOB and that the ŌB ⊗Z Zp -module H 1 (D) is isomorphic to ŌB ⊗Z Zp (this operation might enlarge V and e). Let {tHα |α ∈ J} be the set of tensors of T(N )[ p1 ] ´ that are crystalline realizations of Hodge cycles on the generic fibre of AV . Let Get Qp be the subgroup of GL H 1 (H)[ p1 ] that fixes vHα for all α ∈ J. The group schemes GB(k) , ´ GRe [ p1 ] , and Get Qp are forms of the Mumford–Tate group of AV ×Spec(V ) Spec(C) (see [De2, Sect. 3]) and therefore (as AV ×Spec(V ) Spec(C) has complex multiplication) are tori. The set {tHα |α ∈ J} includes the endomorphisms of N [ p1 ] that are crystalline realizations of elements of ŌB . Thus ŌB ⊗Z Zp is a Zp -subalgebra of EndZp (H 1 (H)) which as a Zp submodule is a direct summand. (a) We show that the assumption that either the condition 5.1.3 (i) or the condition 5.1.3 (ii) holds leads to a contradiction. We can assume that the condition 5.1.3 (ii) ∼ holds i.e., there exists an isomorphism ρ : (M, (tα )α∈J ) → (H 1 (H) ⊗Zp W (k), (vHα )α∈J ). ´ We write ρφρ−1 = t(1H 1 (H) ⊗ σ), where t ∈ Get Qp (B(k)). Thus t is an endomorphism of H 1 (H) ⊗Zp W (k) whose Hodge slopes are 1 and 0 with the same multiplicity d and which centralizes ŌB . Thus we have t ∈ ŌB ⊗Z W (k) = W (k)1M + pOB ⊗Z W (k) and det(t) ∈ pd Gm (W (k)). But as the ŌB ⊗Z Zp -module H 1 (D) is isomorphic to ŌB ⊗Z Zp , the determinant of t is either a unit or divisible by p2d . Contradiction. (b) We assume that d = 1 and that the elliptic curve A is supersingular. Then G is the group scheme of invertible elements of the W (k)-algebra End(M )∩Lie(GB(k) ) and thus it is a smooth, affine group scheme over W (k). We look at the ŌB -module M . The element √ X := p −p of ŌB is an endomorphism of (M, φ) such that X 2 = −p3 . Using a W (k)-basis {e1 , e2 } for M such that φ(e1 ) = e2 and φ(e2 ) = pe1 , it is easy to see that the endomorphism X of (M, φ) must be divisible by p. Therefore we have inclusions ŌB ֒→ OB ֒→ End(M, φ). As OB ⊗Z W (k) is a discrete valuation ring, the OB ⊗Z W (k)-module M is free of rank 1. Thus the ŌB -module M/pM is isomorphic to OB /pOB ⊗Fp k. But we have functorial 1 identifications M/pM = HdR (H/V ) ⊗V k = (N/F 1 (Re )N ) ⊗V k = N ⊗Re k. 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Adrian Vasiu Department of Mathematical Sciences, Binghamton University Binghamton, New York 13902-6000, U.S.A. e-mail: adrian@math.binghamton.edu fax: 1-607-777-2450 54 View publication stats

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