STRATIFICATIONS OF HILBERT MODULAR VARIETIES

Transcription

1 Unspecified Journal Volume 00, Number 0, Xxxx XXXX, Pages S????-????(XX) STRATIFICATIONS OF HILBERT MODULAR VARIETIES E Z GOREN AND F OORT Abstract We consider g-dimensional abelian varieties in characteristic p, with a given action of O L - the ring of integers of a totally real field L of degree g A stratification of the associated moduli spaces is defined by considering the action of O L on a certain canonical subgroup of abelian varieties The properties of this stratification and its relation to other stratifications, eg by Newton polygons, are studied 1 Introduction There are two important stratifications of the moduli space A = A g,1 F p of principally polarized abelian varieties of dimension g in characteristic p One is given by Newton polygons The other is given by the structure of the p-torsion sub group scheme We will refer to it as the Ekedahl-Oort stratification, see [5] Similar constructions can be made for other Shimura varieties of Hodge type In this paper we consider a certain stratification of Hilbert modular varieties in positive characteristic It is a refinement of the Ekedahl-Oort stratification on these varieties We will examine its relation to the Newton polygon stratification The stratification is obtained as follows Fix a totally real field L of degree g over Q Let O L be the ring of integers of L and let p be a rational prime that is inert in L Fix an integer n greater or equal to 3 and prime to p Denote by F a finite field obtained from a field of p g elements by adjoining a primitive n-th root of unity We consider quadruples X = (X, λ, ι, α) over a field k of characteristic p containing F, consisting of an abelian variety X over k, a principal polarization λ of X, an embedding ι of O L into the endomorphisms of X fixed by the Rosati involution associated to λ, and a full symplectic level-n structure α We shall denote the moduli space of such data X by M n It is a regular, irreducible, g-dimensional variety over F It is a fine moduli scheme Given such X, the cotangent space of X is a free O L k -module of rank 1 Let us denote the embeddings of O L /p into F by {σ 1,, σ g }, ordered in such a way that Frobenius composed with σ i is σ i+1 Let V denote the Verschiebung map Consider ker(v : H 0 (Ω 1 X) H 0 (Ω 1 X)) It is a k-vector space and an O L -module on which O L /p acts by a set of characters τ, each with multiplicity one We call τ the type of X and denote it by τ(x) The stratification is obtained by the type 1991 Mathematics Subject Classification Primary 11F41, 14G35, 14K10 1 c 0000 (copyright holder)

2 2 E Z GOREN AND F OORT Our main results are as follows : For every set of characters τ {σ 1,, σ g } there exists a closed subscheme W τ of M n, which is universal with respect to the property τ(x) τ We establish the following theorems: 1 W τ is locally irreducible and is locally linear 2 The generic point of every component of W τ has type τ 3 dim(w τ ) = g τ 4 W τ W σ = W τ σ 5 The variety W 0 τ, defined as W τ \ σ τ W σ, is quasi-affine 6 The components of all the W τ s have a natural structure of a simplicial complex 7 Let E be the Hodge bundle over M n, and let W a = τ =a W τ We obtain formulae expressing W a, considered as an element of the Chow ring, as a combination of certain tautological Chern classes In particular, W 1 = (p 1)c 1 (E) 8 Let τ(x) = τ(y), then X[p] = Y [p] as principally polarized group schemes with O L -structure 9 The possible p-divisible groups up to isogeny are G l/g +G (g l)/g for 0 l g, and g G 1/2 Let β 0 < β 1 < be the possible Newton polygons 10 We say that a type is spaced if it contains no two consecutive σ i s Let λ(τ) = max { τ : τ τ, τ is spaced} (except for g odd and τ = {1,, g}, where we put λ(τ) = (g + 1)/2) The generic point of every component of W τ has Newton polygon equal to β λ(τ) 11 We define N i to be the closed subscheme where the Newton polygon is weakly above β i We conjecture that dim(n i ) is equal to g i, and prove this in special cases (leaving a more systematic discussion for a future paper) Those cases imply in particular the weak Grothendieck conjecture for principally polarized p-divisible groups with real multiplication We remark that as far as local computations are concerned, as long as p is unramified, the assumption that p is inert in L is completely technical We hope to discuss the general case in a future paper The results of this paper seem fundamental to the study of Hilbert modular forms mod p and p-adic Hilbert modular forms from a geometric point of view See [9] Acknowledgments The first named author thanks Utrecht university for its hospitality during a visit in autumn 1997 and thanks the second named author for the invitation The visit, during which this work was done, was part of the special project on Algebraic curves and Riemann surfaces, geometry and arithmetic funded by the NWO (The Dutch organization of pure research) 2 Local Structure 21 Basic definitions Fix a totally real field L of degree g over Q Let d L be the different ideal of L Let O L be the ring of integers of L and let p be a fixed rational prime that is inert in L Fix an integer n greater or equal to 3 and prime to p Denote by F a finite field obtained from a field of p g elements by adjoining a primitive n-th root of unity

3 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 3 For any ring R of characteristic p, let W (R) denote the ring of infinite Witt vectors with values in R and let σ : W (R) W (R), τ : W (R) W (R), be, respectively, the ring homomorphism Frobenius and the additive map Verschiebung Write Emb(O L, W (F)) = {σ 1,, σ g }, where σ σ i = σ i+1 We abuse notation and write also Emb(O L /p, F) = {σ 1,, σ g } We shall usually denote abelian varieties by the letters X, Y We denote by X[p] the p-torsion sub group scheme of X and by X(p) the p-divisible group We use X t to denote the dual abelian variety Let f(x) and a(x) be the usual f-number and a-number of X That is, for X defined over a field k of characteristic p, p f(x) = X[p]( k), a(x) = dim k (Hom k (α p, X[p])) We denote by α(x) the maximal α p -elementary subgroup of X The order of α(x) is p a(x) We denote by D(X(p)) the contravariant Dieudonné module of X(p) and by D(X(p)) the covariant Dieudonné module of X(p) The same notation applies to a finite commutative p-torsion group scheme G Let M n be the moduli space of quadruples X = (X, λ, ι, α)/s consisting of: A scheme S over F A principally polarized abelian scheme (X, λ) over S An embedding of rings O L End(X) λ, where End(X) λ denotes the endomorphisms of X/S, as an abelian scheme, that are fixed by the Rosati involution associated to λ An isomorphism of symplectic O L -modules α : O L /no L d 1 L /nd 1 L X[n](S) This implies that X[n]/S is a constant group scheme over S, and the last isomorphism can be interpreted as an isomorphism of constant group schemes over S We consider M n as a scheme over F It is known to be a fine moduli scheme It is a regular, irreducible, g-dimensional variety over F We denote by M the coarse moduli scheme where we take no level structure That is, the coarse moduli scheme of triples (X, λ, ι) We remark that the level structure α plays no significant role beyond enabling us to identify the universal deformation ring of X with the completion of the local ring of the corresponding point in M n Given a point t M n (S) we let X t = (X t, λ t, ι t, α t ) denote the corresponding object obtained by pulling back the universal object over M n We remark that by [4], Corollaire 29, H 0 (Ω 1 X/S ) is a locally free O L/p O S -module of rank 1

4 4 E Z GOREN AND F OORT Definition 21 Let X = (X, λ, ι, α)/k be as above with k an algebraically closed field We define the type of X, τ(x) Z/gZ, as follows : The action of O L /p on D(α(X)) is given by a(x) different characters { σ i1,, σ ia(x) } We let τ(x) = { i 1,, i a(x) } Notation Given k matrices B 1,, B k of size l l and some i such that 1 i k, we denote by d i (B 1,, B k ) the matrix A in M k (M l ), which is everywhere zero, except for A i,1 = B 1, A i+1,2 = B 2,, A i 1,k = B k For example, for i = 1 we get a diagonal matrix with the matrices B 1,, B k on the diagonal 22 Review of equi-characteristic deformation theory We quickly review some of the definitions and results of equi-characteristic deformation theory of abelian varieties and p-divisible groups Our main references are [19], [2], [22] and [25] Further references are cited there From abelian varieties to p-divisible groups Definition 22 Let S be a scheme A p-divisible group of height h (synonym: a Barsotti-Tate group of height h) over S is an inductive system of finite flat commutative group schemes over S, 0 = G 0 G 1 G n (closed immersions), such that G i is of order p hi and G i = Ker(p i : G i+1 G i+1 ) Given such a p-divisible group G one can construct its Serre dual G t, see [21], Section 76 One defines the codimension of G as the dimension of G t We have dim(g) + codim(g) = height(g) Example 23 Let X S be an abelian scheme of relative dimension g, then X(p) def = lim X[p i ], i is a p-divisible group of height 2g and dimension g Theorem 24 (Serre-Tate) Let S be a scheme such that p is locally nilpotent in O S Let I O S be a nilpotent ideal, S 0 = Spec(O S /I) and X 0 S 0 be an abelian scheme The functor { } { } X S abelian scheme G S p divisible group (21) with f : X S S 0 X0 with φ : G S S 0 X0 (p) given by X X(p) is an equivalence of categories We refer to [12], Section 121, and [15], Section V23, for more details regarding this theorem

5 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 5 Remark 25 A refined statement of Theorem 24, taking into account polarizations and endomorphisms, is valid Its formulation is clear We will usually apply this for the case where S is a characteristic p scheme, ie p = 0 on S, to study equi-characteristic deformations From p-divisible groups to Dieudonné modules Let R be a commutative ring with 1 having characteristic p At the moment R is arbitrary Later on we will be interested in particular in the case where R is a quotient of k[[t 1,, t g ]] by a prime ideal, where k is a field of characteristic p Consider the non-commutative Cartier ring C(R) = Cart p (R) as in [19] p 416 For R a perfect ring C(R) = W (R)[F ][[V ]], where W (R) is the Witt ring over R and F and V satisfy the following relations: 1 F V = V F = p; 2 F α = α σ F, V α σ = αv, V αf = α τ, α W (R) Definition 26 A Dieudonné module M over C(R) is a left C(R)-module that satisfies the following: 1 V is injective on M and M/V M is a free R-module 2 M is complete and separated in the V -adic topology We recall that a commutative formal group γ over R is a group functor { } R algebras of finite {commutative groups}, length as R modules which is represented by a formal R-scheme Γ Moreover, γ is formally smooth if for every such R-algebra R 1 and for every ideal I 1 of R 1, such that I 2 1 is the zero ideal, the natural map from γ(r 1 ) to γ(r 1 /I 1 ) is surjective Example 27 Let X Spec(R) be an abelian scheme The connected component of the p-divisible group X(p) is a commutative, formally smooth, formal group Theorem 28 There is a covariant functor, { } commutative, formally smooth, {Dieudonné C(R) modules}, formal groups over R which is an equivalence of categories, given by Γ C(Γ) Moreover, Γ is finite dimensional over R if and only if C(Γ) is finitely generated over C(R) The tangent space of Γ is canonically isomorphic with C(Γ)/V C(Γ) Consider a class of C(R)-modules M given by dividing the free object (22) n+d i=1 C(R)e i by the relations: 1 F e j = n+d i=1 α ije i, j = 1,, d 2 e j = V ( n+d i=1 α ije i ), j = d + 1,, n + d for some α ij W (R) C(R) One can prove that under certain conditions, eg R perfect and V topologically nilpotent, that this defines a Dieudonné C(R)-module Any Dieudonné C(R)- module M, where R is a local artinian ring or a complete local noetherian ring,

6 6 E Z GOREN AND F OORT whose reduction is the Dieudonné module of a p-divisible group, can be written in this form ([18], Section1) One says it is displayed In this case (α ij ) is invertible If R is perfect, M is generated as a W (R)- module by e 1, e 2g, and if R is a perfect field this is a basis for M as a W (R)-module Using the notation A B (α ij ) =, C D one finds that Frobenius is given by (A C ) pb pd The module associated to G t is given by the basis f 1,, f n+d, with relations 1 f j = V ( n+d i=1 δ ijf i ), j = 1,, n 2 F f j = n+d i=1 δ ijf i, j = n + 1,, n + d We have the relation (see [18], p504) (δ ij ) = t (α ij ) 1 A principal quasi-polarization λ : G G t of a p-divisible group G (ie, λ = λ t and λ is an isomorphism), is equivalent to an isomorphism with the same properties β : C(G) C(G t ), Studying deformations of Dieudonné modules From [18], Theorem 1 and the methods of [19], Section 1, one can deduce the following (see also [22], Section 2): Let k be a perfect field of characteristic p Let G be a p-divisible group over k of height h and dimension d Let Let R = k[[t ij : 1 i d, 1 j h d]] A B (α ij ) = C D be a matrix giving a Dieudonné module D in displayed form over C(R), as in (22), where n = h d Let T ij denote the Teichmüller lift of t ij to the Witt ring W (R) and let T = (T ij ) The matrix (A ) + T C B + T D (23) C D is giving a display for the universal deformation of G over Spec(R) Suppose further that λ is a principal quasi-polarization, thus n = d, and that e 1,, e n+d are chosen a symplectic basis That is, for i smaller then j, we have < e i, e j > equals zero, unless j equals i + d and then < e i, e j > is equal one In

7 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 7 this case, the universal deformation with principal polarization is given by dividing by the ideal generated by t ij t ji for all i, j Remark 29 Suppose that D is the Dieudonné module of the p-divisible group of a principally polarized g-dimensional abelian variety (X, λ) Let D be the contravariant Dieudonné module We have the well known diagram 0 H 0 (Ω 1 X ) H1 dr (X) H1 (O X ) 0 V D/pD D/pD D/V D Lie(X) Lie(X t ) The Hasse-Witt matrix is a matrix describing the action of Frobenius on H 1 (O X ) It is dual to the matrix of Verschiebung acting on H 0 (Ω 1 X t ) in the diagram 0 H 0 (Ω 1 X ) H dr t 1 (X) H 1 (O X t) 0 V D/pD D/pD D/V D, Lie(X t ) Lie(X) hence equal (for the right choice of bases), to the matrix of Frobenius on the term Lie(X) Note that when writing the module in a displayed form, e g+1,, e 2g span H 0 (Ω 1 X t ) and e 1,, e g project to a basis of Lie(X) Therefore: The matrix A + T C appearing in (23) read mod p is the Hasse-Witt matrix of the deformation corresponding to T Remark 210 The theory of displays was originally developed by D Mumford (unpublished) It was exposed and developed by P Norman and Norman Oort in [18] and [19] It was further studied by T Zink in a greater generality, applicable for non equi-characteristic situations (see, eg, [25], Introduction, Theorem 9) The theory developed by Zink is more suitable for treating the case of nonperfect base rings R We refer the interested reader to [25] Introduction, Definition 11, Page 16, Definition 16, Equation 16, Example 19, Proposition 110 (and the following remark), Formula 231 and Page 60, for a quick overview of how the theories are tied We remark that the computations are virtually the same due to the fact that both objects are defined by the same equations (see [25], Equation 16) Below, the computations done for R = k[[t 1,, t g ]] and similar rings can be interpreted as taking place in its perfect closure, or in terms of displays in the sense of [25] 23 The type and local deformations Let t M n (k) be a geometric point Let D be the covariant Dieudonné module of X t with the induced O L -structure and perfect alternating pairing <, > We have a decomposition (24) given by O L W (k) = W (k), i Z/gZ a 1 (σ 1 (a),, σ g (a))

8 8 E Z GOREN AND F OORT The ring O L W (k) acts on D and induces a decomposition (25) D = D i i Z/gZ The following lemma is essentially known (see [24], Proposition 35) We include the proof for completeness Lemma 211 The decomposition (25) has the following properties: (i) F (D i ) D i+1, dim k (D i+1 /F (D i )) = 1 (ii) V (D i ) D i 1, dim k (D i 1 /V (D i )) = 1 (iii) O L acts on D i via σ i (iv) The pairing <, > induces on each D i a perfect alternating pairing <, > i and D i D j for i j Proof Part (iii) follows straight from the definition For clarity, we denote the action of an element r O L on x D by [r]x Note that for every two elements r and s in O L W (k), we have < [r]x, [s]y >=< x, [rs]y > Choosing r, s to be orthogonal idempotents giving the projections onto D i, D j respectively, (iv) follows Let x D i then [r]x = σ i (r)x Every element r O L induces an isogeny [r] : D D of Dieudonné modules Hence [r]f x = F [r]x = F σ i (r)x = σ σ i (r)x = σ i+1 (r)x Parts (i) and(ii) follow Definition 212 An admissible basis for D, given the polarization and the O L - structure, is a basis {X 1,, X g, Y 1,, Y g } for D, such that (i) {X i, Y i } is a symplectic basis for D i ; (ii) Y i V (D i+1 ) Let A B C D be the corresponding display for such a basis Note that for suitable a i, b i, c i, d i, we have: A = d 2 (a 2,, a g, a 1 ), B = d 2 (b 2,, b g, b 1 ), C = d 2 (c 2,, c g, c 1 ), D = d 2 (d 2,, d g, d 1 ) We have a universal display over C(R) as in (23) Proposition 213 Let R = k[[t ij ]]/(t ij t ji ) Then the maximal closed subscheme of Spec(R) to which the O L -action and the level-n structure extends, is given by H = Spec(H), where H = R/(t ij : i j) Proof Since n is prime to p, the level structure extends to any deformation and uniquely We can therefore disregard it It is enough to prove that the O L -action extends to this closed subscheme, because it is well known that the local deformation ring of the Hilbert modular variety is regular on g parameters Let D H be the display derived from (23) by putting t ij = 0 for i j Then D H = D H,i,

9 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 9 where D H,i is obtained from D i by extending scalars to W (H), with the naturally given action of W (k) on each component The action of O L is defined via the map O L i W (k), a (σ 1 (a),, σ g (a)) This is a map of C(H)-modules if and only if it commutes with Frobenius That is, if and only if (26) where M 1 M σ 2 = M 2 M 1, A + T C p(b + T C) M 1 =, C pd d1 (σ M 2 = 1 (a),, σ g (a)) 0 0 d 1 (σ 1 (a),, σ g (a)) It is easy to verify that (26) holds Theorem 214 Let t M n (k) be a geometric point of type τ There is a choice of parameters t 1,, t g, such that the universal local deformation ring of X t is isomorphic to Spec(k[[t 1,, t g ]]), and such that for every type ρ τ the property of being of type containing ρ is given by the closed regular g ρ dimensional formal subscheme defined by the ideal (t i : i ρ) Proof The universal Dieudonné module D H is displayed by the matrix A + T C B + T C, C D and A + T C = d 2 (a 2 + T 2 c 2,, a n + T g c g, a 1 + T 1 c 1 ), C = d 2 (c 2,, c g, c 1 ) By Remark 29, the matrix (A ) + T C 0 C 0 (mod p) is the matrix of Frobenius on H dr 1 (X), where X is the universal deformation over Spec(k[[t 1,, t g ]]), and the matrix A + T C = d 2 (a 2 + c 2 T 2,, a g + c g T g, a 1 + c 1 T 1 ), read mod p, is the Hasse-Witt matrix Note that a i = 0 (mod p), if and only if F (D i 1 ) = V (D i+1 ), if and only if i τ(x) Since, a i 0 (mod p) implies that a i + t i c i is invertible (mod p), the theorem follows Definition 215 For τ Z/gZ, let W τ be the closed subscheme of M n, with the reduced induced structure, having the property that for every geometric point t W τ we have τ(x t ) τ We let Wτ 0 = W τ \ W τ τ τ ; τ τ Corollary 216 The subscheme W τ is a locally irreducible and regular scheme In particular different components of W τ do not intersect and every component is regular

10 10 E Z GOREN AND F OORT Proof This follows directly from the theorem Corollary 217 For every two types τ and ρ, W τ W ρ = W τ ρ Proof Clearly we have equality at least on geometric points, which implies that W τ W ρ W τ ρ The only question is of multiplicity and this is supplied by Theorem 214 Definition 218 Let X/M n be the universal principally polarized abelian scheme with real multiplication by O L and symplectic level-n structure Let E be the relative cotangent bundle, E = e (Ω 1 X/M n ), where e : M n X is the zero section Let λ 1 be the first Chern class of E Equivalently, of det(e) See Section 41 for more on this subject Proposition 219 (compare [5]) The variety W 0 τ is quasi-affine Proof It is well known that λ 1 is an ample line bundle on M n Indeed, the sections of kλ 1 are modular forms of weight k and give the Baily-Borel compactification of M n, see [7], Chapter X, Section 3 Hence λ 1 is ample on W τ (see also [16], Introduction and Theorem VII32) We prove next that it is torsion on W 0 τ : Let (G, <, >) be the restriction to W 0 τ, of the universal p-torsion sub group scheme, ie, X[p], together with its perfect alternating form coming from the polarization on X The results of Section 32 show that (G, <, >) has the same isomorphism type for every geometric point of W 0 τ It follows that there exists a finite cover f : T W 0 τ, such that (G W 0 τ T, <, > W 0 τ T ), is constant (one follows the argument of [5] or [20], Section 5 See also Remark 39) Therefore, its cotangent space, (Lie(G W 0 τ T )) = (E W 0 τ ) W 0 τ T, is constant as well and so is its determinant Hence f λ 1 is trivial and since f is finite, λ 1 is torsion We conclude that for certain m greater or equal to 1, the line bundle mλ 1 is trivial on Wτ 0 and very ample on W τ It follows that we can represent mλ 1 by an effective divisor D contained in W τ Wτ 0 Hence W τ D is quasi-affine and so is Wτ 0 Before stating the next corollary, it may benefit the discussion to formulate the following principle which we use repeatedly below Remark 220 Let S be an integral scheme and let ξ be its generic point Let t be a non-singular geometric point of S Let X/S be an abelian scheme, and denote by X ξ and X t the corresponding fibres We may read many of the properties of X ξ from the local deformation theory of X t in the following manner Let T be the formal neighborhood of t in S Let η be the generic point of T, then X η is obtained from X ξ by base change of fields Hence, eg, X η and X ξ have the same a-number, f-number, Newton polygon etc

11 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 11 Corollary 221 The generic point of every component of W τ has type τ Proof Let C be a component of W τ It follows from Proposition 219 that C contains a point c of a strictly larger type τ It follows from Theorem 214 that we can find a local deformation of c whose generic point is of type τ and using the local irreducibility of C, given by Corollary 216, the assertion follows Corollary 222 Every component of W τ contains a superspecial point Proof Let C be a component of W τ It follows from Proposition 219 that C contains a point c of a strictly larger type τ From Theorem 214 we know that locally at c the subscheme W τ is irreducible and is contained in W τ It follows that W τ contains a component of W τ and the result follows by induction Corollary 223 For every type τ, dim(w τ ) = g τ Proof By Corollary 222 every component of W τ contains a superspecial point Using Theorem 214 and Corollary 216, the result follows 3 The Type and the p-torsion 31 The Ekedahl-Oort stratification Consider the coarse moduli space of principally polarized abelian varieties of dimension g in characteristic p, and denote it by A Given a geometric point t of A, we get a couple (X t [p], <, >), where X t is the abelian variety parameterized by t and <, >: X t [p] X t [p] µ p is the perfect alternating pairing obtained from the polarization λ t on X t Ekedahl & Oort proved that over an algebraically closed field, there are finitely many isomorphism classes of such couples and that every isomorphism class is locally closed The stratification itself is formed by taking carefully chosen unions of such locally closed sets Since the details are a bit lengthy, we refer the reader to [20], [5] and [21] for more on this fascinating story 32 The type determines the Ekedahl-Oort strata in M n Circular groups,swords and dwords We follow [14] and [21] extracting what we need Let k be an algebraically closed field of characteristic p Let w = a 1 a s be a word in the letters {F, V }, ie, a i is an element of {F, V } Define, M w = s k x i, i=1 and make M w into a (contravariant) Dieudonné module by defining F x i = 0, x i = V x i+1 if a i = V, F x i = x i+1, V x i+1 = 0 if a i = F, where by x s+1 we mean x 1 Thinking about the generators as the hours marks on a clock, F acts clockwise and V acts counter clockwise, where a i = F (resp, a i = V ) is telling us that F is not zero (resp, is zero) on x i Note that the action of V is then almost imposed

12 12 E Z GOREN AND F OORT on us, if we wish to have Im(V ) = Ker(F ) and Im(F ) = Ker(V ) Hence, we call such a Dieudonné module a circular module and we call the corresponding group C w a circular group It is killed by p, and indeed satisfies Im(V ) = Ker(F ), Im(F ) = Ker(V ) Note that C w depends only on w mod cyclic permutations Another easy property is that C D w = C w D, where C D w is the dual group to C w, and w D is the word obtained from w by changing every F to V and every V to F A dword (dual word) of length 2g is a couple (w, w D ), where w is a word of length g Given a dword z = (w, w D ), we let C z = C w C w D A sword (special word) is a circular word w = a 1 a 2g with the property a i a i+g for all i, and for some 1 i 2g, we have a i = V, a i+1 = F (here, as below the indices are read in a cyclic way) Example 31 Consider the sword w = F V V F One finds that C w is a group scheme of order p 4 killed by F 3, by V 3 and by F V = p It is isomorphic, over an algebraically closed field, to the p-torsion of any supersingular, but not superspecial, abelian surface If we consider z = (F V, V F ) instead, then we get the p-torsion of a superspecial abelian surface The group C z is killed by F 2, by V 2 and by F V = p In particular, C z is not isomorphic to C w Remark 32 Note that a circular group C is the kernel of multiplication by p on a p-divisible group G, ie, G[p] = C Conversely, for every p-divisible group G, the kernel of multiplication by p is a direct sum of circular groups In this respect, our conventions are a slight variation on [21], Section 13 We also remark that a circular group C w is of local-local type, if and only if, both the letters F and V appear in w For example, the word w = V V gives the Dieudonné module of µ 2 p, while the word w = F F gives the constant group scheme (Z/pZ) 2 Lemma 33 Let z be a dword or sword of length 2g Then C z has a naturally defined O L -structure and O L -polarization, ie, a perfect alternating pairing such that and for every a O L <, >: C z C z k, < F x, y >=< x, V y > σ, < ax, y >=< x, ay > Proof We indicate how to define the pairing and O L -structure leaving the verification of the details to the reader Assume that z = (w, w D ) is a dword and that M w = g k x i, M w D = g k y i i=1 i=1

13 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 13 We let M z = g (M z ) i, (M z ) i = k x i + k y i, i=1 and note that it has a natural O L k = i W (k)-structure and hence, via O L O L k, a (σ 1 (a),, σ g (a)), also an O L -structure The pairing is defined by decreeing and < x i, y i >= < y i, x i >= 1, (M z ) i (M z ) j, i j If z is a sword, we basically do the same We let and we put M = 2g k x i, i=1 (M w ) i = k x i + k x i+g, etc < x i, x i+g >= 1, 1 i g, Remark 34 Note that if z is a sword then the O L -structure on C z depends on z and not only on z up to cyclic permutations The same remark applies to dwords Proposition 35 1 Let z = (w, w D ) be a dword, then C w is O L -invariant and indecomposable as an O L -group In this case, the decomposition C z = C w C w D is the unique decomposition up to isomorphism of C z into indecomposable O L -groups 2 Let z be a sword, then C z is indecomposable as an O L -group Remark 36 Note that we are not claiming that C z or C w are simple O L -groups This is not the case For example, when g equals 2, a supersingular point t of M n with a-number equal 1, gives a counter-example In this case X t [p], which by Theorem 38 below is C z for a suitable sword z in fact z is F V V F has a family of O L -subgroups of order p 2 parameterized by P 1 Proof 1 Assume that z = (w, w D ) is a dword, where w = a 1 a g It is clear that C w is O L -invariant Assume that C w = H K is a non-trivial decomposition into O L -groups We look at the Dieudonné modules: M w = g (M w ) i = g k x i = g (H i K i ), i=1 i=1 i=1 where H = D(H) and K = D(K) We see that for a suitable i, H i = k x i and H i+1 = 0 Thus K i = 0 and K i+1 = k x i+1 If a i = F, we get H i+1 = F H i 0, a contradiction If a i = V, we get K i = V K i+1 0, a contradiction 2 Assume that z is a sword, say z = a 1 a 2g Assume that C z = H K, a sum of two non-trivial O L -invariant subgroups, and let H and K be the corresponding Dieudonné module We have (31) M w = g M i = g k x i + k x i+g = g (H i K i ) i=1 i=1 i=1

14 14 E Z GOREN AND F OORT Let Φ be the kernel of Frobenius on M w It is the image of V Similarly, the image of F is the kernel of V For every i, the submodule Φ i is one dimensional and Φ is O L -invariant We establish the following points: For every i, we have both H i 0 and K i 0 Indeed, if H i 0 then H i+1 0 Because, either F H i 0, or F H i = 0 and then we have H i = V M i+1 Then, if H i+1 = 0, we get H i = V K i+1, a contradiction Since, at least for one i, H i 0, we get that this holds in fact for every i Similarly for K For every i, either H i = Φ i or K i = Φ i Indeed, either V H i+1 0 or V K i+1 0 Let τ = {1 i 1 < < i a g} be the indices such that F M i 1 = V M i+1 Assume that for some j, H ij = Φ ij, then K ij+1 = Φ ij+1 Similarly, if K ij = Φ ij, then H ij+1 = Φ ij+1 The idea is this Since H ij = Φ ij, we must have F K ij 0 and hence it is the kernel of V on (M w ) ij+1 If i j+1 i j + 1, then F K ij V M ij+2 = Φ ij+1 Thus, F 2 K ij 0, etc This stops exactly at the first s such that F s K V M Thus, K ij+1 = Φ ij+1 To finish the proof of irreducibility we just note that for any sword w, a(c w ) is odd (then the last gives the contradiction) This is combinatorics: We assume, to simplify notation, that in the word w = a 1 a 2g, we have a 1 = F and a 2g = V Note that a(c w ) is the number of time V F appears in the circular word w More precisely, i + 1 belongs to the type, if and only if a i a i+1 Write the elements of w in two rows a 1 a 2 a 3 a g a g+1 a g+2 a g+3 a 2g Go along this brick road from left to right, jumping from a V brick to a V brick You start at a g+1 and end at a 2g The number of times you jump to the upper row is thus the number of times you jump back to the lower row On the other hand each jump counts a couple V F either on the upper or on the lower row (but just on one of them each time) Thus we get an even number of couples V F But we still have to consider a g a g+1 and a 2g a 1 which are just F V and V F respectively Hence the total number of couples V F is odd Finally, our method of proof shows that in fact the decomposition into indecomposable O L -groups is unique up to the obvious isomorphisms Lemma 37 Let z and t be each either a sword or a dword, then C z = Ct as O L -groups, if and only if both z and t are swords and z = t or z = t D, or both z = (w, w D ) and t = (v, v D ) and w = v or w = v D Proof If we have such kind of equalities clearly we get isomorphic polarized O L - groups (by construction) Conversely, Proposition 35 implies that z and t are either both swords or both dwords If both are dwords, we are done because, obviously,

15 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 15 C w, together with the decomposition into one dimensional spaces M w = M i, determines w We are thus left with the case of swords: An isomorphism as O L -groups implies an isomorphism C z = Ct, h = h i : (M z ) i (M t ) i, where (M z ) i, (M t ) i are as in (31) Say w = a 1 a 2g and t = b 1 b 2g For some i we are given that a i = V, a i+1 = F This implies that the i + 1 component of C z /(F C z + V C z ) = C t /(F C t + V C t ) is non-zero ( i + 1 belongs to the type ) Thus b i b i+1 = V F or b i+g b i+g+1 = V F Assume the first case holds, else take t D The idea is that both words are strings of letters having the same letter in the i-th place such that the letter changes, say a j a j+1, if and only if either a j a j+1 = F V, or a j a j+1 = V F That is, if and only if a j a j+1 = V F, or a j+g a j+g+1 = V F Equivalently, if and only if the j + 1 component of C z /(F C z + V C z ) = C t /(F C t + V C t ) is non-zero ( j + 1 belongs to the type ) Hence z equals t The type determines the Ekedahl-Oort stratification in M n We prove that the type determines the Ekedahl-Oort stratification More precisely, Theorem 38 Let t M n (k) be a geometric point: 1 Suppose that τ(x t ) is odd There exists a unique sword w, depending only on τ(x t ), such that X t [p] is isomorphic to C w with the O L -structure and polarization 2 Suppose that τ(x t ) is even There exists a unique dword z, depending only on τ(x t ), such that X t [p] is isomorphic to C z with the O L -structure and polarization Here uniqueness is taken in the sense of Lemma 37 Proof We may assume that τ, otherwise just take the dword (F F, V V ) We may also assume that 1 τ, since this just affects, eg, for a sword, the place i where a i a i+1 = V F We first define letters a 1,, a 2g Let We let τ = {1 < i 2 < < i a g} (a 1,, a i2 1) = (F,, F ), (a i2,, a i3 1) = (V,, V ),, (a ia,, a g ) = (X,, X), where X = F or V (depending of course on the parity of τ) Then, we just define a i+g equals F if a i = V, and a i+g equals V, if a i = F We consider the case of τ(x t ) = a even In this case X equals V Let w be the word a 1 a g and let z be the dword (w, w D ) Proving an isomorphism D(X t [p]) = M z,

16 16 E Z GOREN AND F OORT amounts to providing, for every i, a symplectic basis x i, y i of D(X t [p]) i, with the property if a i = F, and if a i = V F x i = x i+1, y i = V y i+1, x i = V x i+1, F y i = y i+1, In what follows x i, y i, will denote elements in D(X t [p]) i That may very well involve a tacit claim that this is possible! We choose the x i, y i in the following way: Start by taking some non-zero y 1 Ker(F ) and x 1 such that < x 1, y 1 >= 1 Hence x 1 Ker(F ) Then choose: definition of x i definition of y i a i a i+g i x 1 y 1 ker(f ) F V 1 x 2 = F x 1 y 2 ker(f ) : V y 2 = y 1 F V 2 x i2 1 = F x i2 2 y i2 1 ker(f ) : V y i2 1 = y i2 2 F V i 2 1 x i2 = F x i2 1 y i2 : V y i2 = y i2 1 V F i 2 x i2 +1 ker(f ) : V x i2 +1 = x i2 y i2 +1 = F y i2 V F i x i2 +2 ker(f ) : V x i2 +2 = x i2 +1 y i2 +2 = F y i2 +1 V F i V F i 3 1 x i3 : V x i3 = x i3 1 y i3 = F y i3 1 F V i 3 x ia = F x ia 1 y ia : V y ia = y ia 1 V F i a x ia +1 ker(f ) : V x ia +1 = x ia y ia +1 = F y ia V F i a+1 x g ker(f ) : V x g = x g 1 y g = F y g 1 V F g x 1 : V x 1 = x g y 1 = F y g Note that for every r we have, by induction, < x r, y r >= 1 Hence, x r and y r form a symplectic basis to D(X t [p]) r Note that y 1 Im ( F D(Xt [p]) g ) = Ker ( V D(Xt [p]) 1 ) = Ker ( F D(Xt [p]) 1 ) Therefore, using that k is algebraically closed, by taking a suitable multiple of y 1 we may assume that y 1 = y 1 Since < x 1, y 1 >=< x 1, y 1 >= 1 we have x 1 x 1 Ker(F ) Thus we may also assume that x 1 = x 1 This finishes the proof in the case of even a The case of odd a is similar enough to be left to the reader We just hint that the required word w is a 1 a 2g, the construction goes the same, and one argues that by a suitable modification we can get x 1 = y 1, y 1 = x 1

17 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 17 Remark 39 The proof of Theorem 38 suggests another method of proving the assertion, made in the proof of Proposition 219, about the existence of a finite covering T W 0 τ, such that, denoting by G the universal p-torsion sub group scheme over W 0 τ, G T Wτ 0 is constant The problem, in terms of the proof, is to find sections {x i, y i } satisfying the above relations Locally on the base, this amounts to taking certain p l 1 roots, for various l s, of functions 4 The Global Structure 41 The Hodge bundle and W τ In this section we follow some ideas of [17] and [8] Let π : (X, Λ, I, A) M n be the universal object over M n We will abuse notation and write X for (X, Λ, I, A) Let e : M n X be the identity ( section, and let ) E = e Ω 1 X/M n = π Ω 1 X/M n be the Hodge bundle over M n It is a locally free sheaf of rank g over M n Recall that a modular form of (equi-)level k is a section of ω k, where ω def = det(e) It is known that ω is an ample line bundle on M n, see [7], Chapter X, Section 3, and [16], Introduction, Theorem 11 Consider the subvariety of M n defined by def W r = τ: τ =r We want to explain how it is a related to c r (E), where c r denotes the r-th Chern class of a vector bundle This gives a global interpretation to our stratification To make our relations exact we continue to take a symplectic full level-n structure, where n is at least 3 and prime to p, and work with M n The result for the coarse moduli space M where we take no level structure involves studying the ramification of the natural morphism from M n to M To see the problem, notice that when g equals 1, W 1 is the set of supersingular elliptic curves, and deg(c 1 ) = 1/24 This can be deduced from the following facts: W τ ω = E, ω 2 = Ω 1 Mn (cusps) The right formula is Deuring s mass formula, 1 Aut(E) = p 1 = (p 1) deg(c 1 ) 24 E supersingular

18 18 E Z GOREN AND F OORT Recall that we consider the regular scheme M n as a scheme over the finite field F containing F p g In particular we have a decomposition of O L O Mn induced from the decomposition O L /p F = g F, a 1 (σ 1 (a),, σ g (a)) i=1 Thus, over M n, the Hodge bundle E is a direct sum of g line bundles: E = L 1 L g Given any locally free sheaf K over O L O Mn, we will denote by K i component In particular, L i = E i its i-th Let X (p) be the Frobenius image of X Recall that it is defined by the following cartesian square: X (p) X, F abs Mn M n where F abs is the absolute Frobenius, defined as the identity on the underlying topological space of M n, and as raising to the p-th power on the structure sheaf Let T be the relative cotangent sheaf of X (p) restricted to the origin It is easy to verify that T = E (p), where for any O Mn -sheaf K we define K (p) as the O Mn -sheaf obtained by twisting by Frobenius O Mn O Mn, K (p) = O Mn OM n K The definition is compatible under the dictionary between locally-free sheaves and vector bundles The Verschiebung morphism X (p) /M n X/M n, which is O L -equivariant, induces a linear O L O Mn -map V : E T This can be justified by either interpreting E and T as cohomology objects via their embedding in the first de Rham cohomology, or by interpreting them as the cotangent sheaves We thus get, for every i, an O L O Mn -linear map V : E i T i, where O L acts via σ i Since E = E i, it follows from the definitions that Thus, we get an O L O Mn -linear map T i = E (p) i 1 V : E i E (p) i 1

19 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 19 Furthermore, over any perfect O Mn -algebra R, in particular, for every geometric point of M n, we get a σ 1 -linear map, V : E i E i 1, by twisting with respect to σ 1 That is, over R, E i 1 = E (p) i 1 R R Therefore, from this point of view, W {i 1} is the degeneracy locus of the linear map V : E i E (p) i 1 To wit, for every geometric point t, (D (α(x t ))) i = (H ) 0 Ω 1 /V H 0 Ω 1 = (E X t X t i Xt ) / (V E i+1 Xt ) i Thus, i belongs to τ(x t ), if and only if is the zero map V : E i+1 Xt E i Xt Let l i = c 1 (L i ) be the first Chern class of the line bundle L i, then (41) c r (E) = l i1 l ir 1 i 1 < <i r g Applying the Thom-Porteous formula, see [6], Chapter 14, we conclude that in the Chow ring, CH Q (M n), W {i} = c 1 (L (p) i L i+1 ) = c 1 (L p i L i+1 ) = pc 1 (L i ) c 1 (L i+1 ) = pl i l i+1 Using Corollary 217 we conclude the following theorem Theorem 41 The following equalities hold in CHQ (M n): r W {i1,,i r} = plij l ij +1, W r = Corollary 42 In CH Q (M n), I Z/gZ I = r j=1 (pl j l j+1 ) j I W 1 = (p 1)c 1 (E) Corollary 43 The subspace of CH Q (M n) spanned by the tautological classes {l 1,, l g } is equal to the one spanned by {W 1,, W g }

20 20 E Z GOREN AND F OORT 42 A simplicial complex In this section, we show that the components of the W τ s form a simplicial complex, where every top-dimensional simplex is of dimension g 1 and corresponds to a superspecial point Definition 44 Define a graph T with g + 1 levels, whose vertices at the i-th level are the irreducible components of the W τ s with τ = i Edges in this graph exist only between consecutive levels Two vertices v, v, of levels i and i+1 respectively, corresponding to components C and C are connected if and only if C C Theorem 45 The graph T is a colored simplicial complex, in which the faces of dimension i are the vertices of level i + 1 in T In particular, M n, which is equal to W, corresponds to the empty set and the superspecial locus of M n corresponds to the g 1-dimensional simplices Every maximal simplex is of dimension g 1 The vertices are colored according to their type and simplices intersect only along faces having the same coloring Remark 46 One may call T the intersection complex of all the W τ s Proof All the assertions follow from the following points: The intersection of a component of W τ and a component of W σ is either empty or a component of W τ σ This follows from Corollary 217 For every τ, every component C of W τ and every τ τ there exists a unique component C of W τ such that C C Take any geometric point t of C Theorem 214 shows that we may deform X t = (X t, λ t, ι t, α t ) to a quadruple X = (X, λ, ι, α) such that τ(x) = τ Hence t lies on some component C of W τ But the local irreducibility of W τ, given by Corollary 216, implies C is unique For every τ, every component C of W τ and every τ τ, there is a component C of W τ such that C C By Corollary 222, C has a superspecial point Use the previous point, or repeat the argument Corollary 47 Let H be a family of prime-to-p Hecke operators acting transitively on the superspecial points in M n Then H acts transitively on the components of W τ for every τ Proof The prime-to-p Hecke operators act on the colored simplicial complex T, ie, they also preserve the type The result follows by descending induction on the dimension By assumption it holds for the top simplices Assume it holds for faces of dimension i+1 and take two faces of dimension i with the same color, ie two components C 1 and C 2 of W τ, where τ equals i Let l τ, τ = τ l Let D 1 and D 2 be components of type τ, such that D j C j, j = 1, 2 By induction, there exists a Hecke operator T, such that D 1 T (D 2 ), hence T (C 2 ) contains a component C 1 of W τ containing D 1 Since there is a unique such component we get that C 1 = C 1

21 STRATIFICATIONS OF HILBERT MODULAR VARIETIES 21 It is thus natural to study the representation of the prime-to-p Hecke algebra on the complex T as well as its homology as an abstract simplicial complex! This could be viewed as a generalization of the classical study of the action of the Hecke operators on the zero-dimensional complex of supersingular elliptic curves (the case g = 1 in our discussion) As an indication of the interesting information that this complex encrypts, we mention the following Theorem 48 (See [1]) Let g = 2 Assume that p > 2 and n 3 The number of components of W {i}, for i equals 1 or 2, is 1 2 [M n : M] ζ L ( 1), and the number of components of W {1,2} is p [M n : M] ζ L ( 1) 2 5 Newton Polygons 51 This section is devoted to some results concerning the Newton polygon stratification of M n and a conjecture on the dimensions of the various strata We hope to return to this conjecture in a subsequent paper Below, we give certain partial results, which imply, in particular, what one may call the weak Grothendieck conjecture for M n : Let β and γ be two symmetric Newton polygons with the same initial and end points (0, 0), (2g, g), respectively Assume that β γ (see Definition 53 below) The weak Grothendieck conjecture, for abelian varieties, asserts that there exists an integral scheme S, with a generic point η and a geometric point t S, and an abelian scheme X over S, such that the Newton polygon of X η is β and the Newton polygon of X s is γ The strong Grothendieck conjecture is obtained by specifying the isomorphism class of X t It is clear how to formulate such conjectures for any PEL Shimura variety (though not their validity), under the obvious assumption that β and γ appear for some geometric points of this Shimura variety, or how to formulate it for p-divisible groups Theorem 59 below implies that the weak Grothendieck conjecture holds for the Hilbert modular variety M n We remark that Grothendieck formulated his conjecture, in the setting of p-divisible groups, in a letter to Barsotti, reproduced as the Appendix to [10] Given Theorem 24, this is essentially the weak Grothendieck conjecture for abelian varieties A stronger conjecture, regarding a sequence of polygons, was formulated by Koblitz in [13], p 215 For more extensive discussion and results in this direction, including proofs of the weak versions, we refer the reader to [22], Section 6 Clearly, the Grothendieck conjecture is about the variation of the Newton polygon along the local deformation ring of an abelian variety (possibly with some extra structure) It is clear therefore that any local approach to this conjecture is intimately related to Conjecture 55

22 22 E Z GOREN AND F OORT 52 The possible p-divisible groups Theorem 51 Let t be a geometric point of M n The p-divisible group X t (p) is isogenous to or (under the convention ka/kb = k a/b) G l/g + G (g l)/g, 0 l g, g G 1/2 Proof Let t be a geometric point of M n and let (X t, λ t, ι t, α t ) be the corresponding object Let D be contravariant Dieudonné module of X t (p) We have a decomposition as in (25) D = D i i Z/gZ Note that (D, F g ) is a σ g -crystal, which decomposes as a direct sum of g isogenous rank-2 σ g -crystals, (D, F g ) = (D i, F g ) i Z/gZ The isogeny being, essentially, F itself It follows that if the slopes of, say, (D 1, F g ) are α and β, then the slopes of (D, F ) are α/g and β/g, each appearing with multiplicity g On the other hand, 2 ord p ( F g ) = D1 pg, and its slopes have denominator at most 2 (see [11], Formula 131) Remark 52 The converse also holds, ie, every such isogeny class is obtained from some point of M n This follows from Theorem 522 below Definition 53 We say, for two Newton polygons β, γ, with the same initial and end points, that β γ if every point of β is not below γ We say that β > γ if in addition β γ Let β 0 < β 1 < < β s be the possible Newton polygons of points of M n given by Theorem 51 (eg, β 0 is the ordinary polygon and β s is the supersingular polygon, s = [(g + 1)/2]) Given a geometric point t of M n, we let N t be the Newton polygon of (X t, λ t, ι t, α t ) For every i such that 0 i s, let N i be the closed subscheme where N t β i Remark 54 The reader should note that our definition of the order between Newton polygon is the opposite of a frequently used definition (see for example [23], Section 3) That is, in our definition β γ is what is often denoted γ β Our choice is based on the intuition that under specialization Newton polygons go up and up is bigger then down The other choice is based on the intuition that if γ β then the dimension of the Newton strata of γ is smaller or equal to that of β Conjecture 55 dim(n i ) = g i Remark 56 We intend to discuss this conjecture in a future paper Some partial results of independent interest are given below See, in particular, Theorems 59 and 522

23 STRATIFICATIONS OF HILBERT MODULAR VARIETIES The cyclic case The following lemma is useful to us (see also [22], Lemma 35) Lemma 57 Let k be a perfect field of characteristic p Let (M, F ) be a rank two σ i -crystal with ord p ( 2 F ) = g Suppose that F is given with respect to some basis by the matrix m1 m m = 2, m 3 m 4 then the first Newton slope of (M, F ) is given by { ( )} g min 2, ord m σ i p m σi 3 (51) 1 + m 4 = min (under the convention 0/0 = 1) m 3 { ( g 2, ord p m 1 + m σi 4 Proof Since the argument is symmetric, we just prove the first formula the one involving m 3 If m 3 = 0, then the result is clear If m 3 0, then, denoting the basis vectors by X and Y, we may consider the cyclic submodule of finite index generated by X and Y = m 1 X + m 3 Y With respect to this basis, Frobenius is given by i 0 mσ 3 m 3 (m 1 m 4 m 2 m 3 ) 1 m σi m 1 + m σ i 3 4 The characteristic polynomial is x 2 m σ i (m σi m 4 )x + mσ i 3 (m 1 m 4 m 2 m 3 ) m 3 m 3 The lemma follows from [3], Lemma 2 p82, Lemma 3 p84 (or rather their proofs) m 3 m 2 m σi 2 )} Remark 58 A typical application of Lemma 57 shall be as follows: We will be interested in knowing the variation of the Newton polygon along a certain formal subvariety of the deformation space of (X, λ, ι, α) or of an RM crystal (defined in Section 54 below) Using displays, specifically (22) and Proposition 213, we would know the effect of Frobenius, F, on a deformation parameterized by this subvariety Using the argument in the proof of Theorem 51, it will be enough to study the effect of F g on some D i Here Lemma 57 becomes handy Theorem 59 Let t M n (k) be a geometric point such that a(x t ) = 1 Assume that the Newton polygon of X t is β l There is a choice of parameters t 1,, t g, such that the local equi-characteristic universal deformation ring of X t is given by k[[t 1,, t g ]], and such that the following holds: For every r l the closed formal scheme, where the Newton polygon is weakly above β r, is given by Proof Let t 1 = = t r = 0 D = g D i i=1