Crystalline Boundedness Principle Adrian Vasiu, U of Arizona 04/17/02; refinements 5/27/03; revisions 5/25/04 and 5/27/05; final version 12/15/05 6 0 Abstract. We prove that an F-crystal (M,ϕ) over an algebraically closed field k of 0 characteristic p > 0 is determined by (M,ϕ) mod pn, where n ≥ 1 depends only on the 2 rank of M and on the greatest Hodge slope of (M,ϕ). We also extend this result to n a triples (M,ϕ,G), where G is a flat, closed subgroup scheme of GLM whose generic fibre J is connected and has a Lie algebra normalized by ϕ. We get two purity results. If C 4 is an F-crystal over a reduced F -scheme S, then each stratum of the Newton polygon p ] stratification of S defined by C, is an affine S-scheme (a weaker result was known before T for S noetherian). The locally closed subscheme of the Mumford scheme A defined N d,1,Nk by the isomorphismclass ofa principallyquasi-polarizedp-divisiblegroup overk of height . h 2d, is an affine A -scheme. t d,1,Nk a m R´esum´e. Nous prouvons qu’un F-cristal (M,ϕ) d´efini sur un corps k alg´ebriquement [ closdecaract´eristiquep > 0estd´etermin´epar(M,ϕ)modpn, ou` n ≥ 1d´ependseulement 5 du rang de M et de la plus grand pente de Hodge de (M,ϕ). On ´etend ce r´esultat aux v 9 triplets (M,ϕ,G), ou` G est un sous-groupe ferm´e et plat de GLM dont la fibre g´en´erique 9 est connexe et a une alg`ebre de Lie normalis´ee par ϕ. Nous obtenons deux r´esultats 1 de puret´e. Si C est un F-cristal sur un F -sch´ema r´eduit S, alors chaque strate de la 5 p 0 stratificationdu polygonede Newtonde S d´efini par Cest un S-sch´ema affine (un r´esultat 2 moins g´en´eral ´etait d´eja` connu pour S noeth´erien). Le sous-sch´ema localement ferm´e du 0 / sch´ema de Mumford A d´efini par la classe d’isomorphisme d’un groupe p-divisible h d,1,Nk at principalement quasi-polaris´e sur k de hauteur 2d est un Ad,1,Nk-sch´ema affine. m Key words: F-crystals, schemes, affine group schemes, abelian schemes, p-divisible : v groups, Newton polygons, and stratifications. i X MSC 2000: 11G10, 11G18, 14F30, 14G35, and 20G25. r a §1 Introduction ¯ Let p ∈ N be a prime. Let k be a perfect field of characteristic p. Let k be an algebraic closure of k. Let W(k) be the Witt ring of k. Let B(k) := W(k)[1] be the field p of fractions of W(k). Let σ := σ be the Frobenius automorphism of k, W(k), and B(k). k A group scheme H over Spec(W(k)) is called integral if H is flat over Spec(W(k)) and H is connected (i.e. if the scheme H is integral). Let Lie(H ) be the Lie algebra B(k) B(k) over B(k) of H . If H is smooth over Spec(W(k)), let Lie(H) be the Lie algebra over B(k) W(k) of H. If O is a free module of finite rank over some commutative Z-algebra R, let GL be the group scheme over Spec(R) of linear automorphisms of O. O 1 ¯ Let (r,d) ∈ N×(N∪{0}), with r ≥ d. Let D be a p-divisible group over Spec(k) of height r and dimension d. It is well known that if d ∈ {0,1,r−1,r}, then: (*) D is uniquely determined up to isomorphism by its p-torsion subgroup scheme D[p]. But (*) does not hold if 2 ≤ d ≤ r − 2. In 1963 Manin published an analogue of (*) for 2 ≤ d ≤ r −2 but unfortunately he separated it into three parts (see [28, p. 44, 3.6, and 3.8] and below). Only recently, this paper and [36] contain explicit analogues of (*) for 2 ≤ d ≤ r − 2. The two main reasons for this delay in the literature are: (i) the widely spread opinion, which goes back more than 40 years, that p-divisible groups involve an infinite process, and (ii) the classification results of [28, p. 44] were rarely used. Our point of view is that F-crystals in locally free sheaves of finite rank over many Spec(k)-schemes Y involve a bounded infinite process. In this paper we give meaning to ¯ this point of view for the case Y = Spec(k). We start with few definitions. 1.1. Definitions. (a) By a latticed F-isocrystal with a group over k we mean a triple (M,ϕ,G), where M is a free W(k)-module of finite rank, where ϕ is a σ-linear automorphism of M[1], and where G is an integral, closed subgroup scheme of GL , p M such thattheLiesubalgebra Lie(G )ofEnd(M[1])isnormalizedbyϕ. Herewedenote B(k) p also by ϕ the σ-linear (algebra) automorphism of End(M[1]) that takes e ∈ End(M[1]) p p into ϕ ◦ e ◦ ϕ−1 ∈ End(M[1]). If G = GL , then often we do not mention G and we p M omit “with a group”. (b) By anisomorphism between two latticed F-isocrystalswitha group (M ,ϕ ,G ) 1 1 1 and (M ,ϕ ,G ) over k we mean a W(k)-linear isomorphism f : M →˜M such that 2 2 2 1 2 ϕ ◦f = f ◦ϕ and the isomorphism GL →˜GL induced by f, takes G onto G . 2 1 M1 M2 1 2 The pair (M[1],ϕ) is called an F-isocrystal over k. If we have pM ⊆ ϕ(M) ⊆ M, p then the pair (M,ϕ) is called a Dieudonn´e module over k. For g′ ∈ G(B(k)) let g′ϕ be the σ-linear automorphism of M[1] that takes x ∈ M[1] into g′(ϕ(x)) ∈ M[1]. The triple p p p (M,g′ϕ,G) is also a latticed F-isocrystal with a group over k. Often thereexistsa“good”classMofmotivesoverk thathasthefollowingproperty. The crystalline realization of any motive M in M is naturally identified with (M,g ϕ) M for some g ∈ G(W(k)) and moreover G is the identity component of the subgroup M B(k) of GL that fixes some tensors of the tensor algebra of M[1] ⊕ Hom(M[1],B(k)) M[1] p p p which do not depend on M and which are (expected to be) crystalline realizations of motives over k that are intrinsically associated to M. For instance, see [40, §5 and §6] for contexts that pertain to classes of H1 motives of abelian varieties over Spec(k) which are associated to k-valued points of a (fixed) good integral model of a Shimura variety of Hodge type. The paper [40] and many previous ones (like [25]) deal with particular cases of such triples (M,ϕ,G)’s: the pair (M,ϕ) is a Dieudonn´e module over k, the group scheme G is reductive, and there exists a semisimple element s ∈ G(B(k)) whose ϕ eigenvalues are 1 and p and such that ϕs−1 is a σ-linear automorphism of M. Any ϕ good classification of the triples (M,g ϕ,G) up to isomorphisms defined by elements of M G(W(k)), is often an important tool toward the classification of motives in M. 2 Classically, one approaches the classification of all triples (M,gϕ,G) with g ∈ G(W(k)), up to isomorphisms defined by elements of G(W(k)), in two steps. The first step aims to classify (M[1],gϕ,G )’s up to isomorphisms defined by elements p B(k) of G(B(k)). The second steps aims to use the first step in order to study (M,gϕ,G)’s. A systematic and general approach to the first step was started in [24], which works in the context in which the group G is reductive, k = k¯, and the pair (M[1],G ) B(k) p B(k) has a Q structure (M ,G ) with respect to which ϕ becomes g (1 ⊗σ) for some p Qp Qp ϕ MQp g ∈ G(B(k)); thus, inordertoclassify(M[1],gϕ,G )’suptoisomorphismsdefinedby ϕ p B(k) elements ofG(B(k)), oneonlyhastodescribe theimageG oftheset {gg |g ∈ G(W(k))} ϕ ϕ in the set B(G ) of σ-conjugacy classes of elements of G (B(k)) = G(B(k)). Even if Q Q p p ¯ k = k, in general such Q structures do not exist (for instance, they do not exist if the p group G is commutative and (Lie(G ),ϕ) has non-zero slopes). B(k) B(k) One can define two natural equivalence relations I and R on the set underlying ϕ ϕ the group G(W(k)) as follows. A pair (g ,g ) ∈ G(W(k))2 belongs to I (resp. to R ) if 1 2 ϕ ϕ and only if there exists g ∈ G(W(k)) (resp. g ∈ G(B(k))) such that g g ϕ = g ϕg . 12 12 12 1 2 12 The set of isomorphism classes of (M,gϕ,G)’s (up to isomorphisms defined by elements of G(W(k))) is in natural bijection to the quotient set G(W(k))/I . The quotient set ϕ G(W(k))/R is a more general version of the above type of sets G . In general, the ϕ ϕ natural surjective map G(W(k))/I ։ G(W(k))/R is not an injection and some of ϕ ϕ its fibres have the same cardinality as k. In general, one can not “recover” (M,gϕ,G) and its reductions modulo powers of p from the equivalence class [g] ∈ G(W(k))/I and ϕ from the triple (M[1],gϕ,G ). The last two sentences explain why in this paper, for p B(k) the study of the quotient set G(W(k))/I and of (reductions modulo powers of p of) ϕ (M,gϕ,G)’s, we can not appeal to the results of [24], [37], etc. In addition, the language of latticed F-isocrystals is more general and more suited for reductions modulo powers of p, for endomorphisms, for deformations, and for functorial purposes than the language of either σ-conjugacy classes or equivalence classes of I . ϕ If g , g , g ∈ G(W(k)) satisfy g g ϕ = g ϕg , it is of interest to keep track of the 1 2 12 12 1 2 12 greatest number n ∈ N ∪ {0} such that g and 1 are congruent mod pn12. As the 12 12 M relation I is not suitable for this purpose, it will not be used outside this introduction. ϕ The set {(M,gϕ,G)|g ∈ G(W(k))} is in natural bijection to G(W(k)). Any set of the form {(M,gϕ,G)|g ∈ G(W(k))} will be called a family of latticed F-isocrystals with a group over k. This paper is a starting point for general classifications of families of ¯ latticedF-isocrystalswithagroupoverk. Thefactthatsuchclassificationsareachievable is supported by the following universal principle. ¯ 1.2. Main Theorem A (Crystalline Boundedness Principle). Suppose k = k. Let (M,ϕ,G) be a latticed F-isocrystal with a group over k. Then there exists a number n ∈ N∪{0} that is effectively bounded from above and that has the property that for fam any pair (g,g ) ∈ G(W(k))2 such that g is congruent mod pnfam to 1 , there exist n n M fam fam isomorphisms between (M,gϕ,G) and (M,g gϕ,G) which are elements of G(W(k)). n fam Thus the equivalence class [g] ∈ G(W(k))/I depends only on g mod pnfam; this ϕ 3 supports our bounded infinite process point of view. If G = GL and (M,ϕ) is a M Dieudonn´e module over k, then Main Theorem A is a direct consequence of [28, p. 44, 3.6, and 3.8]. By a classical theorem of Dieudonn´e (see [7, Thms. 3 and 5], [28, §2], [5, Ch. IV, §4], or [14, Ch. III, §6]), the category of p-divisible groups over Spec(k) is antiequivalent to the category of Dieudonn´e modules over k. Thus we get a new proof of the following result which in essence is due to Manin and which is also contained in [36]. 1.3. Corollary. There exists a smallest number T(r,d) ∈ N ∪ {0} such that any ¯ p-divisible group D over Spec(k) of height r and dimension d, is uniquely determined up to isomorphism by its pT(r,d)-torsion subgroup scheme D[pT(r,d)]. Upper bounds of T(r,d) are effectively computable in terms of r. 1.4. On the proof of Main Theorem A. The proof of Main Theorem A (see 3.1) relies on what we call the stairs method. The method is rooted on the simple fact that for any t ∈ N and every y,z ∈ End(M), the two automorphisms 1 + pty and M 1 +ptz of M commute mod p2t. To outline the method, we assume in this paragraph M that G is smooth over Spec(W(k)). Let m ∈ N∪{0} be the smallest number for which there exists a W(k)-submodule E of Lie(G) that contains pm(Lie(G)) and that has a W(k)-basis {e ,e ,...,e } such that for l ∈ {1,...,v} we have ϕ(e ) = pnle , where π 1 2 v l π(l) is a permutation of the set {1,...,v} and where n ’s are integers that have the following l stairs property. For any cycle (l ,...,l ) of π, the integers n ,...,n are either all non- 1 q l1 lq negative or all non-positive. The existence of m is implied by Dieudonn´e’s classification of F-isocrystals over k (see [28, §2]). In general, the W(k)-submodule E is not a Lie subalgebra of Lie(G). For any g˜ ∈ G(W(k)) congruent mod p2m+t to 1 , there exists M e˜ ∈ E such that the elements g˜ and 1 + pm+te˜ of GL (W(k)) are congruent mod M M p2m+1+t. Due to this and the stairs property, for p ≥ 3 there exists an isomorphism between (M,g˜ϕ,G) and (M,ϕ,G) which is an element g˜ ∈ G(W(k)) congruent mod 0 pm+t to1 (see 3.1.1). Ifp = 2, thena slight variantofthisholds. Exponentialmaps(see M 2.6) substitute from many points of view the classical Verschiebung maps of Dieudonn´e modules; for instance, one can choose g˜ to be an infinite product of exponential elements 0 of the form ∞ pi(m+1)ei, where e ∈ E. See 2.2 to 2.4 for the σ-linear preliminaries that Pi=0 i! are necessary for the estimates which give us the effectiveness part of Main Theorem A. These estimates provide inductively upper bounds of m in terms of dim(G ) and of B(k) the s-number and the h-number of the latticed F-isocrystal (Lie(G),ϕ) over k (see 2.2.1 (e) for these two non-negative integers which do not change if ϕ is replaced by gϕ). 1.5. Complements, examples, and applications. See 3.2 for interpretations and variants of Main Theorem A in terms of reductions modulo powers of p; in particular, see 3.2.4 for the passage from Main Theorem A to Corollary 1.3. In 3.3 we improve (in many cases of interest) the upper bounds (of n , etc.) we obtain in 3.1.1 to 3.1.5. fam In §4 we include four examples. It is well known that if the p-divisible group D is ordinary, then D is uniquely determined up to isomorphism by D[p] and moreover D has ¯ a unique lift to Spec(W(k)) (called the canonical lift) that has the property that any endomorphism of D lifts to it. Example 2 identifies the type of latticed F-isocrystals 4 ¯ with a group over k to which the last two facts generalize naturally (see 4.3.1 and 4.3.2). Example 4 shows that if r = 2d, d ≥ 3, and the slopes of the Newton polygon of D are 1 and d−1, then D is uniquely determined up to isomorphism by D[p3] (see 4.5). d d In §5 we list four direct applications of Main Theorem A and of 3.2. First we present the homomorphism form of Main Theorem A (see 5.1.1). Second we define transcendental ¯ degrees of definition for many classes of latticed F-isocrystals with a group over k (see 5.2). When the transcendental degrees of definition are 0, we also define (finite) fields of definition. In particular, Theorem 5.2.3 (when combined with Lemma 3.2.2) implies that it is possible to build up an atlas and a list of tables of isomorphism classes of p-divisible ¯ groups (endowed with certain extra structures) over Spec(k) that are definable over the spectrum of a fixed finite field F , which are similarin nature to the atlas of finite groups pq (see [3]) and to the list of tables of elliptic curves over Spec(Q) (see [4]). Let N ∈ N \ {1,2} be relatively prime to p. Let A be the smooth, quasi- d,1,N projective Mumford moduli scheme over Spec(F ) that parametrizes isomorphism classes p of principally polarized abelian schemes with level-N structure and of relative dimension d over Spec(F )-schemes (see [33, Thms. 7.9 and 7.10]). Third we apply the principally p quasi-polarized version of Corollary 1.3 (see 3.2.5) to get a new type of stratification of A . Here the word stratification is used in a wide sense (see 2.1.1) which allows the d,1,N number of strata to be infinite. The strata we get are defined by isomorphism classes of principally quasi-polarized p-divisible groups of height 2d over spectra of algebraically closed fields of characteristic p; they areregular andequidimensional (see 5.3.1and 5.3.2). Moreover, this new type of stratification of A satisfies the purity property we define d,1,N in 2.1.1, i.e. its strata are affine A -schemes (see 5.3.1 and 5.3.2). Variants of 1.3, d,1,N 3.2.5, 3.2.6, and 5.3.2 but without its purity property part, are also contained in [36]. Fourth we get a new proof (see 5.4) of the “Katz open part” of the Grothendieck– Katz specialization theorem for Newton polygons (see [22, 2.3.1 and 2.3.2]). The main goal of §6 is to prove the following result (see 6.1 and 6.2). 1.6. Main Theorem B. Let C be an F-crystal in locally free sheaves of finite rank over a reduced Spec(F )-scheme S. Then the Newton polygon stratification of S defined p by C satisfies the purity property (i.e. each stratum of it is an affine S-scheme). A variant of Main Theorem B was obtained first in [10, 4.1], for the particular case when S is locally noetherian. The fact that the variant is a weaker form of Main Theorem B is explained in 6.3 (a). The main new idea of §6 is: Newton polygons are encoded in the existence of suitable morphisms between different evaluations of F-crystals (viewed without connections) at Witt schemes of (effectively computable) finite lengths. The proof of Main Theorem B combines this new idea with the results of Katz (see [22, 2.6 and 2.7]) on isogenies between F-crystals of constant Newton polygons over spectra of ¯ (perfections of) complete, discrete valuation rings that are of the form k[[x]]. Acknowledgements. We would like to thank U. of Utah and U. of Arizona for good conditions in which to write this paper and D. Ulmer, G. Faltings, and the referee for valuable comments. 5 §2 Preliminaries See 2.1 forour mainnotationsandconventions. See 2.2 forfew definitions and simple propertiesthatpertaintolatticedF-isocrystalswithagroupoverk. Inparticular,in2.2.2 wedefine Dieudonn´e–Fontaine torsionsandvolumesoflatticedF-isocrystals. Inequalities andestimatesonsuchtorsionsaregatheredin2.3and2.4(respectively); theyareessential for examples and for the effectiveness part of 1.2. In 2.5 we apply [42] to get Z structures p ¯ for many classes of latticed F-isocrystals with a group over k. In 2.6 and 2.7 we include group scheme theoretical properties that are needed in §3 and §4. In 2.8 we present complements on the categories M(W (S)) we will introduce in 2.1. In 2.9 we recall two q results of commutative algebra. Subsections 2.8 and 2.9 are not used before 5.4. For Newton polygons of F-isocrystals over k we refer to [22, 1.3]. 2.1. Notations and conventions. Byw wedenoteanarbitraryvariable. Ifq ∈ N, let F be the field with pq elements. If R is a commutative F -algebra, let W(R) be the pq p Witt ring of R and let W (R) be the ring of Witt vectors of length q with coefficients q in R. We identify R = W (R). Let Φ be the canonical Frobenius endomorphism of 1 R either W(R) or W (R); we have Φ = σ = σ. Let R(pq) be R but viewed as an R- q k k q algebra via the q-th power Frobenius endomorphism Φ : R → R. If R is reduced, let R Rperf := ind.lim. R(pq) be the perfection of R. q∈N Let M(W (R)) be the abelian category whose objects are W (R)-modules endowed q q with Φ -linear endomorphisms and whose morphisms are W (R)-linear maps that re- R q spect the Φ -linear endomorphisms. We identify M(W (R)) with a full subcategory of R q M(W (R)) and thus we can define M(W(R)) := ∪ M(W (R)). q+1 q∈N q If S is a Spec(F )-scheme, in a similar way we define W (S), Φ , M(W (S)), and p q S q M(W(S)). We view W (S) as a scheme and by a W (S)-module we mean a quasi- q q coherent moduleoverthestructurering sheafO ofW (S). Theformal scheme W(S) Wq(S) q is used only as a notation. If S = Spec(R), then we identify canonically M(W (R)) = q M(W (S)) and M(W(R)) = M(W(S)). If t ∈ {1,...,q} and ∗(q) is a morphism of q M(W (S)), let ∗(t) be the morphism of M(W (S)) that is the tensorization of ∗(q) with q t W (S). Let Stop be the topological space underlying S. All crystals over S (i.e. all t crystals on Berthelot’s crystalline site CRIS(S/Spec(Z ))) are in locally free sheaves of p finite rank. An F-crystal C over S comprises from a crystal M over S and an isogeny Φ∗(M) → M of crystals over S; let hC ∈ N ∪ {0} be the smallest number such that S phC annihilates the cokernel of this isogeny. We identify an F-crystal (resp. an F- isocrystal) over Spec(k) with a latticed F-isocrystal (M,ϕ) over k that has the property that ϕ(M) ⊆ M (resp. with an F-isocrystal over k as defined in §1). The pulls back of F-crystals C and C over S to an S-scheme S (resp. to an affine S-scheme Spec(R )) ∗ 1 1 are denoted by C and C (resp. by C and C ). S1 ∗S1 R1 ∗R1 Let (M,ϕ,G) be a latticed F-isocrystal with a group over k. We refer to M as its W(k)-module. Letr ∈ N∪{0}betherankofM. Iff andf aretwoZ-endomorphisms M 1 2 of either M or M[1], let f f := f ◦ f . Two Z-endomorphisms of M are said to be p 1 2 1 2 6 congruent mod pq if their reductions mod pq coincide. Let M∗ := Hom(M,W(k)). Let T(M) := ⊕ M⊗t ⊗ M∗⊗u. t,u∈N∪{0} W(k) We denote also by ϕ the σ-linear automorphism of T(M)[1] that takes f ∈ M∗[1] into p p σfϕ−1 ∈ M∗[1] and that acts on T (M)[1] in the natural tensor product way. The p p canonical identification End(M[1]) = M[1]⊗ M∗[1] is compatible with the ϕ actions p p B(k) p (see 1.1 (a) for the action of ϕ on End(M[1])). If O is either a free W(k)-submodule or a p B(k)-vector subspace of T (M)[1] such that ϕ(M) ⊆ M, then we denote also by ϕ the σ- p linear endomorphism of O induced by ϕ. The W(k)-span of tensors v ,...,v ∈ T (M)[1] 1 n p is denoted by < v ,...,v >. The latticed F-isocrystal (M∗,ϕ) over k is called the dual 1 n of (M,ϕ). We emphasize that the pair (M∗,ϕ) involves no Tate twist. A bilinear form on M is called perfect if it defines naturally a W(k)-linear isomorphism M→˜ M∗. Let G˜ be a connected subgroup of GL . As ϕ is a σ-linear automorphism B(k) M[1] p of M[1], the group {ϕg˜ϕ−1|g˜ ∈ G˜ (B(k))} is the group of B(k)-valued points of the p B(k) unique connected subgroup of GL that has ϕ(Lie(G˜ )) as its Lie algebra (see [1, M[1] B(k) p Ch. II, 7.1] for the uniqueness part). So as ϕ normalizes Lie(G ), for g ∈ G(B(k)) we B(k) have ϕgϕ−1 ∈ G(B(k)); in what follows this fact is used without any extra comment. In this paragraph we assume ϕ(M) ⊆ M. We also refer to (M,ϕ,G) as an F- crystal with a group over k. The Hodge slopes of (M,ϕ) (see [22, 1.2]) are the non- negative integers h ,...,h such that the torsion W(k)-module M/ϕ(M) is isomorphic 1 rM to ⊕rM W(k)/(phi). If O is a W(k)-submodule of M such that ϕ(O) ⊆ O, we denote i=1 also by ϕ the σ-linear endomorphism of M/O induced by ϕ. We refer to the triple (M/pqM,ϕ,G ) as the reduction mod pq of (M,ϕ,G). If G = GL , then often we Wq(k) M do not mention G and G and we omit “with a group”. The reduction (M/pqM,ϕ) Wq(k) mod pq of (M,ϕ) is an object of M(W (k)). q If a, b ∈ Z with b ≥ a, let S(a,b) := {a,a+1,...,b}. If l ∈ N, if ∗ is a small letter, and if (∗ ,...,∗ ) is an l-tuple which is either an element of Zl or an ordered W(k)-basis 1 l of some W(k)-module, then we define ∗ for any t ∈ Z via the rule: ∗ := ∗ , where t t u u ∈ {1,...,l}∩(t+lZ). If x ∈ R, let [x] be the greatest integer of the interval (−∞,x]. 2.1.1. Conventions on stratifications. Let K be a field. By a stratification S of a reduced Spec(K)-scheme X (in potentially an infinite number of strata), we mean that: (i) for any field L that is either K or an algebraically closed field that contains K, a set S of disjoint reduced, locally closed subschemes of X is given such that each point L L of X with values in an algebraic closure of L factors through some element of S ; L L (ii) if i : L ֒→ L is an embedding between two fields as in (a), then the reduced 12 1 2 scheme of the pull back to L of any member of S , is an element of S ; so we have a 2 L1 L2 natural pull back injective map S(i ) : S ֒→ S . 12 L1 L2 If the inductive limit of all maps S(i ) exists (resp. does not exist) in the category 12 of sets, then we say that the stratification S has a class which is (resp. is not) a set. 7 Each element of some set S is referred as a stratum of S. We say S satisfies the purity L property if for any field L as in (a), every element of S is an affine X -scheme.1 Thus S L L satisfies the purity property if and only if each stratum of it is an affine X-scheme. If all maps S(i )’s are bijections, then we identify S with S and we say S is of finite type. 12 K 2.2. Definitions and simple properties. In this Subsection we introduce few notions and simple properties that pertain naturally to latticed F-isocrystals. 2.2.1. Complements to 1.1. (a) A morphism (resp. an isogeny) between two latticed F-isocrystals (M ,ϕ ) and (M ,ϕ ) over k is a W(k)-linear map (resp. isomor- 1 1 2 2 phism)f : M [1] → M [1]such thatfϕ = ϕ f andf(M ) ⊆ M . Iff isanisogeny, then 1 p 2 p 1 2 1 2 by its degree we mean pl, where l is the length of the artinian W(k)-module M /f(M ). 2 1 (b) By a latticed F-isocrystal with a group and an emphasized family of tensors over k we mean a quadruple (M,ϕ,G,(t ) ), α α∈J where (M,ϕ,G) is a latticed F-isocrystal with a group over k, where J is a set of in- dices, and where t ∈ T (M) is a tensor that is fixed by both ϕ and G, such that G α B(k) is the subgroup of GL that fixes t for all α ∈ J. If (M ,ϕ ,G ,(t ) ) and M[1] α 1 1 1 1α α∈J p (M ,ϕ ,G ,(t ) ) are two latticed F-isocrystals with a group and an emphasized 2 2 2 2α α∈J family of tensors (indexed by the same set J) over k, by an isomorphism between them we mean an isomorphism f : (M ,ϕ ,G )→˜(M ,ϕ ,G ) such that the W(k)-linear iso- 1 1 1 2 2 2 morphism T(M )→˜ T(M ) induced by f, takes t into t for all α ∈ J. 1 2 1α 2α (c) By a principal bilinear quasi-polarized latticed F-isocrystal with a group over k we mean a quadruple (M,ϕ,G,λ ), where (M,ϕ,G) is a latticed F-isocrystal with a M group over k and where λ : M ⊗ M → W(k) is a perfect bilinear form with the M W(k) properties that the W(k)-span of λ is normalized by G and that there exists c ∈ Z M such that we have λ (ϕ(x),ϕ(y)) = pcσ(λ (x,y)) for all x, y ∈ M. We refer to λ as a M M M principal bilinear quasi-polarization of (M,ϕ,G), (M,ϕ), and (M[1],ϕ). Let G0 be the p Zariski closure inGL of the identity component of the subgroup of G that fixes λ . M B(k) M We refer to (M,ϕ,G0) as the latticed F-isocrystal with a group over k of (M,ϕ,G,λ ). M The quotient group G /G0 is either trivial or isomorphic to G . B(k) B(k) m By an isomorphism between two principal bilinear quasi-polarized latticed F-iso- crystals with a group (M ,ϕ ,G ,λ ) and (M ,ϕ ,G ,λ ) over k we mean an iso- 1 1 1 M1 2 2 2 M2 morphism f : (M ,ϕ ,G )→˜(M ,ϕ ,G ) such that we have λ (x,y) = λ (f(x),f(y)) 1 1 1 2 2 2 M1 M2 for all x, y ∈ M . We speak also about principal bilinear quasi-polarized latticed F- 1 isocrystals with a group and an emphasized family of tensors over k and about isomor- phisms between them; notation (M,ϕ,G,(t ) ,λ ). α α∈J M Iftheformλ isalternating,wedropthewordbilinear(i.e. wespeakaboutprincipal M quasi-polarized latticed F-isocrystals with a group over k, etc.). 1 This is a more practical, refined, and general definition than any other one that relies on codimension 1 statements on complements. See Remark 6.3 (a) below. 8 (d)WesaytheW-conditionholdsforthelatticedF-isocrystalwithagroup(M,ϕ,G) over k if there exists a direct sum decomposition M = ⊕b F˜i(M), where a, b ∈ Z with i=a b ≥ a, such that M = ⊕b ϕ(p−iF˜i(M)) and the cocharacter µ : G → GL defined by i=a m M the property that β ∈ G (W(k)) acts on F˜i(M) through µ as the multiplication by β−i, m factors through G. In such a case we also refer to (M,ϕ,G) as a p-divisible object with a group over k. We refer to the factorization µ : G → G of µ as a Hodge cocharacter m of (M,ϕ,G). For i ∈ S(a,b) let Fi(M) := ⊕i F˜j(M). We refer to the decreasing and j=b exhaustive filtration (Fi(M)) of M as a lift of (M,ϕ,G). If G = GL , we also i∈S(a,b) M refer to (M,ϕ) as a p-divisible object over k. Here “W” stands to honor [42, p. 512] while the notion “p-divisible object” is a natural extrapolation of the terminology “object” introduced in [11, §2]. (e) By the shifting number (to be abbreviated as the s-number) of a latticed F- isocrystal(M,ϕ)overk wemeanthe smallestnumber s ∈ N∪{0}such that ϕ(psM) ⊆ M (equivalently such that ϕ(M) ⊆ p−sM). By the greatest Hodge slope (to be abbreviated as the h-number) of (M,ϕ) we mean the greatest Hodge slope h of (M,psϕ), i.e. the unique number h ∈ N∪{0} such that we have ph−sM ⊆ ϕ(M) and ph−s−1M 6⊆ ϕ(M). We have s = 0 if and only if (M,ϕ) is an F-crystal over k; in this case h is the number h defined in 2.1. We have s = 0 and h ∈ {0,1} if and only if (M,ϕ) is a (M,ϕ) Dieudonn´e module over k. Let s∗ and h∗ be the s-number and the h-number (respectively) of (M∗,ϕ). We have ϕ(M∗) = ϕ(M)∗ ⊆ ps−hM∗ but ϕ(M∗) 6⊆ ps−h+1M∗. Thus s∗ = max{0,h − s}. As (M,ϕ) is the dual of (M∗,ϕ), we also have s = max{0,h∗−s∗}. So if s = 0, then s∗ = h and h∗ ∈ S(0,h). If s > 0, then s = h∗ −s∗ and thus h∗ = s+s∗ = max(s,h). If s = 0, then the s-number and the h-number of (End(M),ϕ) = (M,ϕ)⊗(M∗,ϕ) are at most s+s∗ = h and h+h∗ ≤ 2h (respectively). 2.2.2. Definitions. (a) Let (M,ϕ) be a p-divisible object (M,ϕ) over k. We say (M,ϕ) is a cyclic Dieudonn´e–Fontaine p-divisible object over k if there exists a W(k)- basis {e ,...,e } of M such that for i ∈ S(1,r ) we have an identity ϕ(e ) = pnie , 1 rM M i i+1 where n ,...,n are integers that are either all non-negative or all non-positive. We 1 rM refer to {e ,...,e } as a standard W(k)-basis of (M,ϕ). 1 rM We say (M,ϕ) is an elementary Dieudonn´e–Fontaine p-divisible object over k if it is a cyclic Dieudonn´e–Fontaine p-divisible object over k that is not the direct sum of two or more non-trivial cyclic Dieudonn´e–Fontaine p-divisible objects over k. We say (M,ϕ) is an elementary Dieudonn´e p-divisible object over k if there exists a W(k)-basis{e ,...,e }ofM suchthatfori ∈ S(2,r )wehaveanidentityϕ(e ) = e 1 rM M i i+1 and moreover ϕ(e ) = pn1e for some integer n that is relatively prime to r . 1 2 1 M We say (M,ϕ) is a Dieudonn´e–Fontaine (resp. a Dieudonn´e) p-divisible object over k if it is a direct sum of elementary Dieudonn´e–Fontaine (resp. of elementary Dieudonn´e) p-divisible objects over k. (b) By the Dieudonn´e–Fontaine torsion (resp. volume) of a latticed F-isocrystal 9 (M,ϕ) over k we mean the smallest number T(M,ϕ) ∈ N∪{0} (resp. V(M,ϕ) ∈ N∪{0}) such that there exists a Dieudonn´e–Fontaine p-divisible object ¯ ¯ (M ,ϕ )overk forwhichwehaveanisogeny f : (M ,ϕ ) ֒→ (M⊗ W(k),ϕ⊗σ )with 1 1 1 1 W(k) k¯ the property that pT(M,ϕ)M ⊆ f(M ) (resp. that M/f(M ) has length V(M,ϕ)). By 1 1 replacing Dieudonn´e–Fontaine with Dieudonn´e, in a similar way we define the Dieudonn´e torsion T (M,ϕ) ∈ N∪{0} and the Dieudonn´e volume V (M,ϕ) ∈ N∪{0} of (M,ϕ). + + 2.2.2.1. Remarks. (a) Any (elementary) Dieudonn´e p-divisible object over k is also an (elementary) Dieudonn´e–Fontaine p-divisible object over k. Moreover, any Dieudonn´e–Fontaine p-divisible object over k is definable over F . p (b) The existence of V (M,ϕ) (and thus also of V(M,ϕ), T (M,ϕ), and T(M,ϕ)) + + ¯ is equivalent to Dieudonn´e’s classification of F-isocrystals over k. This and the fact that suitable reductions (modulo powers of p) of p-divisible objects over k are studied systematically for the first time in [14] and [15], explains our terminology. (c) Classically one works only with Dieudonn´e p-divisible objects (as they are uniquely determined by their Newton polygons) and with Dieudonn´e volumes (as they keeptrackofdegreesofisogenies); see[7],[28],[6],[10],etc. ButworkingwithDieudonn´e– Fontaine p-divisible objects and torsions one can get considerable improvements for many practical calculations or upper bounds (like the ones we will encounter in §3). 2.2.3. Lemma. Let K be an algebraically closed field that contains k. Let (M,ϕ) be a Dieudonn´e–Fontaine p-divisible object over k with the property that ϕ(M) ⊆ M. Let r2 h be the h-number of (M,ϕ), let e := max{r ,[ M]}, let k be the composite field of k M M 4 1 and F , and let m ∈ N. We have the following two properties: prM! (a) For any endomorphism f of (M ⊗ W (K),ϕ⊗σ ), the reduc- heM+m W(k) heM+m K tion f mod pm of f is the scalar extension of an endomorphism of (M ⊗ m heM+m W(k) W (k ),ϕ⊗σ ). If (M,ϕ) is a Dieudonn´e p-divisible object over k, then the previous m 1 k1 sentence holds with e being substituted by r . M M (b) Each endomorphism of (M ⊗ W(K),ϕ⊗σ ) is the scalar extension of an W(k) K endomorphism of (M ⊗ W(k ),ϕ⊗σ ). W(k) 1 k1 Proof: We write (M,ϕ) = ⊕s (M ,ϕ) as a direct sum of elementary Dieudonn´e– i=1 i (i) (i) Fontaine p-divisible objects over k. Let {e ,...,e } be a standard W(k)-basis of 1 rMi (M ,ϕ). We check that (a) holds. Let i ∈ S(1,s) and let j ∈ S(1,r ). We write i 0 0 Mi0 f (e(i0)⊗1) = s rMi e(i)⊗β(i0i), where all β(i0i)’s belong to W (K). Let heM+m j0 Pi=1Pj=1 j j0j j0j heM+m r := l.c.m.{r ,r }; it is a divisor of r !. If i = i , then r = r ≤ r ≤ e . Mi0i Mi0 Mi M 0 Mi0i Mi0 M M r2 If i 6= i , then r +r ≤ r and thus we have r ≤ r r ≤ [ M] ≤ e . 0 Mi0 Mi M Mi0i Mi0 Mi 4 M AsfheM+m(ϕrMi0i(ej(i00))⊗1) = (ϕ⊗σK)rMi0i(fheM+m(ej(i00)⊗1)), wehaveanequality (1) pmj(i00i)βj(0i0ji) = pqj(i0i)σKrMi0i(βj(0i0ji)) ∈ WheM+m(K), 10