On period spaces for p-divisible groups

Urs Hartl ^{∗}

April 25, 2008

Abstract

In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine’s filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.

Résumé

Dans leur livre, Rapoport et Zink ont construits des espaces des périodes, rigides analytiques pour les isocristaux filtrés de Fontaine. Egalement ils ont construits des morphismes des périodes entre des espaces modulaires des groupes de Barsotti-Tate et certaines de leurs espaces des périodes. Dans cet article nous déterminons l’image des morphismes des périodes, contribuant ainsi à une question de Grothendieck. Nous présentons des examples montrant que l’image ne coïncide que rarement avec tout l’espace des périodes de Rapoport-Zink.

Mathematics Subject Classification (2000): 11S20, (14G22, 14L05, 14M15)

1 A question of Grothendieck

Fix a Barsotti-Tate group 𝕏 over 𝔽_{p}^{alg} of height h and dimension d. Let W := W (𝔽_{p}^{alg}) be the ring of Witt vectors and let K_{0} := W [\frac{1}{p}]. Let O_{K} be a complete discrete valuation

ring with residue field 𝔽_{p}^{alg} and fraction field K of characteristic 0. To every Barsotti-Tate group X over O_{K} with 𝕏 ≅ X ⊗_{O_{K}} 𝔽_{p}^{alg} =: X_{𝔽_{p}^{alg}} the theory of Grothendieck-Messing [19] associates an extension

\xymatrix{0 \ar[r] & (Lie X^{∨})K^{∨} \ar[r] & 𝔻(𝕏)_{K} \ar[r] & Lie X_{K} \ar[r] & 0}

where 𝔻(𝕏)_{K} is the crystal of Grothendieck-Messing evaluated on K. We denote by F the Grassmannian of (h − d)-dimensional subspaces of 𝔻(𝕏)_{K_{0}}.

Problem 1.1. (A. Grothendieck [12, Remarque 3, p. 435]) Describe the subset of F formed by the points (Lie X^{∨})K^{∨} where X is any deformation of 𝕏 over any complete discrete valuation ring O_{K} with residue field 𝔽_{p}^{alg} and fraction field K of characteristic 0.

∗The author acknowledges support of the Deutsche Forschungsgemeinschaft in form of DFG-grant HA3006/2-1

This problem is yet unsolved. However, a contribution was made by Rapoport and Zink [20] in the following way. Set D := 𝔻(𝕏)_{K_{0}} and φ_{D} := 𝔻(Frob_{𝕏})_{K_{0}}. Let L_{K} ∈ F be a point defined over a complete, rank one valued extension K of K_{0} (which not necessarily is discrete). View L_{K} as a K-subspace of D_{K} := D ⊗_{K_{0}} K. One defines the Newton slope t_{N}(D, φ_{D}, L_{K}) := ord_{p}(det φ_{D}) and the Hodge slope t_{H}(D, φ_{D}, L_{K}) := dim_{K} L_{K} − dim_{K} D_{K}. Following Fontaine [11] and Rapoport-Zink [20, 1.18], the point L_{K} ∈ F is called weakly admissible if

t_{N}(D, φ_{D}, L_{K}) = t_{H}(D, φ_{D}, L_{K}) = −d  and
t_{N}(D′, φ_{D}|_{D′}, L_{K} ∩ D′_{K}) ≥ t_{H}(D′, φ_{D}|_{D′}, L_{K} ∩ D′_{K})

for all φ_{D}-stable K_{0}-subspaces D′ ⊂ D. Let F^{rig} be the rigid analytic space, and F^{an} the K_{0}-analytic space in the sense of Berkovich [3, 4] associated with F.

Theorem 1.2. (Rapoport-Zink [20, Proposition 1.36]) The set F_{wa}^{rig} := { L_{K} ∈ F^{rig} : L_{K} is weakly admissible } is an open rigid analytic subspace.

The space F_{wa}^{rig} is an example for the p-adic period domains constructed more generally in [20, Proposition 1.36] for arbitrary filtered isocrystals. The proof of Rapoport and Zink even shows that F_{wa}^{rig} is the rigid analytic space associated to an open K_{0}-analytic subspace F_{wa}^{an} ⊂ F^{an}; see [14, Proposition 1.3]. The period domain F_{wa}^{rig} contains the set of Grothendieck’s problem 1.1. To explain this let Nilp_{W} be the category of W -schemes on which p is locally nilpotent. For an S ∈ Nilp_{W} we set S̄ := V(p) ⊂ S.

Theorem 1.3. (Rapoport-Zink [20, Theorem 2.16]) The functor G : Nilp_{W} → Sets

S ⟼ { isomorphism classes of pairs (X, ρ) where X is a Barsotti-Tate

group over S and ρ : 𝕏S̄ → XS̄ is a quasi-isogeny }

is pro-representable by a formal scheme locally formally of finite type over W.

To G one can associate a rigid analytic space G^{rig} by Berthelot’s construction [5], and also a K_{0}-analytic space G^{an}. Rapoport and Zink [20, 5.16] construct a period morphism π̆_{1}^{rig} : G^{rig} → F_{wa}^{rig} as follows. By the theory of Grothendieck-Messing [19], the universal Barsotti-Tate group X over G gives rise to an extension

\xymatrix{0 \ar[r] & (Lie X^{∨})G^{rig}^{∨} \ar[r] & 𝔻(XḠ)_{G^{rig}} \ar[r] & Lie X_{G^{rig}} \ar[r] & 0}

of locally free sheaves on G^{rig}. The quasi-isogeny ρ : XḠ → 𝕏Ḡ induces by the crystalline nature of 𝔻( .) an isomorphism 𝔻(ρ)_{G^{rig}} : 𝔻(XḠ)_{G^{rig}} ⟶^{∼} 𝔻(𝕏)_{G^{rig}} and the image 𝔻(ρ)_{G^{rig}}(Lie X^{∨})G^{rig}^{∨} defines a G^{rig}-valued point of F^{rig}. By [20, 5.27] the induced morphism G^{rig} → F^{rig} factors through F_{wa}^{rig}. This is the period morphism π̆_{1}^{rig}. The same construction also gives a K_{0}-analytic period morphism π̆_{1}^{an} : G^{an} → F^{an}.

Theorem 1.4. (Rapoport-Zink, Colmez-Fontaine, Breuil, Kisin) The set of Grothendieck’s Problem 1.1 is contained in the subset F_{wa}^{rig}. The latter also equals the image of the period morphism π̆_{1}^{rig} : G^{rig} → F^{rig} in the sense of sets.

Proof. Let L_{K} = (Lie X^{∨})K^{∨} be a point in Grothendieck’s set, given by a Barsotti-Tate group X over O_{K} with 𝕏 ≅ X_{𝔽_{p}^{alg}}. Over O_{K}/(p) this isomorphism lifts by rigidity [9, Appendix] to a quasi-isogeny ρ : 𝕏_{O_{K}/(p)} → X_{O_{K}/(p)} and (X, ρ) gives a point of G(O_{K}). By construction π̆_{1}^{rig}(X, ρ)_{K} = L_{K}. So the point L_{K} belongs to the image of π̆_{1}^{rig}, which in turn lies in F_{wa}^{rig}. It remains to show that every K-valued point L_{K} ∈ F_{wa}^{rig} for K/K_{0} finite lies in the image of π̆_{1}^{rig}. By the theorem of Colmez-Fontaine [8] the filtered isocrystal (D, φ_{D}, L_{K}) is admissible, i.e. arises from a crystalline p-adic Galois representation V. By Breuil [6, Theorem 1.4] there is a Barsotti-Tate group X over O_{K} and an isomorphism T_{p}X_{K} ⊗_{ℤ_{p}} ℚ_{p} ≅ V if p > 2. Kisin [18, Corollary 2.2.6] extended Breuil’s theorem to p = 2 and reproved the Colmez-Fontaine Theorem. Fontaine’s functor D_{cris} transforms the isomorphism T_{p}X_{K} ⊗_{ℤ_{p}} ℚ_{p} ≅ V into an isomorphism ρ̄_{∗} : (𝔻(X)_{K_{0}}, 𝔻(Frob_{X})_{K_{0}}, (Lie X^{∨})K^{∨}) ⟶^{∼} (D, φ_{D}, L_{K}) of filtered isocrystals. This defines a quasi-isogeny ρ̄ : 𝕏 → X_{𝔽_{p}^{alg}} which again by rigidity lifts to a quasi-isogeny ρ : 𝕏_{O_{K}/(p)} → X_{O_{K}/(p)}. So L_{K} lies in the image of the period morphism.

Unfortunately an open rigid analytic subspace of F^{rig}, like the image of π̆_{1}^{rig}, is not uniquely determined by its underlying set of points. This however is true for K_{0}-analytic spaces. Therefore it is natural to ask: What is the image of π̆_{1}^{an}? Examples of Genestier-Lafforgue and the author (see Example 3.6 below) show that this image is in general smaller than F_{wa}^{an}. As [20, 1.37 and 5.53] indicate, Rapoport and Zink were aware of this phenomenon. We determine the image of π̆_{1}^{an} in Section 3. But before we need to recall some of the rings of Fontaine Theory.

2 Some of Fontaine’s rings

Let K/K_{0} be a complete, rank one valued field extension, let C be the completion of an algebraic closure of K, and let O_{C} be its valuation ring. Starting out from C various rings are defined in Fontaine Theory. For details on their construction see Colmez [7]. We follow his notation.

We let Ẽ^{+} := { u = (u^{(n)})_{n∈ℕ_{0}} : u^{(n)} ∈ O_{C}, (u^{(n+1)})^{p} = u^{(n)} }. With the multiplication uv := (u^{(n)}v^{(n)})_{n∈ℕ_{0}}, the addition u + v := (lim_{m→∞}(u^{(m+n)} + v^{(m+n)})^{p^{m}})_{n∈ℕ_{0}}, and the valuation v_{E}(u) := v_{C}(u^{(0)}), Ẽ^{+} becomes a complete valuation ring of rank one with algebraically closed fraction field, called Ẽ, of characteristic p. Fix primitive p^{n}-th roots of unity ε^{(n)} such that ε := (1, ε^{(1)}, ε^{(2)}, …) is an element of Ẽ^{+}.

Let Ã := W (Ẽ) and consider the automorphism φ = W (Frob_{p}) of Ã. For an element u ∈ Ẽ let [u] ∈ Ã be its Teichmüller representative.

If x = ∑_{i=0}^{∞} p^{i}[x_{i}] ∈ Ã then we set w_{k}(x) := min{ v_{E}(x_{i}) : i ≤ k }. For r > 0 let

Ã^{(0,r]} := \{ x ∈ Ã : lim_{k→+∞} w_{k}(x) + \frac{k}{r} = +∞ \} and let B̃^{(0,r]} := Ã^{(0,r]}[\frac{1}{p}].

On B̃^{(0,r]} there is a valuation defined for x = ∑_{i≫−∞}^{∞} p^{i}[x_{i}] as v^{(0,r]}(x) := min\{ w_{k}(x) + \frac{k}{r} : k ∈ ℤ \} = min\{ v_{E}(x_{i}) + \frac{i}{r} : i ∈ ℤ \}.

Let B̃^{]0,r]} be the Fréchet completion of B̃^{(0,r]} with respect to the family of semi-valuations v^{[s,r]}(x) := min{ v^{(0,s]}(x), v^{(0,r]}(x) } for 0 < s ≤ r. The logarithm t := log[ε] converges to an element in B̃^{]0,1]}.

Let B̃_{rig}^{†} := ⋃_{r>0} B̃^{]0,r]}. The homomorphism φ gives rise to a bicontinuous isomorphism φ : B̃^{]0,r]} ⟶^{∼} B̃^{]0,r/p]} and thus induces an automorphism of B̃_{rig}^{†}. It satisfies φ(t) = p t.

Finally there is a homomorphism θ : B̃^{(0,1]} → C sending ∑_{i≫−∞}^{∞} p^{i}[x_{i}] to ∑_{i≫−∞}^{∞} p^{i}x_{i}^{(0)} which extends by continuity to B̃^{]0,1]}. The element t lies in the kernel of θ.

3 THE CONSTRUCTION OF F_{A}^{AN} 4

Definition 2.1. A φ-module over B̃_{rig}^{†} is a finite free B̃_{rig}^{†}-module M together with a φ-linear automorphism φ_{M} : M → M.

The following structure theorem was proved by Kedlaya [17, Theorem 4.5.7].

Theorem 2.2. Every φ-module M over B̃_{rig}^{†} is isomorphic to ⨁_{i} M_{c_{i},d_{i}} where M_{c,d} =

(B̃_{rig}^{†})^{⊕d}, φ_{M_{c,d}} = ( 0 p^{c} 1 1 0 ) · φ for c, d ∈ ℤ with (c, d) = 1 and d > 0.

One defines the degree of M as deg M := ∑_{i} c_{i}.

3 The construction of F_{a}^{an}

As above let D = 𝔻(𝕏)_{K_{0}} and φ_{D} = 𝔻(Frob_{𝕏})_{K_{0}}. Let (D^{]0,1]}, φ_{D}) := (D, φ_{D}) ⊗_{K_{0}} B̃^{]0,1]} and consider the morphism 1 ⊗ θ : D^{]0,1]} → D ⊗_{K_{0}} C. By a variant of a construction of Berger [1, §II] every point L = L_{K} ∈ F^{an}, with values in a field K as in Section 2, defines a φ-module over B̃_{rig}^{†} as follows.

Proposition 3.1. ([14, Proposition 4.1]) There exists a uniquely determined B̃^{]0,1]}-submodule t D^{]0,1]} ⊂ M_{L}^{]0,1]} ⊂ D^{]0,1]} such that

(1 ⊗ θ)(M_{L}^{]0,1]}) = L_{K} ⊗_{K} C and φ_{D} : M_{L}^{]0,1]} ⟶^{∼} M_{L}^{]0,1]} ⊗B̃^{]0,1]} B̃^{]0,1/p]} is an isomorphism.

In particular M_{L}^{]0,1]} defines a φ-module M_{L} := M_{L}^{]0,1]} ⊗B̃^{]0,1]} B̃_{rig}^{†} over B̃_{rig}^{†}.

The following results are proved in [14].

Theorem 3.2. ([14, Theorem 4.4]) deg M_{L} = t_{N}(D, φ_{D}, L_{K}) − t_{H}(D, φ_{D}, L_{K}).

Consider the subset F_{a}^{an} := { L ∈ F^{an} : M_{L} ≅ M_{0,1}^{⊕h} } of the K_{0}-analytic space F^{an}.

Theorem 3.3. ([14, Theorem 5.2]) The set F_{a}^{an} is an open K_{0}-analytic subspace of F_{wa}^{an}.

Remark 3.4. (on the proof of Theorem 3.3. See also Remark 3.7 below.) The inclusion F_{a}^{an} ⊂ F_{wa}^{an} is seen as follows. If D′ ⊂ D is a φ_{D}-stable K_{0}-subspace then (D′, φ_{D}|_{D′}, L_{K} ∩ D_{K}) defines by Proposition 3.1 a φ-submodule M′_{L} ⊂ M_{L}. Since M_{0,1}^{⊕h} is “semistable” of slope zero we conclude by [17, Lemma 3.4.8] that

t_{N}(D′, φ_{D}|_{D′}, L_{K} ∩ D′_{K}) − t_{H}(D′, φ_{D}|_{D′}, L_{K} ∩ D′_{K}) = deg M′_{L} ≥ deg M_{L} = 0

with equality if D′ = D. Hence F_{a}^{an} ⊂ F_{wa}^{an}. On the other hand Berger’s proof [1] of the Colmez-Fontaine Theorem shows that this inclusion induces a bijection on rigid analytic points (with K/K_{0} finite), or more generally points of F_{wa}^{an} with with values in a discretely valued field K.

Theorem 3.5. F_{a}^{an} is the image of the period morphism π̆_{1}^{an} : G^{an} → F_{wa}^{an}.

Proof. Let L_{K} ∈ F^{an} be a K-valued point in the image of π̆_{1}^{an} with K/K_{0} not necessarily finite. So L_{K} = 𝔻(ρ)_{K}(Lie X^{∨})K^{∨} for a Barsotti-Tate group X over O_{K} and a quasiisogeny ρ : 𝕏_{O_{K}/(p)} → X_{O_{K}/(p)}. Then the Tate module T_{p}X_{K} of X induces an injection M_{0,1}^{⊕h} ≅ T_{p}X_{K} ⊗_{ℤ_{p}} B̃_{rig}^{†} ↪ M_{L} which must be an isomorphism by reasons of degree. This was proved in [14, Proposition 6.1]. Thus L_{K} ∈ F_{a}^{an}.

Conversely let L_{K} ∈ F_{a}^{an}. Then the morphism (B̃_{rig}^{†})^{⊕h} ⟶^{∼} M_{L_{K}} ↪ D ⊗_{K_{0}} B̃_{rig}^{†} is represented, with respect to a K_{0}-basis of D, by a matrix M ∈ M_{h}(B̃_{rig}^{†}) with tM^{−1} ∈ M_{h}(B̃_{rig}^{†}). Then in fact M, tM^{−1} ∈ M_{h}(B̃_{rig}^{+}) ⊂ M_{h}(B_{cris}^{+}) by [2, Proposition I.4.1]. For notation see [2, §I.1] or [14, §2]. So M defines an isomorphism B_{cris}^{⊕h} ⟶^{∼} D ⊗_{K_{0}} B_{cris} compatible with Frobenius, which maps (B_{cris}^{+})^{⊕h} onto the preimage of L_{K} ⊗_{K} C under the map id ⊗θ : D ⊗_{K_{0}} B_{cris}^{+} → D ⊗_{K_{0}} C. This means that (D, φ_{D}, L_{K})^{∨} is admissible in the sense of Faltings’ [10, Definition 1]. Note that Faltings uses contravariant Dieudonné modules. By [10, Theorems 9 and 14], L_{K} = 𝔻(ρ)_{K}(Lie X^{∨})K^{∨} for a Barsotti-Tate group X over O_{K} and a quasi-isogeny ρ : 𝕏_{O_{K}/(p)} → X_{O_{K}/(p)}, hence L_{K} lies in the image of π̆_{1}^{an}.

There are many examples showing that only in rare cases F_{a}^{an} = F_{wa}^{an}. We mention one here. Similar examples are due to A. Genestier and V. Lafforgue.

Example 3.6. Let D = K_{0}^{⊕5} and φ_{D} = ( 0 p^{−3} 1 1 0 )· φ. Then h = 5 , d = 3 and

F = Grass(2, 5). Since the isocrystal (D, φ_{D}) is simple F_{wa}^{an} = F^{an}. Let L = L_{K} ∈ F^{an}. Then

M_{−3,5} = D ⊃ M_{L} ⊃ t D ≅ M_{2,5} .

By Theorem 2.2, M_{L} ≅ ⨁_{i} M_{c_{i},d_{i}} with ∑_{i} d_{i} = rk M_{L} = 5 and ∑_{i} c_{i} = deg M_{L} = 0. Moreover by [17, Lemma 3.4.8] all the weights c_{i}/d_{i} must lie between −3/5 and 2/5. So either M_{L} ≅ M_{0,1}^{⊕5} or M_{L} ≅ M_{−1,2} ⊕ M_{1,3}.

Now one easily checks that Hom_{φ}(M_{−1,2}, D) = Hom_{φ}(M_{−1,2}, M_{−3,5}) =

\{ A = ( \begin{matrix}φ^{5}(x) & x \\ φ^{11}(x) & φ^{6}(x) \\ φ^{17}(x) & φ^{12}(x) \\ φ^{23}(x) & φ^{18}(x) \\ φ^{29}(x) & φ^{24}(x) \end{matrix} ) : x = ∑_{ν∈ℤ}p^{ν}φ^{−10ν}([u]), u ∈ E, 0 < v_{E}(u) < ∞\} .

The bad situation M_{L} ≅ M_{−1,2}⊕M_{1,3} occurs if and only if L_{K} is generated by the columns of such a matrix θ(A) ∈ C^{5×2}, since then the homomorphism A factors through M_{L} and this forbids M_{L} ≅ M_{0,1}^{⊕5} by [17, Lemma 3.4.8]. Since obviously such L_{K} exist, this proves that the inclusion F_{a}^{an} ⊂ F_{wa}^{an} is strict.

Remark 3.7. (on the proof of Theorem 3.3.) One can explain the idea for the proof of Theorem 3.3 by means of this example. Namely the complement F^{an} ∖ F_{a}^{an} is the image of the continuous map from the compact set {u ∈ Ẽ^{+} : 1 ≤ v_{E}(u) ≤ p^{10} } given by u ↦ θ(A).

Acknowledgements. The author is grateful to L. Berger, J.-M. Fontaine, V. Berkovich, A. Genestier, M. Kisin, V. Lafforgue, and M. Rapoport for various helpful discussions and their interest in this work.

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Urs Hartl University of Münster Institute of Mathematics Einsteinstr. 62 D – 48149 Münster Germany E-mail: uhartl@uni-muenster.de