Uniformizing the Stacks of Abelian Sheaves

Urs Hartl

University of Freiburg, Institute of Mathematics, Eckerstr. 1, D-79104 Freiburg, Germany, E-mail: urs.hartl@math.uni-freiburg.de

Summary. Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon–Rapoport–Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects “of dimension 1”. Their higher dimensional generalizations are called abelian sheaves. In the analogy between function fields and number fields, abelian sheaves are counterparts of abelian varieties. In this article we study the moduli spaces of abelian sheaves and prove that they are algebraic stacks. We further transfer results of Čerednik–Drinfeld and Rapoport–Zink on the uniformization of Shimura varieties to the setting of abelian sheaves. Actually the analogy of the Čerednik–Drinfeld uniformization is nothing but the uniformization of the moduli schemes of Drinfeld modules by the Drinfeld upper half space. Our results generalize this uniformization. The proof closely follows the ideas of Rapoport–Zink. In particular, analogies of p-divisible groups play an important role. As a crucial intermediate step we prove that in a family of abelian sheaves with good reduction at infinity, the set of points where the abelian sheaf is uniformizable in the sense of Anderson, is formally closed. Mathematics Subject Classification (2000): 11G09, (11G18, 14L05)

Introduction

In arithmetic algebraic geometry the moduli spaces of abelian varieties are of great importance. For instance they have played a major role in Faltings’ proof of the Mordell conjecture [16], the proof of Fermat’s Last Theorem [7], and the proof of Langlands reciprocity for GL_{n} over non-archimedean local fields of characteristic zero by Harris–Taylor [20]. Therefore their structure and especially their reduction at bad primes is intensively studied. One way to investigate their reduction is through p-adic uniformization. This was begun by Čerednik [6] and Drinfeld [11] and continued by Rapoport–Zink [34]. Čerednik–Drinfeld obtained the uniformization of certain Shimura curves of EL-type by a formal scheme whose associated rigid-analytic space is Drinfeld’s p-adic upper half plane. This formal scheme can be viewed as a moduli space for p-divisible groups which are isogenous to a fixed supersingular p-divisible group. See Boutot–Carayol [5] for a detailed account. Rapoport– Zink generalized these results to the (partial) uniformization of higher dimensional Shimura varieties by more general moduli spaces for p-divisible groups.

In this article we study abelian sheaves as positive characteristic analogues of abelian varieties. We investigate their moduli spaces and prove that these are algebraic stacks. Then our aim is to transfer the above uniformization results to the case of positive characteristic. For the case considered by Čerednik–Drinfeld this was accomplished already by Drinfeld [10]; see below. We use this as a guide line to transfer the results of Rapoport–Zink. We also obtain a partial uniformization of the moduli stacks of abelian sheaves. In the Hilbert-Blumenthal situation a similar uniformization result was obtained by Stuhler [38] using different methods.

Let us explain what abelian sheaves are by first going back 30 years to Drinfeld’s elliptic sheaves. Exploiting the analogy between number fields and function fields, Drinfeld [10, 12] invented the notions of elliptic modules (today called Drinfeld modules) and the dual notion of elliptic sheaves. These structures are analogues of elliptic curves for characteristic p in the following sense. Their endomorphism rings are rings of integers in global function fields of positive characteristic or orders in central division algebras over the later. On the other hand, the moduli spaces are varieties over smooth curves over a finite field. Through these two aspects in which global function fields of positive characteristic come into play, elliptic sheaves and variants of them proved to be fruitful for establishing the Langlands correspondence for GL_{n} over local and global function fields of positive characteristic. See the work of Drinfeld [10, 13, 14], Laumon–Rapoport–Stuhler [31], and Lafforgue [28]. Beyond this the analogy between elliptic modules and elliptic curves is abundant.

In this spirit, Anderson [1] introduced higher dimensional generalizations of Drinfeld’s elliptic modules and called them abelian t-modules. The concept of abelian sheaves is a higher dimensional generalization of elliptic sheaves. Both serve as characteristic p analogues of abelian varieties. Abelian sheaves were studied in various special instances in the past. In this article we intend to give a systematic treatment. The definition of abelian sheaves is as follows. Let C be a smooth projective curve over 𝔽_{q} and let ∞ ∈ C(𝔽_{q}) be a fixed point. For every 𝔽_{q}-scheme S we denote by σ the endomorphism of C ×_{𝔽_{q}} S that acts as the identity on the coordinates of C and as b ↦ b^{q} on the sections b ∈ O_{S}. Now an abelian sheaf of rank r and dimension d over S consists of the following data: a collection of locally free sheaves F_{i} of rank r on C×_{𝔽_{q}} S satisfying a certain periodicity condition. These sheaves are connected by two commuting sets of morphisms Π_{i} : F_{i} → F_{i+1} and τ_{i} : σ^{∗}F_{i} → F_{i+1} such that coker Π_{i} and coker τ_{i} are locally free O_{S}-modules of rank d, supported respectively on ∞ × S and on the graph of a morphism c : S → C called the characteristic of the abelian sheaf. An abelian sheaf of dimension 1 is the same as an elliptic sheaf. In this sense abelian sheaves are higher dimensional elliptic sheaves. The notion of abelian sheaf is dual to the notion of abelian t-module and related to Anderson’s t-motives. In fact, if the characteristic is different from ∞ an abelian sheaf over a field is nothing but a pure t-motive equipped with additional structure at infinity (Section 2). Therefore we like to view abelian sheaves as characteristic p analogues of polarized abelian varieties.

Abelian sheaves with appropriately defined level structure possess moduli spaces which are algebraic stacks locally of finite type over the curve C. The morphism to C is given by assigning to an abelian sheaf its characteristic. We denote the algebraic stacks of abelian sheaves of rank r and dimension d with H-level structures by Ab-Sh_{H}^{r,d}. Here H ⊂ GL_{r}(𝔸_{f}) is a compact open subgroup and 𝔸_{f} are the finite adeles of C. It should be noted that opposed to the case of elliptic sheaves, the stacks of abelian sheaves will in general not be schemes, not even if we add high level structures. This is due to the fact that for every level there are abelian sheaves having non-trivial automorphisms (see Remark 4.2). The non-representability also reflects in the uniformization; see below.

Then our aim is to study the uniformization of these moduli stacks at ∞. Let z be a uniformizing parameter of C at ∞. In the case of elliptic sheaves, Drinfeld [10, 11] showed that the moduli stacks are in fact smooth affine schemes which can be uniformized by a formal scheme Ω^{(r)}. This formal scheme is the characteristic p version of the one used by Čerednik and Drinfeld to uniformize Shimura curves. Correspondingly it is a moduli space for certain formal groups on which multiplication with z is an isogeny. All this was worked out in detail by Genestier [18]. We like to call these formal groups “z-divisible groups”. They play an important role also in our uniformization results.

So let us next explain some facts about z-divisible groups. Naturally these groups are of most use over schemes on which z is not a unit. Therefore we will from now on work over schemes S in Nilp_{𝔽_{q}[[z]]}, the category of 𝔽_{q}[[z]]-schemes on which z is locally nilpotent. However, since it is important to separate the two roles played by z as a uniformizing parameter at ∞ and as an element of O_{S}, we use the symbol z only for the first and we denote the image of z in O_{S} by ζ. Then S belongs to Nilp_{𝔽_{q}[[ζ]]}.

Classically p-divisible groups may be studied via their Dieudonné modules. There is a corresponding notion for z-divisible groups. A Dieudonné 𝔽_{q}[[z]]-module over S is a finite locally free O_{S}[[z]]-module F̂ with a σ-linear endomorphism F such that

1. coker F is locally free as an O_{S}-module,

2. (z − ζ) is nilpotent on coker F,

The theory of z-divisible groups and their Dieudonné 𝔽_{q}[[z]]-modules resembles many facets of the theory of p-divisible groups; see [21].

Also z-divisible groups are related to abelian sheaves through their Dieudonné 𝔽_{q}[[z]]-modules. Namely, the completion of an abelian sheaf over S at ∞ × S is a Dieudonné 𝔽_{q}[[z]]-module. The connection between abelian sheaves and z-divisible groups parallels the situation for abelian varieties. In particular, there is an analogue of the Serre-Tate-Theorem relating the deformation theory of abelian sheaves to the deformation theory of their z-divisible groups.

Finally we come to the uniformization of Ab-Sh_{H}^{r,d} at infinity. We begin by describing the uniformizing spaces. Let r, d, k, ℓ be positive integers with \frac{d}{r} = \frac{k}{ℓ} and k and ℓ relatively prime. Let O_{Δ} be the ring of integers in the central skew field over 𝔽_{q}((z)) of invariant k/ℓ. A special z-divisible O_{Δ}-module over S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} is a z-divisible group E of height rℓ and dimension dℓ with an action of O_{Δ} prolonging the action of 𝔽_{q}[[z]], such that the inclusion 𝔽_{q^{ℓ}} ⊂ O_{Δ} makes Lie E into a locally free 𝔽_{q^{ℓ}} ⊗_{𝔽_{q}} O_{S}-module of rank d. The z-divisible groups associated to abelian sheaves of rank r and dimension d are special z-divisible O_{Δ}-modules. Let 𝔼 be a special z-divisible O_{Δ}-module over Spec 𝔽_{q^{ℓ}}. Then the moduli problem of special z-divisible O_{Δ}-modules which are isogenous to 𝔼 is solved by a formal scheme G locally formally of finite type over Spf 𝔽_{q^{ℓ}}[[ζ]]. The later means that the reduced closed subscheme G_{red} of G is locally of finite type over Spec 𝔽_{q^{ℓ}}.

We fix an abelian sheaf \overline{𝕄} of rank r and dimension d over Spec 𝔽_{q^{ℓ}} whose restriction to C ∖∞ satisfies τ_{i} = Id_{r} ·σ^{∗} and we let 𝔼 be its z-divisible group. The Newton polygon of 𝔼 is a straight line. Let Z be the set of points s of Ab-Sh_{H}^{r,d} ×_{C} ∞ such that the universal abelian sheaf \underline{F}_{s} over s is isogenous to \overline{𝕄} over an algebraic closure of κ(s). It is an important step to show that Z is the set of points over which the Newton polygons of 𝔼 and of the z-divisible group associated to \underline{F}_{s} coincide. This implies that Z is a closed subset. We consider the formal completion Ab-Sh_{H}^{r,d}_{/Z} of Ab-Sh_{H}^{r,d} along Z. It is no longer an algebraic stack, but it is a formal algebraic stack over Spf 𝔽_{q^{ℓ}}[[ζ]]. Formal algebraic stacks are generalizations of algebraic stacks in the same sense as formal schemes generalize usual schemes. (The relevant facts on formal algebraic stacks are collected in an appendix.) Now we can uniformize Ab-Sh_{H}^{r,d}_{/Z} as follows. Being isogenous to the z-divisible group 𝔼 of \overline{𝕄}, the universal special z-divisible O_{Δ}-module on G gives rise to an abelian sheaf on G which we call its algebraization. Let J(Q) be the group of quasi-isogenies of \overline{𝕄}. There are natural embeddings of the group J(Q) into GL_{r}(𝔸_{f}) and into the group of quasi-isogenies of 𝔼. The later group acts on the formal scheme G. We let J(Q) act diagonally on G × GL_{r}(𝔸_{f}). Taking into account level structures we obtain the following

Uniformization Theorem 12.6. There is a canonical 1-isomorphism of formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stacks

Θ : J(Q)\G × GL_{r}(𝔸_{f})/H ⟶^{∼} Ab-Sh_{H}^{r,d}_{/Z} ×_{Spf 𝔽_{q}[[ζ]]} Spf 𝔽_{q^{ℓ}}[[ζ]]

At this point note that the non-representability of the algebraic stacks Ab-Sh_{H}^{r,d} also reflects in the uniformization. Namely, since all unipotent subgroups of J(Q) are torsion, in general a discrete subgroup of J(Q) cannot act fixed point free on G. So the quotients J(Q)\G × GL_{r}(𝔸_{f})/H can only be formal algebraic stacks and not formal algebraic spaces. This phenomenon does not occur in the p-adic uniformization of Shimura-varieties.

We like to mention an interesting aspect of the proof that is also related to the uniformizability of t-motives. Namely by work of Gardeyn [17] the t-motive associated to an abelian sheaf over a complete field extension K of 𝔽_{q}((ζ)) is uniformizable in the sense of Anderson [1] if and only if firstly the abelian sheaf extends to an abelian sheaf over the valuation ring R of a finite extension of K, and secondly its reduction modulo the maximal ideal of R is isogenous to \overline{𝕄}. We show that the second condition is closed. More precisely if \underline{F} is an abelian sheaf or rank r and dimension d over S ∈ Nilp_{𝔽_{q}[[ζ]]}, then the set of points in S over which \underline{F} is isogenous to \overline{𝕄} is closed. This is the key ingredient in the proof of the Uniformization Theorem. It is proved in Section 11 where we avoid the language of stacks and proceed in more down-to-earth terms.

Let us end by explaining the relation of our Uniformization Theorem to the results of Rapoport–Zink [34] and Drinfeld [10]. In the case of elliptic sheaves we have

Ab-Sh_{H}^{r,d}_{/Z} = Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q}[[ζ]] .

This follows from the fact that there is only one polygon between the points (0, 0) and (r, 1) with non-negative slopes and integral break points, namely the straight line. So all special z-divisible O_{Δ}-modules have the same Newton polygon as 𝔼 and therefore Z is all of Ab-Sh_{H}^{r,d} ×_{C} ∞. Moreover in this case G is the formal scheme Ω^{(r)} introduced above and J(Q) ≅ GL_{r}(Q). So we recover Drinfeld’s uniformization theorem

GL_{r}(Q)\Ω^{(r)} × GL_{r}(𝔸_{f})/H ⟶^{∼} M_{H}^{r} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]]

where M_{H}^{r} is the moduli scheme of Drinfeld modules of rank r with H-level structure.

Compared to [34], our uniformization theorem is analogous to the uniformization of the formal completion of Shimura varieties along the most supersingular isogeny class. In this sense uniformizable abelian sheaves correspond to supersingular abelian varieties. There is no doubt that the more general uniformization in [34] of an arbitrary isogeny class of abelian varieties also carries over to the setting of abelian sheaves. Furthermore, we have only described the uniformization at ∞ in this article. But the analogous uniformization results at other places of C should likewise hold. For example in the case of D-elliptic sheaves these were described by Hausberger [24].

Table of Contents

Part One Abelian Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Definition of Abelian Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Relation to Anderson’s t-Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Algebraicity of the Stacks of Abelian Sheaves . . . . . . . . . . . . . . . . . . 10

4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Uniformizing the Stacks of Abelian Sheaves  5

5 Isomorphism Classes Versus Isogeny Classes . . . . . . . . . . . . . . . . . . . 16

Part Two z-Divisible Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Definition of z-Divisible Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7 Dieudonné modules of z-Divisible Groups . . . . . . . . . . . . . . . . . . . . . . 21 8 The Serre-Tate-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 9 Moduli Spaces for z-Divisible Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Part Three Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 11 Closedness of the Uniformizable Locus . . . . . . . . . . . . . . . . . . . . . . . . . 32 12 The Uniformization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 13 Proof of the Uniformization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A Background on Formal Algebraic Stacks . . . . . . . . . . . . . . . . . . . . . . . 43 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Notation

Throughout this article we will denote by 𝔽_{q} the finite field having q elements and characteristic p, C a smooth projective geometrically irreducible curve over 𝔽_{q}, ∞ ∈ C(𝔽_{q}) a fixed point, C′ = C ∖ ∞ A = Γ(C′, O_{C′}) the ring of regular functions on C′, Q = 𝔽_{q}(C) the function field of C, viz. the field of fractions of A, Q_{∞} the completion of Q at ∞, A_{∞} the ring of integers in Q_{∞}, v ∈ M_{f} the finite places of C, i.e. the points of C′, A_{v} the completion of A at the finite place v of C,

Â = ∏_{v∈M_{f}} A_{v}

𝔸_{f} = Q ⊗_{A} Â the finite adeles of C, d, r, k, ℓ positive integers with \frac{d}{r} = \frac{k}{ℓ} and k and ℓ relatively prime, Δ the central skew field over Q_{∞} of invariant k/ℓ, O_{Δ} its ring of integers. All schemes, as well as their products and morphisms between them, are supposed to be over Spec 𝔽_{q}. If X is a scheme we let Sch_{X} be the category of X-schemes. For two schemes X and Y we write X × Y for their product over Spec 𝔽_{q}. A similar notation will be employed to the tensor product over 𝔽_{q}. If i : Y ↪ X is a closed immersion of schemes and F is a quasi-coherent sheaf on X we denote the restriction i^{∗}F by F|_{Y}. As is customary, we will use the term vector bundle for a locally free coherent sheaf on a scheme.

Starting with Section 6 we denote by ζ an indeterminant over 𝔽_{q}, 𝔽_{q}[[ζ]] the ring of formal power series in ζ, Spf 𝔽_{q}[[ζ]] the formal scheme which is the formal spectrum of 𝔽_{q}[[ζ]], Nilp

𝔽_{q}[[ζ]]

the category of schemes over Spf 𝔽_{q}[[ζ]], viz. the category of schemes over Spec 𝔽_{q}[[ζ]] on which ζ is locally nilpotent.

From Section 6 on all schemes will be in Nilp_{𝔽_{q}[[ζ]]}.

Let S be a scheme. We denote by

σ_{S} : S → S  its Frobenius endomorphism which acts as the identity on points and as the q-power map on the structure sheaf,

C_{S} = C × S σ = id_{C} ×σ_{S} the endomorphism of C_{S} that acts as the identity on the coordi- nates of C and as b ↦ b^{q} on the elements b ∈ O_{S}.

For a divisor D on C we denote by O_{C_{S}}(D) the invertible sheaf on C_{S} whose sections have divisor ≥ −D. If F is a coherent sheaf on C_{S} we set F(D) := F ⊗_{O_{C_{S}}} O_{C_{S}}(D). This notation applies in particular to the divisor D = n · ∞ for an integer n.

Part One Abelian Sheaves

1 Definition of Abelian Sheaves

Let S be a scheme and fix a morphism c : S → C. Let J be the ideal sheaf on C_{S} of the graph of c.

Definition 1.1. An abelian sheaf \underline{F} = (F_{i}, Π_{i}, τ_{i}) of rank r, dimension d, and characteristic c over S is a ladder of vector bundles F_{i} on C_{S} of rank r and injective homomorphisms Π_{i}, τ_{i} of O_{C_{S}}-modules (i ∈ ℤ) of the form

⋯ ⟶ F_{i−1} ⟶^{Π_{i−1}} F_{i}  ⟶^{Π_{i}} F_{i+1} ⟶^{Π_{i+1}} ⋯
↑^{τ_{i−2}}  ↑^{τ_{i−1}}  ↑^{τ_{i}}
⋯ ⟶ σ^{∗}F_{i−2} ⟶^{σ^{∗}Π_{i−2}} σ^{∗}F_{i−1} ⟶^{σ^{∗}Π_{i−1}} σ^{∗}F_{i} ⟶^{σ^{∗}Π_{i}} ⋯

subject to the following conditions (for all i ∈ ℤ):

1. the above diagram is commutative,

2. the morphism Π_{i+ℓ−1} ∘ … ∘ Π_{i} identifies F_{i} with the subsheaf F_{i+ℓ}(−k · ∞) of F_{i+ℓ},

3. the cokernel of Π_{i} is a locally free O_{S}-module of rank d,

4. the cokernel of τ_{i} is a locally free O_{S}-module of rank d and annihilated by J^{d}.

A morphism between two abelian sheaves (F_{i}, Π_{i}, τ_{i}) and (F′_{i}, Π′_{i}, τ′_{i}) is a collection of morphisms F_{i} → F′_{i} which commute with the Π’s and the τ’s.

Let us make a few remarks. By condition 2 the cokernel of Π_{i} is supported at ∞. Moreover, due to the periodicity condition 2 we have τ_{i+ℓn} = τ_{i} ⊗ id_{O_{C_{S}} (kn)} for all n ∈ ℤ. Finally, the reader should be aware that we allow that the ideal sheaf J acts non-trivially on coker τ_{i}. In this respect our abelian sheaves are more general than the abelian sheaves studied so far in the literature. As such we like to mention elliptic sheaves [12, 2] which we discuss later, and D-elliptic sheaves [31] which are abelian sheaves equipped with an action of an order D in a central division algebra over Q.

Definition 1.2. We denote by Ab-Sh^{r,d}(S) the category whose objects are the abelian sheaves of rank r and dimension d over S and whose morphisms are the isomorphisms of abelian sheaves. If S′ → S is a morphism of schemes the pullback of an abelian sheaf over S is an abelian sheaf over S′. This defines a fibered category Ab-Sh^{r,d} over the category of 𝔽_{q}-schemes, which is a stack for the fppf-topology. The functor which assigns to an object of Ab-Sh^{r,d}(S) the characteristic c : S → C defines a 1-morphism of stacks

Ab-Sh^{r,d} → C .

Next we introduce level structures on abelian sheaves. Let I ⊂ C′ = C ∖ ∞ be a finite closed subscheme and let \underline{F} = (F_{i}, Π_{i}, τ_{i}) be an abelian sheaf of rank r over S. Then the restrictions F_{i}|_{I×S} are all isomorphic via the morphisms Π_{i}. We call this restriction \underline{F}|_{I×S}. The same holds for the morphisms τ_{i}. So we obtain a morphism

τ|_{I×S} : σ^{∗}\underline{F}|_{I×S} ⟶ \underline{F}|_{I×S}

which we consider as a σ-linear map of \underline{F}|_{I×S} to itself. In this article we always assume that the characteristic c(S) of (F_{i}, Π_{i}, τ_{i}) is disjoint from I. Due to this assumption, τ|_{I×S} is an isomorphism. We consider the functor of τ-invariants of \underline{F}|_{I×S}

(\underline{F}|_{I})^{τ} : Sch_{S} ⟶  O_{I} − modules
T/S ⟼ ker H^{0}(I × T, τ|_{I×T} − id_{\underline{F}|_{I×T}}) .

In Böckle–Hartl [3, Theorem 2.5] the following fact is proved.

Proposition 1.3. The functor (\underline{F}|_{I})^{τ} is representable by a finite étale scheme over S which is a GL_{r}(O_{I})-torsor.

Definition 1.4. An I-level structure on (F_{i}, Π_{i}, τ_{i}) over S is an isomorphism

η̄: (\underline{F}|_{I×S}, τ|_{I×S}) ⟶^{∼} (O_{I×S}^{r}, Id_{r} ·σ^{∗})

from \underline{F}|_{I×S} to O_{I×S}^{r} that commutes with the σ-linear endomorphisms τ|_{I×S} on one and Id_{r} ·σ^{∗} on the other side.

If the characteristic c(S) meets I then \underline{F} does not possess any I-level structure.

Proposition 1.5. Let \underline{F} be an abelian sheaf over a field. Then the automorphism group of \underline{F} is finite.

Proof. We may assume that the base field is algebraically closed. Let η̄ be an I-level structure on \underline{F} = (F_{i}, Π_{i}, τ_{i}). It induces a group homomorphism

α_{I} : Aut(\underline{F}) → Aut(O_{I×S}^{r}, Id_{r} ·σ^{∗}) = GL_{r}(O_{I}) .

The later group is finite. We claim that α_{I} is injective for some sufficiently large finite subscheme I ⊂ C′. From this the proposition will follow. To establish the claim let (f_{i} : F_{i} → F_{i})_{i} be an automorphism of \underline{F}. Note that if f_{0} is the identity then f_{i} must also be the identity for all i. We now consider a finite flat morphism π : C → ℙ_{𝔽_{q}}^{1}. Since the map π_{∗} : Aut_{C}(F_{0}) → Aut_{ℙ^{1}}(π_{∗}F_{0}) is injective we may assume C = ℙ^{1}. Then the vector bundle π_{∗}F_{0} decomposes

π_{∗}F_{0} = ⨁_{i=1}^{s} O_{ℙ^{1}}(n_{i})

for uniquely determined integers n_{1} ≥ … ≥ n_{s}. We let P ∈ ℙ^{1} be a point and set I = (n_{1} − n_{s} + 1) · P. Then α_{I} is injective. ⊓ ⊔

Next we want to give a different definition of I-level structures. Note that via the natural isomorphism (\underline{F}|_{I})^{τ} ⊗_{O_{I}} O_{I×S} ⟶^{∼} \underline{F}|_{I×S} the I-level structures η̄ on \underline{F} over a connected S correspond bijectively to the isomorphisms of O_{I}-modules

η̄′ : (\underline{F}|_{I})^{τ} (S) ⟶^{∼} O_{I}^{r} .

We use this observation to define more general level structures by introducing the adelic point of view. Let \underline{F} be an abelian sheaf of rank r over S. We define the functor

(\underline{F}|Â)^{τ} : Sch_{S} ⟶ Â − modules
T/S ⟼ lim_{⟵_{I}} (\underline{F}|_{I})^{τ} (T) ,

where the limit is taken over all finite closed subschemes I ⊂ C′. Assume that S is connected and choose an algebraically closed base point ι : s → S. Due to Proposition 1.3 we may view (\underline{F}|Â)^{τ} as the Â[π_{1}(S, s)]-module (ι^{∗}\underline{F}|Â)^{τ} (s).

We consider the set IsomÂ((\underline{F}|Â)^{τ} , Â^{r}) := IsomÂ((ι^{∗}\underline{F}|Â)^{τ} (s), Â^{r}) of isomorphisms of Â-modules. Via its natural action on Â^{r} the group GL_{r}(Â) acts on this set from the left. Via its action on (ι^{∗}\underline{F}|Â)^{τ} (s) the group π_{1}(S, s) acts on it from the right.

Definition 1.6. Let H ⊂ GL_{r}(Â) be a compact open subgroup. An H-level structure on \underline{F} over S is an H-orbit in IsomÂ((\underline{F}|Â)^{τ} , Â^{r}) which is fixed by π_{1}(S, s). (Because of this later condition the notion of level structure is independent of the chosen base point.)

In particular if H = H_{I} = ker(GL_{r}(Â) → GL_{r}(O_{I})) an H-level structure is nothing else than an I-level structure. Note that as before an H-level structure can only exist if the characteristic c(S) does not meet the set of places v of C for which H_{v} ≠ GL_{r}(A_{v}).

Definition 1.7. Let Ab-Sh_{H}^{r,d}(S) be the category whose objects are the abelian sheaves of rank r and dimension d together with an H-level structure over S and whose morphisms are the isomorphisms of abelian sheaves which respect the level structures. Analogous to Definition 1.2, this defines a stack Ab-Sh_{H}^{r,d} over C.

For H = H_{∅} = GL_{r}(Â) the definition of H-level structure is vacuous. Therefore we have Ab-Sh_{H_{∅}}^{r,d} = Ab-Sh^{r,d}. We will show in Section 3 that these stacks are algebraic over C.

There is a free action of the group ℤ on these stacks given on objects by the map

[n] : (F_{i}, Π_{i}, τ_{i}) ↦ (F_{i+n}, Π_{i+n}, τ_{i+n}) .

Example 1.8 (Drinfeld Modules and Elliptic Sheaves). Let d = 1 and let H = H_{I}. Then an abelian sheaf is what was called an elliptic sheaf in Blum– Stuhler [2]. We consider the open substack

Ab-Sh_{H}^{r,1} ×_{C} C′

of abelian sheaves with characteristic disjoint from ∞. It is shown in [2, Theorem 3.2.1] that there is a 1-isomorphism between the stack Dr-Mod_{I}^{r} of Drinfeld A-modules of rank r with I-level structure and the open and closed substack of Ab-Sh_{H}^{r,1} ×_{C} C′ consisting of those (F_{i}, Π_{i}, τ_{i}) with deg(F_{0}|_{C_{s}}) = 1 − r for each algebraically closed point s of S.

On the other hand in Ab-Sh_{H}^{r,1} we always have deg(F_{i}|_{C_{s}}) = deg(F_{0}|_{C_{s}}) + i. Using the free action of ℤ on Ab-Sh_{H}^{r,1} we thus obtain a 1-isomorphism of stacks

Ab-Sh_{H}^{r,1} ×_{C} C′ ≅ ℤ × Dr-Mod_{I}^{r} .

The first factor gives the degree of F_{0}.

In fact if I ≠ ∅ the stack Dr-Mod_{I}^{r} is a smooth affine scheme of finite type over 𝔽_{q}. See for example [2, Theorem 2.3.8]. So Ab-Sh_{H}^{r,1} ×_{C} C′ is a smooth scheme locally of finite type in this case.

We will give another example in Section 4 which shows that for d > 1 one can neither expect that Ab-Sh_{H}^{r,d} is a scheme, nor that it is smooth over C′.

2 Relation to Anderson’s t-Motives

We will show that abelian sheaves with characteristic disjoint from ∞ are the same as polarized A-motives, a variant of Anderson’s [1] t-motives. Let S be a scheme and fix a characteristic morphism c : S → C′ disjoint from ∞. Let Γ(c) ⊂ C_{S} be the graph of c and let J be the ideal sheaf defining Γ(c).

Definition 2.1. A (pure) polarized A-motive (F, τ) of rank r, dimension d, and characteristic c over S consists of a vector bundle F on C_{S} of rank r and a morphism of coherent sheaves τ : σ^{∗}F → F(k · ∞) such that

1. the cokernel of τ is supported on Γ(c) ∪ ∞, the part supported on Γ(c) is annihilated by J^{d},

2. for every i = 1, …, ℓ the image of τ^{i} : σ^{i ∗}F → F(ik · ∞) lies in F(k · ∞) and τ^{ℓ} : σ^{ℓ ∗}F → F(k · ∞) is a local isomorphism at ∞,

3. locally at ∞, F is contained in the image τF.

Here we denote by τ^{i} the composition (τ ⊗ id_{O((i−1)k·∞)}) ∘ … ∘ σ^{(i−1) ∗}τ mapping

σ^{i ∗}F ⟶ σ^{(i−1) ∗}F(k · ∞) ⟶ … ⟶ σ^{∗}F((i − 1)k · ∞) ⟶ F(ik · ∞) .

Lemma 2.2. Conditions 1 and 2 imply that τ is injective and that the part of coker τ which is supported on Γ(c) (respectively on ∞) is a locally free O_{S}-module of rank d (respectively (ℓ − 1)d).

Proof. Consider the exact sequences of O_{S}-modules induced by τ

\xymatrix{0 \ar[r] & ker τ \ar[r] & σ^{∗}F \ar[r] \ar[d]^{τ} & im τ \ar[r] & 0 \\ 0 \ar[r] & im τ \ar[r] & F(k · ∞) \ar[r] & coker τ \ar[r] & 0 .}

For a point s ∈ S we tensor them with the residue field κ(s) at s to obtain

\xymatrix{0 \ar[r] & Tor_{1}^{O_{S}}(im τ, κ(s)) \ar[r] & (ker τ)_{s} \ar[r] & (σ^{∗}F)_{s} \ar[r]^{β} \ar[d]^{τ⊗ id} & (im τ)_{s} \ar[r] & 0 \\ 0 \ar[r] & Tor_{1}^{O_{S}}(coker τ, κ(s)) \ar[r] & (im τ)_{s} \ar[r]^{α} & F(k · ∞)_{s} \ar[r] & (coker τ)_{s} \ar[r] & 0 .}

We claim that τ ⊗ id is injective. Indeed consider a point on C_{s} and its local ring which is a PID. Then over this PID τ ⊗ id is a morphism between finite free modules of the same rank with torsion cokernel. Hence by the elementary divisor theorem τ ⊗ id = αβ must be injective.

Now the surjectivity of β shows that α is injective. Therefore Tor_{1}^{O_{S}}(coker τ, κ(s)) vanishes and by the local criterion for flatness [15, 6.8], coker τ is a flat O_{S}-module. This in turn implies the flatness of im τ. Hence (ker τ)_{s} is identified with the kernel of β. However, β is an isomorphism and thus (ker τ)_{s} is zero. By Nakayama’s lemma ker τ is zero and τ is injective. Finally let G be the part of coker τ which is supported on Γ(c). Since the characteristic is disjoint from ∞, G is a direct summand of coker τ. Then condition 2 implies that coker τ^{ℓ} = ⨁_{i=0}^{ℓ−1} σ^{i ∗}G and the lemma follows. ⊓ ⊔

If S = Spec K is the spectrum of a field, A = 𝔽_{q}[t], and (F, τ) is a polarized Amotive over S then the K[t, τ]-module Γ(C′_{S}, F) is a pure t-motive of weight d/r as defined by Anderson [1]. Compared to Anderson’s definition however, our polarized A-motive contains additional data at infinity which rigidifies the structure of the A-motive. Since Anderson’s t-motives serve as characteristic p analogues of abelian varieties this may justify our terminology.

As in Section 1 we can define H-level structures on polarized A-motives for compact open subgroups H ⊂ GL_{r}(Â). Correspondingly we obtain the stack Pol_{H}^{r,d} of polarized A-motives with H-level structure. This is a stack over C′.

Theorem 2.3. There is a 1-isomorphism of stacks

Ab-Sh_{H}^{r,d} ×_{C} C′ ≅ Pol_{H}^{r,d} .

Proof. We construct two mutually 2-inverse 1-morphisms T and T′ between the stacks in question. First consider the 1-morphism T : Ab-Sh_{H}^{r,d} ×_{C} C′ → Pol_{H}^{r,d} which assigns to an abelian sheaf (F_{i}, Π_{i}, τ_{i}) over S with characteristic disjoint from ∞ the polarized A-motive of rank r and dimension d consisting of

F := F_{0} , τ := Π_{ℓ−1} ∘ … ∘ Π_{1} ∘ τ_{0} : σ^{∗}F → F(k · ∞) .

Clearly this construction is also compatible with level structures.

Conversely let (F, τ) be a polarized A-motive of rank r and dimension d over S. For 0 ≤ i ≤ ℓ we set

F_{i} := F + … + τ^{i}F ⊂ F(k · ∞) .

Then F_{i} is equal to F outside ∞ due to the definition of τ. And locally at ∞ it is isomorphic to σ^{i ∗}F due to condition 3 of Definition 2.1. Therefore F_{i} is locally free on C_{S}. Let Π_{i} : F_{i} → F_{i+1} be the inclusion and let τ_{i} : σ^{∗}F_{i} → F_{i+1} be the morphism τ. Again by condition 3, coker Π_{i} is supported at ∞ and coker τ_{i} is supported on Γ(c) and annihilated by J^{d}. By Lemma 2.2 these cokernels are locally free O_{S}-modules of rank d. Furthermore, by condition 2 we have F_{ℓ} = F(k · ∞). For arbitrary i ∈ ℤ we take n ∈ ℤ such that 0 ≤ i − nℓ < ℓ and define

F_{i} := F_{i−nℓ} ⊗_{O_{C_{S}}} O_{C_{S}}(nk · ∞) , Π_{i} := Π_{i−nℓ} ⊗ id , τ_{i} := τ_{i−nℓ} ⊗ id .

Then (F_{i}, Π_{i}, τ_{i}) is an abelian sheaf of rank r and dimension d over S. This construction is also compatible with level structures and defines a 1-morphism T′ : Pol_{H}^{r,d} → Ab-Sh_{H}^{r,d} ×_{C} C′. One easily proves that T and T′ are mutually 2-inverse. ⊓ ⊔

Remark 2.4. Example 1.8 together with Theorem 2.3 shows that every Drinfeld module carries a canonical polarization. This parallels the situation for elliptic curves.

3 Algebraicity of the Stacks of Abelian Sheaves

In this section we will prove the following theorem.

Theorem 3.1. Ab-Sh_{H}^{r,d} is an algebraic stack in the sense of Deligne–Mumford [9], locally of finite type over C.

Note that the example in Section 4 below shows that it will in general not be smooth over C. Nevertheless, if the characteristic is disjoint from ∞ and one stratifies the stack according to the isomorphy type of the O_{C}-modules coker Π_{i} and coker τ_{i} then each stratum will be smooth over C′.

More precisely, we let z be a uniformizing parameter on C at ∞ and we fix a flag G_{•} of 𝔽_{q}[z]/z^{k}-submodules

(3.1)  (0) ⊂ G_{1} ⊂ … ⊂ G_{ℓ−1} ⊂ G_{ℓ} = (𝔽_{q}[z]/z^{k})^{⊕r}

such that the successive quotients all have dimension d over 𝔽_{q}. We say that an abelian sheaf \underline{F} of rank r and dimension d over S has isomorphy type G_{•} of Π if for every point s ∈ S the flag of O_{C_{s}}-modules

(0) ⊂ F_{1}/F_{0} ⊂ … ⊂ F_{ℓ−1}/F_{0} ⊂ F_{ℓ}/F_{0} = F_{0}(k · ∞)/F_{0}

is isomorphic to the flag G_{•} ⊗_{𝔽_{q}} κ(s).

To fix the isomorphy type of τ let S be a scheme with characteristic morphism c : S → C. We denote by J ⊂ O_{C_{S}} the ideal sheaf defining the graph of c, by Γ_{d} the closed subscheme of C_{S} defined by J^{d}, and by O_{d} the structure sheaf O_{C_{S}}/J^{d} of Γ_{d}. We fix integers

(3.2)  (e_{1} ≤ e_{2} ≤ … ≤ e_{r}) = \underline{e}

between 0 and d with e_{1} + … + e_{r} = d. We say that an abelian sheaf \underline{F} of rank r and dimension d over S has has isomorphy type \underline{e} of τ if for every point s ∈ S the O_{C_{s}}-module coker τ_{−1} is isomorphic to

⨁_{ν=1}^{r} O_{C_{s}}/ J^{e_{ν}} O_{C_{s}} .

Note that if the characteristic is disjoint from ∞ then the cokernels of all τ_{i} are isomorphic.

We consider the locally closed subset (see Laumon–Moret-Bailly [30, §5]) of Ab-Sh_{H}^{r,d} consisting of the points over which the universal abelian sheaf has isomorphy types G_{•} of Π and \underline{e} of τ. We give this subset the reduced induced structure ([30, 4.10]) and obtain a substack Ab-Sh_{H}^{r,d,G_{•},\underline{e}} of Ab-Sh_{H}^{r,d}.

Theorem 3.2. The substack Ab-Sh_{H}^{r,d,G_{•},\underline{e}} ×_{C} C′ is smooth over C′ of relative dimension ∑_{μ>ν}(e_{μ} − e_{ν}) if non-empty.

The proof of these theorems follows Laumon–Rapoport–Stuhler [31]. For a given compact open subgroup H ⊂ GL_{r}(Â) consider a closed subscheme I ⊂ C′ which is supported on the places v for which H_{v} ≠ GL_{r}(A_{v}) and satisfies

H ⊃ H_{I} := ker(GL_{r}(Â) → GL_{r}(O_{I})) .

Then Ab-Sh_{H_{I}}^{r,d} is finite étale over Ab-Sh_{H}^{r,d} by Proposition 1.3. In fact it is even a H/H_{I}-torsor. Hence Ab-Sh_{H}^{r,d} is a quotient of Ab-Sh_{H_{I}}^{r,d} in the sense of stacks; cf. [30, 4.6.1]. Therefore it suffices to prove the two theorems for H = H_{I}. We assume this situation from now on.

We will cover Ab-Sh_{H}^{r,d} by open substacks corresponding to stable vector bundles with additional level structure. To be precise we proceed as follows. Let F be a locally free sheaf of rank r on C_{S}. An I-level structure on F is an isomorphism

η̄: F|_{I×S} ⟶^{∼} O_{I×S}^{r} of O_{I×S}-modules.

Definition 3.3. We say that the pair (F, η̄) is stable if for all algebraically closed points s ∈ S and all locally free O_{C_{s}}-modules G properly contained in F_{s}, we have

\frac{deg(G) − deg(I)}{rk(G)} < \frac{deg(F_{s}) − deg(I)}{rk(F_{s})}

(compare Seshadri [37, 4.I. Définition 2]).

We denote by Ab-Sh_{H,st}^{r,d} the open substack of Ab-Sh_{H}^{r,d} of those abelian sheaves for which (F_{0}, η̄) is stable. We will show that Ab-Sh_{H,st}^{r,d} is representable by a disjoint union of quasi-projective schemes of relative dimension d(r − 1) over C ∖ I if I ≠ ∅. For this purpose we need to introduce some additional stacks.

We denote by Vec_{I}^{r} the stack classifying locally free sheaves of rank r on C with I-level structure and by Vec_{I,st}^{r} the substack of such locally free sheaves which are stable. Seshadri [37, 4.III] proves the following fact (see also [31, 4.3]). Note that there is a typing error in the formula for the dimension in [37].

Proposition 3.4. If I ≠ ∅ the stack Vec_{I,st}^{r} is representable by a disjoint union of quasi-projective schemes over 𝔽_{q} which are smooth of dimension r^{2}(g − 1 + deg I). Here g is the genus of C.

Definition 3.5. Let S be a scheme and let (F_{i}, Π_{i}) be a sequence of locally free sheaves on C_{S} as in the first row of the Definition 1.1 of an abelian sheaf. Suppose that (F_{i}, Π_{i}) satisfies the conditions 2 and 3 of Definition 1.1. An I-level structure on (F_{i}, Π_{i}) is a collection of I-level structures η̄_{i} on the F_{i} that are compatible with the morphisms Π_{i}. We denote by Seq_{I}^{r,d} the stack classifying sequences (F_{i}, Π_{i}) as above together with I-level structures η̄_{i} and by Seq_{I,st}^{r,d} the open substack of those sequences for which F_{0} with its level structure is stable.

Lemma 3.6. The natural 1-morphism

Seq_{I}^{r,d} → Vec_{I}^{r} , (F_{i}, Π_{i}, η̄_{i}) ↦ (F_{0}, η̄_{0})

is representable by a closed subscheme of a flag variety. The strata Seq_{I}^{r,d,G_{•}} with fixed isomorphy type G_{•} of Π are smooth over Vec_{I}^{r} if non-empty. In particular Seq_{I,st}^{r,d} and Seq_{I,st}^{r,d,G_{•}} are representable by a disjoint union of quasi-projective schemes (respectively quasi-projective and smooth schemes) over 𝔽_{q} if I ≠ ∅.

Proof. Let F_{0} on C_{S} be given corresponding to a 1-morphism S → Vec_{I}^{r}. Due to the periodicity condition 2 the sequence (F_{i}, Π_{i}) corresponds to a flag of length ℓ of O_{C_{S}}-submodules

(3.3)  (0) ⊂ F_{1}/F_{0} ⊂ … ⊂ F_{ℓ−1}/F_{0} ⊂ F_{ℓ}/F_{0} = F_{0}(k · ∞)/F_{0}

such that the successive quotients are all locally free O_{S}-modules of rank d. Hence the first assertion follows.

To prove the statement about Seq_{I}^{r,d,G_{•}} note that locally on S the sheaf F_{ℓ}/F_{0} is isomorphic to O_{S}[z]/z^{k}. By the elementary divisor theorem a point Spec K of Seq_{I}^{r,d} belongs to Seq_{I}^{r,d,G_{•}} if and only if the flag (3.3) is conjugate to G_{•} under GL_{r}(K[z]/z^{k}). Therefore Seq_{I}^{r,d,G_{•}} is relatively representable over Vec_{I}^{r} by the homogeneous space

GL_{r}(𝔽_{q}[z]/z^{k})/ Stab(G_{•}) .

The group GL_{r}(𝔽_{q}[z]/z^{k}) is the Weil restriction R_{(𝔽_{q}[z]/z^{k})/𝔽_{q}} GL_{r} and hence a smooth connected algebraic group over 𝔽_{q}. The stabilizer of G_{•} corresponds to a closed algebraic subgroup defined over 𝔽_{q}. Thus the above homogeneous space is a smooth algebraic variety over 𝔽_{q}. From this the lemma follows. ⊓ ⊔ Definition 3.7. We let Hecke_{I}^{r,d} be the stack classifying the commutative diagrams with I-level structures

⋯ ⟶ F_{i−1} ⟶^{Π_{i−1}} F_{i} ⟶^{Π_{i}} F_{i+1} ⟶^{Π_{i+1}} ⋯
↑^{t_{i−2}}  ↑^{t_{i−1}}  ↑^{t_{i}}
⋯ ⟶ F′_{i−2} ⟶^{Π′_{i−2}} F′_{i−1} ⟶^{Π′_{i−1}} F′_{i} ⟶^{Π′_{i}} ⋯

such that the sequences (F_{i}, Π_{i}) and (F′_{i}, Π′_{i}) with their I-level structures belong to Seq_{I}^{r,d} and such that the t_{i} satisfy conditions 1 and 4 of Definition 1.1 and respect the I-level structures. Assigning to such a diagram over S the morphism S → C ∖ I on whose graph the cokernels of the t_{i} are supported, defines a 1-morphism of stacks Hecke_{I}^{r,d} → C ∖ I.

The above stacks fit into the following 2-cartesian diagram of stacks

(3.4)

Ab-Sh_{H}^{r,d}  ⟶  Seq_{I}^{r,d}
↓  ↓^{( id,σ_{Seq} )}
Hecke_{I}^{r,d}

(1^{st} row, 2^{nd} row)
⟶ Seq_{I}^{r,d} × Seq_{I}^{r,d}
↓
C ∖ I

On the stack Hecke_{I}^{r,d} the cokernel of t_{−1} is a quotient of F_{0} which is locally free of rank d over the base and supported on the graph of the characteristic morphism c. We analyze this property. Let T be the stack

(C ∖ I) × Seq_{I}^{r,d} .

On C × T consider the ideal sheaf J defining the graph Γ(c) of the characteristic morphism c : T → C ∖ I ⊂ C (see [30, §12]). We denote by Γ_{d} the closed substack of C × T defined by J^{d} and by O_{d} the sheaf O_{C×T} /J^{d}. The quotients of F_{0} that are supported on Γ(c) and are locally free over T of rank d are classified by Grothendieck’s Quot-scheme relative to T [FGA, n^{o} 221, Théorème 3.1]

Quot_{F_{0}⊗O_{d}/Γ_{d}/T}^{d} .

It is a stack projective over T .

Lemma 3.8. The 1-morphism

Hecke_{I}^{r,d} → Quot_{F_{0}⊗O_{d}/Γ_{d}/T}^{d} × Seq_{I}^{r,d}

given by the cokernel of t_{−1} and the second row, is representable by a closed immersion. Over C ∖ (I ∪ ∞) the 1-morphism

Hecke_{I}^{r,d} → Quot_{F_{0}⊗O_{d}/Γ_{d}/T}^{d}

obtained by projection onto the first factor is a 1-isomorphism. In particular, the 1-morphism Hecke_{I}^{r,d} → Seq_{I}^{r,d} × Seq_{I}^{r,d} from (3.4) is representable by a quasiprojective morphism.

Proof. The substack Hecke_{I}^{r,d} of Quot_{F_{0}⊗O_{d}/Γ_{d}/T}^{d} × Seq_{I}^{r,d} is defined by the following conditions 1. F′_{−1} equals the kernel of the morphism from F_{0} to the universal quotient,

2. for each i = −ℓ, …, −2 the sheaf F′_{i} is contained in the intersection of F_{i+1} and F′_{−1} which we view as subsheaves of F_{0} via Π_{−1} ∘ … ∘ Π_{i+1} and t_{−1},

3. if we let t_{i} be the inclusion F′_{i} ⊂ F_{i+1} then coker t_{i} is annihilated by J^{d},

4. t_{−1} is compatible with the I-level structures on F′_{−1} and F_{0}.

Namely by descending induction on i the short exact sequences of O_{S}-modules

[figure: 0; coker t_{i−1}; coker(Π_{i} ∘ t_{i−1}); coker Π_{i}; 0; 0; coker Π′_{i−1}; coker(t_{i} ∘ Π′_{i−1}); coker t_{i}; 0]

imply that coker t_{i} is a locally free O_{S}-module of rank d. Now clearly the above conditions are represented by a closed immersion.

Over C ∖ (I ∪ ∞) defining F′_{i} as the intersection F_{i+1} ∩ F′_{−1} in conditions 1 and 2 automatically gives an object (F′_{i}, Π′_{i}) in Seq_{I}^{r,d}. This proves that the projection onto the first factor is a 1-isomorphism there. ⊓ ⊔

Proof (of Theorem 3.1). Recall that we have assumed H = H_{I}. Considering the diagram (3.4) we conclude from the previous lemmas that Ab-Sh_{H,st}^{r,d} is representable by a disjoint union of quasi-projective schemes over C ∖ I if I ≠ ∅.

Now we let I ⊂ I′ ⊂ C′ be two finite closed subschemes with I′ ≠ ∅ and we set H′ = H_{I′}. By restricting I′-level structures to I-level structures we obtain a 1-morphism of stacks

r_{I′,I} : Ab-Sh_{H′}^{r,d} → Ab-Sh_{H}^{r,d} .

Over C ∖ I′ this 1-morphism is a torsor under the finite group

G_{I′,I} := ker(GL_{r}(O_{I′}) → GL_{r}(O_{I})) ≅ H/H′

due to Proposition 1.3. Since r_{I′,I}^{−1}(Ab-Sh_{H,st}^{r,d}) ⊂ Ab-Sh_{H′,st}^{r,d}, the open substack Ab-Sh_{H′,st}^{r,d} which is stable under G_{I′,I}, gives as a quotient in the sense of stacks an open substack

Ab-Sh_{H′,st}^{r,d} / G_{I′,I} ⊂ Ab-Sh_{H}^{r,d}

that contains Ab-Sh_{H,st}^{r,d} ×_{C∖I} C∖I′. It is an algebraic stack in the sense of Deligne– Mumford. If we let I′ vary among finite closed subschemes of C′ containing I these open substacks cover Ab-Sh_{H}^{r,d}, since every vector bundle becomes stable for a sufficiently high level structure. This proves Theorem 3.1 except for the assertion on the dimension which follows from Theorem 3.2. ⊓ ⊔

Proof (of Theorem 3.2). We denote by Quot\underline{e} the reduced, locally closed substack of Quot_{F_{0}⊗O_{d}/Γ_{d}/T}^{d} consisting of those points for which the universal quotient is isomorphic to

⨁_{ν=1}^{r} O_{C_{s}}/ J^{e_{ν}} O_{C_{s}} .

We claim that Quot\underline{e} is smooth over T of relative dimension ∑_{μ>ν}(e_{μ}−e_{ν}). Indeed, locally on T the sheaf F_{0} ⊗O_{d} is isomorphic to O_{d}^{r}. Let H ⊂ O_{d}^{r} be the kernel of the morphism from O_{d}^{r} to the universal quotient on Quot\underline{e}. The condition on the isomorphy type of the universal quotient implies that H is conjugate to ⊕_{ν=1}^{r}J^{e_{ν}} /J^{d} under GL_{r}(O_{d}). Therefore Quot\underline{e} is locally isomorphic to the homogeneous space

GL_{r}(O_{d}) / Stab(⊕_{ν=1}^{r}J^{e_{ν}} /J^{d}) .

As in the proof of Lemma 3.6 this homogeneous space is smooth and the claim follows. The theorem can now be deduced using Lemma 3.8 and applying [31, Lemma 4.2] to diagram (3.4). ⊓ ⊔

4 An Example

In this section let C = ℙ_{𝔽_{q}}^{1} and A = 𝔽_{q}[t]. Let I = V(t) ⊂ C and H = H_{I} and set z = \frac{1}{t}. Then C ∖ I = Spec 𝔽_{q}[z]. We consider the case where d = r = 2, k = ℓ = 1 and describe the algebraic stack Ab-Sh_{H}^{2,2}. It decomposes

Ab-Sh_{H}^{2,2} = ∐_{n∈ℤ} Ab-Sh_{H}^{2,2}(n)

into the open and closed substacks on which the vector bundle F_{0} has degree n. The shift by 1 from 1.7 yields a 1-isomorphism Ab-Sh_{H}^{2,2}(n) → Ab-Sh_{H}^{2,2}(n + 2). So it suffices to describe Ab-Sh_{H}^{2,2}(n) for n = 0, 1. We want to treat the case n = 0 here.

Let M_{I}^{2,2} be the scheme

Spec 𝔽_{q}[ζ][a_{μν} : 1 ≤ μ, ν ≤ 2] / (a_{11} + a_{22} + 2ζ , a_{11}a_{22} − a_{12}a_{21} − ζ^{2}) .

We view (a_{μν}) as a 2 × 2 matrix with trace −2ζ and determinant ζ^{2}. Mapping z to ζ defines a morphism c : M_{I}^{2,2} → C. On S = M_{I}^{2,2} we set for i ∈ ℤ

F_{i} = O_{C_{S}}(i · ∞)^{⊕2} and τ_{i} = (1 + t(a_{μν})) · σ^{∗} : σ^{∗}F_{i} → F_{i+1} and

we let Π_{i} : F_{i} → F_{i+1} be the morphism induced by the inclusion O_{C_{S}} ⊂ O_{C_{S}}(∞). Due to the trace and determinant condition on the matrix (a_{μν}), the data (F_{i}, Π_{i}, τ_{i}) is an abelian sheaf of rank 2, dimension 2, and characteristic c over S.

We want to define an I-level structure on \underline{F} = (F_{i}, Π_{i}, τ_{i}). Note that \underline{F}|_{I×S} is canonically isomorphic to O_{S}^{2} with τ|_{I×S} = Id_{2} ·σ^{∗}. Hence the identity morphism on O_{S}^{2} defines an I-level structure η̄ on \underline{F}.

Proposition 4.1. The 1-morphism M_{I}^{2,2} → Ab-Sh_{H}^{2,2} induced by (\underline{F}, η̄) identifies M_{I}^{2,2} with the (representable) open substack of Ab-Sh_{H}^{2,2} on which the underlying vector bundle with level structure (F_{0}, η̄) is stable (Definition 3.3) and has degree zero.

We will see below that (F_{0}, η̄) is stable and of degree zero if and only if F_{0} ≅ O_{C_{S}}^{2}.

Proof. Let T be a scheme together with a characteristic morphism c′ : T → Spec 𝔽_{q}[z] disjoint from I. Denote the image c′^{∗}(z) in O_{T} by ζ′. Let (F′_{i}, Π′_{i}, τ′_{i}, η̄′) be an abelian sheaf of rank 2, dimension 2, and characteristic c′ over T with Ilevel structure such that (F′_{0}, η̄′) is stable and of degree zero. We have to exhibit a uniquely defined morphism f : T → M_{I}^{2,2} such that f^{∗}(\underline{F}, η̄) ≅ (F′_{i}, Π′_{i}, τ′_{i}, η̄′). Since the morphisms Π′_{i} identify F′_{i} with F′_{0}(i · ∞) it suffices to concentrate on F′_{0} and τ′_{0}.

We claim that the stability condition implies F′_{0} ≅ O_{C_{T}}^{2} globally on T. Indeed let π : C_{T} → T be the projection onto the second factor. We first show that π_{∗}F′_{0} is locally free of rank 2 on T and that π^{∗}π_{∗}F′_{0} → F′_{0} is an isomorphism. Let s ∈ T be an algebraically closed point. The stability implies that every invertible O_{C_{s}}-module G ⊂ F′_{0}|_{C_{s}} has degree at most 0. Hence F′_{0}|_{C_{s}} ≅ O_{C_{s}}^{2} and thus H^{1}(C_{s}, F′_{0}|_{C_{s}}) = (0). By the theorem on cohomology and base change [23, III.12.11] this implies that R^{1}π_{∗}F′_{0} vanishes and that π_{∗}F′_{0} is locally free of rank 2 on T. Moreover π^{∗}π_{∗}F′_{0} → F′_{0} is an isomorphism in the fiber over s and hence on all of T by Nakayama. Now the level structure η̄′ induces an isomorphism π_{∗}F′_{0} ≅ (π^{∗}π_{∗}F′_{0})|_{I×T} ⟶^{∼} O_{T}^{2} . From this our claim follows.

As τ′_{0} maps σ^{∗}F′_{0} into F′_{1} = F′_{0}(∞), it is represented with respect to a basis of F′_{0} by a matrix

τ′_{0} = (U_{0} + tU_{1}) · σ^{∗} with U_{0} ∈ GL_{2}(O_{T}) and U_{1} ∈ M_{2}(O_{T}) .

Identifying F′_{0}|_{I×T} with O_{T}^{2} , we can express η̄′ by a matrix in GL_{2}(O_{T}). There is a uniquely defined change of basis of F′_{0} such that this matrix becomes the identity. Then we also have U_{0} = Id. The condition on coker τ′_{0} now implies that

det(Id +tU_{1}) = (1 − ζ′t)^{2} .

Then the required morphism f : T → M_{I}^{2,2} is given by f^{∗}(a_{μν}) = U_{1} and f^{∗}(ζ) = ζ′. ⊓ ⊔

Remark 4.2. From this example one sees that Ab-Sh_{H}^{r,d} need not be smooth over C. Namely, M_{I}^{2,2} is not smooth at the points with a_{12} = a_{21} = 0, a_{11} = a_{22} = −ζ. The reason for this is that at these points the O_{C_{s}}-module coker τ_{0} is isomorphic to (κ(s)[z]/(z − ζ))^{⊕2} whereas at all other points it is isomorphic to κ(s)[z]/(z − ζ)^{2}. Compare with Theorem 3.2.

This example also shows that in general one cannot hope that the stacks Ab-Sh_{H}^{r,d} are schemes. Namely, for any level I there are abelian sheaves of rank 2 and dimension 2 with an I-level structure that have non-trivial automorphisms. Indeed, let I = V(a) ⊂ C′ for an a ∈ 𝔽_{q}[t] with deg a = n. Let c : S → C ∖ I be arbitrary and denote the image of z in O_{S} by ζ. Let f ∈ O_{S}[t] have degree ≤ n + 1. Then the abelian sheaf with

F_{i} = O_{C_{S}}((i + n) · ∞) ⊕ O_{C_{S}}(i · ∞) , τ_{i} = ( \begin{matrix}1 − ζt & f \\ 0 & 1 − ζt \end{matrix} ) · σ^{∗}

admits an I-level structure after a finite étale extension of S. It has non-trivial automorphisms compatible with this level structure of the form

( \begin{matrix}1 & xa \\ 0 & 1 \end{matrix} )

for x ∈ 𝔽_{q}.

As a consequence Ab-Sh_{H}^{r,d} is not quasi-compact in general. Indeed recall from the proof of Theorem 3.1 the covering of Ab-Sh_{H}^{r,d} by open substacks

U_{I} := Ab-Sh_{H_{I},st}^{r,d} / (H/H_{I})

where I ⊂ C′ runs through all finite subschemes with H_{I} ⊂ H. Note that U_{I} ⊂ U_{I′} if I ⊂ I′. Now if Ab-Sh_{H}^{r,d} were quasi-compact this covering would have a finite refinement. So Ab-Sh_{H}^{r,d} would equal a single U_{I} for a large enough I. From this we could deduce that

Ab-Sh_{H_{I}}^{r,d} = Ab-Sh_{H_{I},st}^{r,d}

is in fact a scheme. Since for r = d = 2 the later is not the case, Ab-Sh_{H}^{2,2} is not quasi-compact.

5 Isomorphism Classes Versus Isogeny Classes

The general yoga that mediates between isomorphism classes of abelian varieties over ℤ_{p}-schemes and prime-to-p isogeny classes of such can be transfered to abelian sheaves. This will allow us to define H-level structures for arbitrary compact open subgroups H ⊂ GL_{r}(𝔸_{f}). We begin by defining the notion of isogeny for abelian sheaves.

Definition 5.1. A morphism between abelian sheaves (F_{i}, Π_{i}, τ_{i}) and (F′_{i}, Π′_{i}, τ′_{i}) over S is called an isogeny if

1. all morphisms F_{i} → F′_{i} are injective,

2. coker(F_{i} → F′_{i}) is supported on D × S for an effective divisor D ⊂ C, and 3. coker(F_{i} → F′_{i}) is locally free of finite rank as an O_{S}-module.

The isogeny is called finite (or prime to ∞) if for all i the support of coker(F_{i} → F′_{i}) is disjoint from ∞. A (finite) quasi-isogeny between (F_{i}, Π_{i}, τ_{i}) and (F′_{i}, Π′_{i}, τ′_{i}) is a (finite) isogeny between (F_{i}, Π_{i}, τ_{i}) and (F′_{i}(D), Π′_{i}, τ′_{i}) for some effective divisor D ⊂ C (respectively D ⊂ C ∖ ∞).

Note that an isogeny between two abelian sheaves can only exist if they both have the same rank and dimension. The following proposition is evident. It justifies our definition of quasi-isogenies.

Proposition 5.2. Let α : \underline{F} → \underline{F}′ be an isogeny of abelian sheaves over S. Then there exists an effective divisor D ⊂ C and an isogeny α^{∨} : \underline{F}′ → \underline{F}(D) with α ∘ α^{∨} and α^{∨} ∘α being the isogenies induced by the inclusion O_{C} ⊂ O_{C}(D). If D is chosen minimal then D and α^{∨} are uniquely determined.

Example 5.3. Let \underline{F} = (F_{i}, Π_{i}, τ_{i}) be an abelian sheaf of rank r and dimension d over S. Then the collection of the Π_{i} defines an isogeny

(Π_{i}) : \underline{F}[1] → \underline{F}

where [1] denotes the shift by 1 (cf. 1.7). Similarly let P be a point on C and let S ∈ Nilp_{A_{P}} via the characteristic morphism c : S → C. Then the collection of the τ_{i} defines an isogeny

(τ_{i}) : σ^{∗}\underline{F}[1] → \underline{F} .

For abelian varieties there is a general principle relating isomorphism classes of abelian varieties with level structure over ℤ_{p}-schemes to prime-to-p isogeny classes of such. This principle also applies to abelian sheaves. We need to introduce the analogue for ℤ_{p} in our situation.

Notation 5.4. The local ring O_{C,∞} is a discrete valuation ring. Let z be a uniformizing parameter. We identify the completion of O_{C,∞} with 𝔽_{q}[[z]] and Q_{∞} with 𝔽_{q}((z)). Let ζ be an indeterminant over 𝔽_{q} and denote by 𝔽_{q}[[ζ]] the ring of formal power series in ζ. Fix the characteristic morphism c : Spec 𝔽_{q}[[ζ]] → C defined by c^{∗}(z) = ζ. Clearly this morphism identifies 𝔽_{q}[[z]] with 𝔽_{q}[[ζ]]. However, since z − ζ does not necessarily act trivially on coker τ_{i} we use two different symbols to separate the two roles played by z as a uniformizing parameter at ∞ and as an element of O_{S}. From now on all base schemes for abelian sheaves will be schemes over Spec 𝔽_{q}[[ζ]]. We consider the base change of our stacks

Ab-Sh_{H}^{r,d} ×_{C} Spec 𝔽_{q}[[ζ]] .

For the sake of brevity we denote them again by Ab-Sh_{H}^{r,d}. This should not cause confusion since from now on we work entirely in the local situation over Spec 𝔽_{q}[[ζ]].

Working with isogeny classes instead of isomorphism classes we have to modify our definition of H-level structures. The new definition will have the additional advantage that it extends to arbitrary compact open subgroups H ⊂ GL_{r}(𝔸_{f}) which are not necessarily contained in GL_{r}(Â). Let \underline{F} be an abelian sheaf of rank r over S. We define the functor

(\underline{F}|_{𝔸_{f}})^{τ} : Sch_{S} ⟶ 𝔸_{f} − modules
T/S ⟼ 𝔸_{f} ⊗Â (\underline{F}|Â)^{τ} (T) .

Assume that S is connected. Choosing an algebraically closed base point ι : s → S we may view (\underline{F}|_{𝔸_{f}})^{τ} as the 𝔸_{f}[π_{1}(S, s)]-module (ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s).

We consider the set Isom_{𝔸_{f}} ((\underline{F}|_{𝔸_{f}})^{τ} , 𝔸_{f}^{r}) := Isom_{𝔸_{f}} ((ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s), 𝔸_{f}^{r}) of isomorphisms of 𝔸_{f}-modules. Via its natural action on 𝔸_{f}^{r} the group GL_{r}(𝔸_{f}) acts on this set from the left. Via its action on (ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s) the group π_{1}(S, s) acts on it from the right.

Definition 5.5. Let H ⊂ GL_{r}(𝔸_{f}) be a compact open subgroup. A rational Hlevel structure on \underline{F} over S is an H-orbit in Isom_{𝔸_{f}} ((\underline{F}|_{𝔸_{f}})^{τ} , 𝔸_{f}^{r}) which is fixed by π_{1}(S, s). (Again the later condition implies that the notion of level structure is independent of the chosen base point.)

Every quasi-isogeny α : \underline{F} → \underline{F}′ induces an isomorphism

(α|_{𝔸_{f}})^{τ} : (ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s) ⟶^{∼} (ι^{∗}\underline{F}′|_{𝔸_{f}})^{τ} (s)

and thus carries rational H-level structures on \underline{F} to rational H-level structures on \underline{F}′.

Theorem 5.6. If H ⊂ GL_{r}(Â) is a compact open subgroup then the stack Ab-Sh_{H}^{r,d} is canonically 1-isomorphic to the stack X whose category of S-valued points has

as objects: all pairs (\underline{F}, γ̄) consisting of an abelian sheaf \underline{F} of rank r and dimension

d and a rational H-level structure γ̄ on \underline{F} over S and as morphisms: all finite quasi-isogenies that are compatible with the rational H-

level structures.

Proof. Let S be a Spec 𝔽_{q}[[ζ]]-scheme and let (\underline{F}, η̄) be an object of Ab-Sh_{H}^{r,d}(S). I.e. \underline{F} is an abelian sheaf and η̄ is an H-level structure on \underline{F} over S (in the sense of Definition 1.6). Then 𝔸_{f} ⊗Â η̄ is a rational H-level structure on \underline{F}. This defines a canonical 1-morphism of stacks f : Ab-Sh_{H}^{r,d} → X.

For it to be a 1-isomorphism we have to show that f(S) : Ab-Sh_{H}^{r,d}(S) → X (S) is an equivalence of categories for all S. Let (\underline{F} = (F_{i}, Π, i, τ_{i}), γ̄) be an object of X (S). Choose an algebraically closed base point ι : s → S and a representative

γ : (ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s) ⟶^{∼} 𝔸_{f}^{r} .

of γ̄. There is an element a ∈ A with γ^{−1}(aÂ^{r}) ⊂ (ι^{∗}\underline{F}|Â)^{τ} (s). Then the sheaf

(ι^{∗}\underline{F}|Â)^{τ} (s) / γ^{−1}(aÂ^{r})

on C_{s} has finite length and support disjoint from ∞. Via the identification of (\underline{F}|Â)^{τ} with (ι^{∗}\underline{F}|Â)^{τ} (s) this sheaf can be viewed as a quotient sheaf of F_{i} for all i. Its support is of the form D × S where D ⊂ C′ is the divisor of zeros of a. Note that D × S is disjoint from the characteristic of \underline{F}. Therefore the kernel of this quotient map is an abelian sheaf (F′_{i}, Π′_{i}, τ′_{i}) of rank r and dimension d over S which is isogenous to \underline{F}. Consider the abelian sheaf \underline{F}′(D) := (F′_{i}(D), Π′_{i}, τ′_{i}) and the induced finite quasi-isogeny α from \underline{F}′(D) to \underline{F}. By construction γ ∘ (α|_{𝔸_{f}})^{τ} induces an isomorphism

η̄: (\underline{F}′(D)|Â)^{τ} ⟶^{∼} Â^{r} .

This yields an H-level structure η̄ on \underline{F}′(D). Clearly this construction is independent of the choice of γ.

The pair (\underline{F}′(D), 𝔸_{f} ⊗Â η̄) is isogenous to (\underline{F}, γ̄) by construction. This proves that the functor f(S) is essentially surjective. Analyzing the above construction further shows that it is also fully faithful. Hence f is a 1-isomorphism of stacks. ⊓ ⊔

The reader should note that if v is a finite place of C, the analogous statement holds for abelian sheaves over A_{v}-schemes and prime-to-v isogenies.

The above theorem enables us to define the stacks of abelian sheaves with Hlevel structure for arbitrary compact open subgroups H ∈ GL_{r}(𝔸_{f}).

Definition 5.7. Ab-Sh_{H}^{r,d} is the stack over Spec 𝔽_{q}[[ζ]] whose category of S-valued points has

as objects: all pairs (\underline{F}, γ̄) consisting of an abelian sheaf \underline{F} of rank r and dimension

d and a rational H-level structure γ̄ on \underline{F} over S and as morphisms: all finite quasi-isogenies that are compatible with the rational H-

level structures.

For varying H the stacks Ab-Sh_{H}^{r,d} form a projective system of stacks. The transition maps Ab-Sh_{H′}^{r,d} → Ab-Sh_{H}^{r,d} for H′ ⊂ H are representable by finite étale morphisms of schemes. We define a right action of GL_{r}(𝔸_{f}) on this projective system by letting g ∈ GL_{r}(𝔸_{f}) act through the 1-isomorphisms

g : Ab-Sh_{H}^{r,d} ⟶^{∼} Ab-Sh_{gHg^{−1}}^{r,d}

which are defined by (\underline{F}, γ̄) ↦ (\underline{F}, \overline{g^{−1}γ}) on S-valued points. Using these 1isomorphisms it follows from Theorem 3.1 that all the stacks Ab-Sh_{H}^{r,d} are algebraic in the sense of Deligne–Mumford and locally of finite type over Spec 𝔽_{q}[[ζ]].

Part Two z-Divisible Groups

Our ultimate goal in this article is to study the uniformization of the stacks Ab-Sh_{H}^{r,d} at ∞. Classically, this corresponds to the p-adic uniformization of moduli spaces of abelian varieties. For those uniformization questions an indispensable tool are the associated p-divisible groups. In the same manner we are thus lead to the idea of “zdivisible groups”. These groups were studied in detail in Hartl [21]. But they already appeared in special cases in the work of Drinfeld [11], Genestier [18], Laumon [29], Taguchi [39] and Rosen [36]. For the uniformization of Ab-Sh_{H}^{r,d} these z-divisible groups are of equal importance than p-divisible groups are for abelian varieties. Therefore the next few sections are devoted to them. We first review some facts from [21] in Sections 6 and 7.

As z-divisible groups are of most use over schemes on which z is not a unit we will from now on work over Spf 𝔽_{q}[[ζ]] (see Notation 5.4). Since we want to relate z-divisible groups to abelian sheaves, we should also consider abelian sheaves over Spf 𝔽_{q}[[ζ]]-schemes. Hence we right away introduce the base change

Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q}[[ζ]] .

This is no longer an algebraic stack. But it is a formal algebraic Spf 𝔽_{q}[[ζ]]-stack (A.5). Formal algebraic stacks are related to algebraic stacks in the same way as formal schemes are related to usual schemes. For the necessary background we refer the reader to the appendix.

6 Definition of z-Divisible Groups

We continue to work in the local situation introduced in Notation 5.4. In particular z is a uniformizing parameter at ∞ and we identify Q_{∞} with 𝔽_{q}((z)). Let Δ be the central skew field over Q_{∞} of invariant k/ℓ and let O_{Δ} be its ring of integers. We identify O_{Δ} with the 𝔽_{q}-algebra 𝔽_{q^{ℓ}}[[z, Π]] of non-commutative power series subject to the relations

Π^{ℓ} = z^{k} , z Π = Π z , z λ = λ z , Π λ^{q} = λ Π for all λ ∈ 𝔽_{q^{ℓ}} .

Denote by Nilp_{𝔽_{q}[[ζ]]} the category of schemes over Spec 𝔽_{q}[[ζ]] on which ζ is locally nilpotent. From now on in this article the scheme S will be in Nilp_{𝔽_{q}[[ζ]]}.

Definition 6.1. Let R be a unitary ring. We define an R-module scheme over S to be a flat commutative S-group scheme E together with a unitary ring homomorphism R → End_{S} E. It is called finite of order d if it is so as an S-group scheme. A morphism of R-module schemes is a morphism of the underlying S-group schemes which is compatible with the R-action.

For an R-module scheme E over S we define its coLie module Lie^{∗}E as the O_{S}-module of invariant differentials. It is canonically isomorphic to e^{∗}Ω_{E/S} where e : S → E is the zero section. We have Lie E = Hom_{S}(Lie^{∗}E, O_{S}) as O_{S}-module.

The additive group scheme 𝔾_{a,S} is an example for an 𝔽_{q}-module scheme over S. Likewise every S-group scheme which locally on S is isomorphic to 𝔾_{a,S}^{d} for some integer d ≥ 0 is an 𝔽_{q}-module scheme. Such a scheme is called an 𝔽_{q}-vector group scheme of dimension d over S. For every a ∈ 𝔽_{q} the endomorphism induced on its coLie module equals the multiplication with a viewed as an element of Γ(S, O_{S}).

Definition 6.2. Let h, d ≥ 1 be integers. A z-divisible group of height h and dimension d over S is an inductive system of finite 𝔽_{q}[[z]]-module schemes over S

E = (E_{1} ⟶^{i_{1}} E_{2} ⟶^{i_{2}} E_{3} ⟶^{i_{3}} …)

such that for each integer n ≥ 1

1. the 𝔽_{q}-module scheme E_{n} can be embedded into an 𝔽_{q}-vector group scheme over S,

2. the order of E_{n} is q^{hn},

3. the following sequence of 𝔽_{q}[[z]]-module schemes over S is exact

0 → E_{n} ⟶^{i_{n}} E_{n+1} ⟶^{z^{n}} E_{n+1} ,

4. (z − ζ)^{d} = 0 on Lie^{∗}E_{n}.

5. d = max{ dim_{κ(s)}(Lie^{∗}E_{n} ⊗_{O_{S}} κ(s)) : s ∈ S, n ≥ 1 }.

A morphism of z-divisible groups over S is a morphism of inductive systems of 𝔽_{q}[[z]]-module schemes.

We set Lie^{∗}E = lim_{⟵} Lie^{∗}E_{n}. Conditions 1 to 4 imply that this is a locally free O_{S}-module. Condition 5 asserts that its rank is the dimension of E.

The reader should observe that we do not require that z − ζ acts trivially on Lie^{∗}E. This is in conformity with the previous sections. In this respect our notion of z-divisible group is more general than the variants considered in [11, 18, 39, 36] and different from the classical case of p-divisible groups.

The group of morphisms Hom_{S}(E, E′) between two z-divisible groups E and E′ over S is a torsion free 𝔽_{q}[[z]]-module. Let \underline{Hom}_{S}(E, E′) denote the sheaf of germs of morphisms on S.

Definition 6.3. The category of z-divisible groups over S up to isogeny has as objects the z-divisible groups over S and as morphisms from E to E′ all global sections of the sheaf \underline{Hom}_{S}(E, E′) ⊗_{𝔽_{q}[[z]]} 𝔽_{q}((z)) on S. An isomorphism in this new category is called a quasi-isogeny. An isogeny between z-divisible groups is a morphism between z-divisible groups which also is a quasi-isogeny.

In particular, for every quasi-isogeny α there exists locally on S an integer n such that z^{n}α is an isogeny. Quasi-isogenies have the following rigidity property.

Proposition 6.4. Let ι : S′ ↪ S be a closed subscheme defined by a sheaf of ideals which is locally nilpotent. Let E and E′ be two z-divisible groups over S. Then every quasi-isogeny from ι^{∗}E to ι^{∗}E′ lifts in a unique way to a quasi-isogeny from E to E′.

Proposition 6.5. Let α : E → E′ be a quasi-isogeny of z-divisible groups over S. Then the functor on Nilp_{𝔽_{q}[[ζ]]}

T ↦ { φ ∈ Hom_{𝔽_{q}[[ζ]]}(T, S) : φ^{∗}α is an isogeny }

is representable by a closed subscheme of S.

Definition 6.6. A z-divisible O_{Δ}-module over S is a z-divisible group E over S with an action O_{Δ} → End_{S} E of O_{Δ}, which prolongs the natural action of 𝔽_{q}[[z]]. A morphism of z-divisible O_{Δ}-modules which is an isogeny of z-divisible groups is called an isogeny.

Definition 6.7. If S belongs to Nilp_{Spf 𝔽_{q^{ℓ}} [[ζ]]} a z-divisible O_{Δ}-module E of height rℓ and dimension dℓ over S is called special if the action of O_{Δ} induced on Lie^{∗}E, makes Lie^{∗}E into a locally free 𝔽_{q^{ℓ}} ⊗ O_{S}-module of rank d.

At the end of the next section we will show that the later are precisely the z-divisible groups that arise from abelian sheaves.

Proposition 6.8. For a z-divisible O_{Δ}-module E over S the condition of being special is represented by an open and closed immersion into S.

Proof. Clearly Lie^{∗}E decomposes into a direct sum ∑_{i=0}^{ℓ−1}(Lie^{∗}E)_{i} of components (Lie^{∗}E)_{i} on which λ ∈ 𝔽_{q^{ℓ}} ⊂ O_{Δ} acts via λ^{q^{i}} ∈ O_{S}. Then E is special if and only if all (Lie^{∗}E)_{i} are locally free O_{S}-modules of rank d. The proposition follows. ⊓ ⊔

7 Dieudonné modules of z-Divisible Groups

We continue with the notation from Section 6. Let S be a scheme in Nilp_{𝔽_{q}[[ζ]]} and consider the completion of C_{S} along the closed subscheme ∞ × V(ζ). Its structure sheaf is the sheaf O_{S}[[z]] on S of formal power series in z. We denote the sheaf O_{Δ} ⊗_{𝔽_{q}[[z]]} O_{S}[[z]] on S by O_{Δ} ˆ⊗O_{S}.

Consider the additive group 𝔾_{a,S} = Spec O_{S}[ξ] over S. On 𝔾_{a,S} we have the Frobenius isogeny F rob_{q} : 𝔾_{a,S} → 𝔾_{a,S} defined by F rob_{q}^{∗}(ξ) = ξ^{q}. Let E = (E_{n}, i_{n}) be a z-divisible group over S. We associate to E the sheaf

M_{E} = lim_{⟵_{n}} Hom_{S}(E_{n}, 𝔾_{a,S})

on S. We make M_{E} into a sheaf of O_{S}[[z]]-modules by letting z act through the isogeny z on E. The σ_{S}-linear multiplication with Frob_{q} on the left defines a morphism

F_{E} : σ^{∗}M_{E} → M_{E} .

If E is moreover a z-divisible O_{Δ}-module then M_{E} becomes an O_{Δ} ˆ⊗O_{S}-module through the action of O_{Δ} from the right. The module M_{E} may be viewed as the analogue of the contravariant Dieudonné module associated to a p-divisible group. See [21] for a general discussion of this analogy. We recall the following facts.

Definition 7.1. A Dieudonné 𝔽_{q}[[z]]-module over S of dimension d and rank r is a sheaf F̂ of O_{S}[[z]]-modules on S equipped with a O_{S}[[z]]-module homomorphism F : σ^{∗}F̂ → F̂ such that locally on S in the Zariski topology

1. F̂ is free of rank r as an O_{S}[[z]]-module,

2. coker F is free of rank d as an O_{S}-module,

3. (z − ζ)^{d} = 0 on coker F.

A morphism between Dieudonné 𝔽_{q}[[z]]-modules is a morphism of sheaves of O_{S}[[z]]- modules which is compatible with F.

Note that F is automatically injective. As for p-divisible groups the z-divisible groups are classified by their Dieudonné 𝔽_{q}[[z]]-modules.

Theorem 7.2. The functor E ↦ (M_{E}, F_{E}) is an anti-equivalence between the category of z-divisible groups of height r and dimension d over S and the category of Dieudonné 𝔽_{q}[[z]]-modules of rank r and dimension d over S. There is a canonical isomorphism Lie^{∗}E = coker F_{E}.

If one seeks a classification of p-divisible groups up to isogeny one works with isocrystals instead of crystals. For z-divisible groups we do the same.

Definition 7.3. A Dieudonné 𝔽_{q}((z))-module over S is a finite locally free O_{S}[[z]][\frac{1}{z}]- module V together with an isomorphism F : σ^{∗}V ⟶^{∼} V.

To every Dieudonné 𝔽_{q}[[z]]-module \underline{F̂} = (F̂, F) over S we associate its Dieudonné 𝔽_{q}((z))-module

\underline{F̂} [\frac{1}{z}] := (F̂ ⊗_{O_{S}[[z]]} O_{S}[[z]][\frac{1}{z}] , F ⊗ id) .

Definition 7.4. A morphism α : \underline{F̂} → \underline{F̂}′ of Dieudonné 𝔽_{q}[[z]]-modules is called an isogeny if α is injective and coker α is a locally free O_{S}-module of finite rank. We define a quasi-isogeny between Dieudonné 𝔽_{q}[[z]]-modules \underline{F̂} and \underline{F̂}′ to be an isomorphism between \underline{F̂} [\frac{1}{z}] and \underline{F̂}′[\frac{1}{z}].

Proposition 7.5. The functor E ↦ (M_{E}, F_{E}) maps isogenies to isogenies and quasi-isogenies to quasi-isogenies.

Now let m/n be a rational number written in lowest terms with n > 0. Then we define the Dieudonné 𝔽_{q}((z))-module V(m/n) over Spec 𝔽_{q} as follows

V = 𝔽_{q}((z))^{n} , F = ( \begin{matrix}0 & … & z^{m} \\ 1 & ... & ⋮ \\ ... & ... \\ 1 & 0 \end{matrix} ) · σ^{∗}: σ^{∗}V → V .

There is the following analogue of Dieudonné’s Theorem [32].

Theorem 7.6. Let K be an algebraically closed field with Spec K ∈ Nilp_{𝔽_{q}[[ζ]]}. Then every Dieudonné 𝔽_{q}((z))-module over Spec K is isomorphic to a direct sum

⨁_{i} V(m_{i}/n_{i}) ⊗_{𝔽_{q}((z))} K((z))

for uniquely determined rational numbers m_{1}/n_{1} ≤ m_{2}/n_{2} ≤ . . ..

This result allows us to define the Newton polygon of a Dieudonné 𝔽_{q}[[z]]-module \underline{F̂} = (F̂, F) over a field K in Nilp_{𝔽_{q}[[ζ]]}. Namely over an algebraically closed extension its Dieudonné 𝔽_{q}((z))-module decomposes as in the theorem. Then the Newton polygon is the polygon which passes through the points

( n_{1} + … + n_{i} , m_{1} + … + m_{i})

for all i and is extended linearly between them. It is independent of the chosen algebraically closed extension. We also define the Hodge polygon as usual by the elementary divisors of the K[[z]]-module coker F. The Hodge polygon lies below the Newton polygon. They both have the same initial point (0, 0) and the same terminal point (rk \underline{F̂}, dim \underline{F̂}). We obtain the analogue of the theorem of Grothendieck–Katz [19, 25].

Theorem 7.7. Let \underline{F̂} be a Dieudonné 𝔽_{q}[[z]]-module of rank r over S and let P be the graph of a continuous real valued function on [0, r] which is linear between successive integers. Then the set of points in S at which the Hodge (respectively Newton) polygon of \underline{F̂} lies above P is Zariski closed.

Note that the stratification of Ab-Sh_{H}^{r,d} considered in Section 3 is related to the stratification according to the Hodge-polygon. This relation comes from Construction 7.13 below.

Definition 7.8. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} be the spectrum of a field. We say that a Dieudonné 𝔽_{q}[[z]]-module over S is isoclinic if its Newton polygon has only a single slope.

Proposition 7.9. Let \underline{F̂} be an isoclinic Dieudonné 𝔽_{q}[[z]]-module of rank r and dimension d over a perfect field K. Then \underline{F̂} is isogenous over K to a Dieudonné 𝔽_{q}[[z]]-module satisfying im F^{r} = z^{d}F̂.

Later we will also need the analogue of Katz’s Constancy Theorem.

Theorem 7.10. Let Spec K[[π]] ∈ Nilp_{𝔽_{q}[[ζ]]} be the spectrum of a power series ring over an algebraically closed field K and let \underline{F̂} be a Dieudonné 𝔽_{q}[[z]]-module over Spec K[[π]]. Suppose that at the two points of Spec K[[π]] the Newton polygons coincide, and that this common Newton polygon has only a single slope. Then \underline{F̂} is isogenous to a constant Dieudonné 𝔽_{q}[[z]]-module (i.e. one obtained by pullback under the morphism Spec K[[π]] → Spec K).

We now want to apply this theory to special z-divisible O_{Δ}-modules. Since for every z-divisible group E there is a canonical isomorphism Lie^{∗}E = coker F_{E} the property of being special reflects on M_{E}. We consider the following class of Dieudonné 𝔽_{q}[[z]]-modules. Let S be in Nilp_{𝔽_{q^{ℓ}} [[ζ]]}.

Definition 7.11. A formal abelian sheaf of rank r and dimension d over S is a sheaf F̂ of O_{Δ} ˆ⊗O_{S}-modules on S together with a morphism of O_{Δ} ˆ⊗O_{S}-modules F : σ^{∗}F̂ → F̂ such that

1. (F̂, F) is a Dieudonné 𝔽_{q}[[z]]-module over S of rank rℓ,

2. coker F is locally free of rank d as an 𝔽_{q^{ℓ}} ⊗ O_{S}-module.

In the situation of special z-divisible O_{Δ}-modules Theorem 7.2 takes the following form.

Theorem 7.12. The functor E ↦ M_{E} is an anti-equivalence between the category of special z-divisible O_{Δ}-modules of height rℓ and dimension dℓ over S and the category of formal abelian sheaves of rank r and dimension d over S.

Next we want to describe the relation with abelian sheaves. We will obtain the analogue of the classical functor which assigns to every abelian variety its p-divisible group.

Construction 7.13. Let S be in Nilp_{𝔽_{q}[[ζ]]} and assume that there is a morphism β : S → Spec 𝔽_{q^{ℓ}}[[ζ]]. Let (F_{i}, Π_{i}, τ_{i}) be an abelian sheaf of rank r and dimension d over S. Consider the completions F̂_{i} = F_{i} ⊗_{O_{C_{S}}} O_{S}[[z]] of the sheaves F_{i} at ∞. Using the periodicity F_{ℓ} ≅ F_{0}(k · ∞) we obtain morphisms

(7.1) Π_{i} : F̂_{i} → F̂_{i+1} for all i = 0, …, ℓ − 2 , z^{k} Π_{ℓ−1} : F̂_{ℓ−1} → F̂_{0} and
(7.2) τ_{i} : σ^{∗}F̂_{i} → F̂_{i+1} for all i = 0, …, ℓ − 2 , z^{k} τ_{ℓ−1} : σ^{∗}F̂_{ℓ−1} → F̂_{0}

We set F̂ = F̂_{0} ⊕ … ⊕ F̂_{ℓ−1} and let the endomorphism Π : F̂ → F̂ be given by the morphisms (7.1). We let λ ∈ 𝔽_{q^{ℓ}} act on F̂_{i} as the scalar β^{∗}λ^{q^{i}} and we let the σ-linear endomorphism F : σ^{∗}F̂ → F̂ be given by the morphisms (7.2). I.e. Π, λ and F are expressed by the block matrices

Π = ( \begin{matrix}0 & … & z^{k}Π_{ℓ−1} \\ Π^{0} & ... & ⋮ \\ & ... & ... \\ Π_{ℓ−2} & 0 \end{matrix} ) , λ = ( \begin{matrix}β^{∗}λ Id_{ℓ} \\ & β^{∗}λ^{q} Id_{ℓ} \\ & & . & . & . \\ & & & & & β^{∗}λ^{q^{ℓ−1}} Id_{ℓ} \end{matrix} ) and

F = ( \begin{matrix}0 & … & z^{k}τ_{ℓ−1} \\ τ^{0} & ... & ⋮ \\ & ... & ... \\ τ_{ℓ−2} & 0 \end{matrix} ) .

In this way (F̂, F) is a formal abelian sheaf of rank r and dimension d over S. By Theorem 7.12 we obtain a functor from Ab-Sh^{r,d}(S) to the category of special z-divisible O_{Δ}-modules of height rℓ and dimension dℓ over S.

Moreover, every isogeny of abelian sheaves over S induces an isogeny of the associated formal abelian sheaves and an isogeny of the associated special z-divisible O_{Δ}-modules.

8 The Serre-Tate-Theorem

Classically the Serre-Tate-Theorem relates the deformation theory of an abelian variety in characteristic p to the deformation theory of its p-divisible group. In the case of abelian sheaves the same principle prevails. We begin with the analogue of the following classical construction. Let A be a fixed abelian variety over a scheme S ∈ Nilp_{ℤ_{p}} and let A[p^{∞}] be its p-divisible group. Then to every pair (X, α̂) consisting of a p-divisible group X over S and an isogeny α̂ : A[p^{∞}] → X of p-divisible groups there exists a uniquely determined abelian variety α̂_{∗}A and a p-isogeny α : A → α̂_{∗}A which induces α̂ on p-divisible groups. This construction can be applied to abelian sheaves as well.

Proposition 8.1. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} and let \underline{F} be an abelian sheaf of rank r and

dimension d over S. Let \underline{F̂} be the associated formal abelian sheaf and let α̂ : \underline{F̂}′ → \underline{F̂} be a quasi-isogeny. Then there exists an abelian sheaf \underline{F}′ and a quasi-isogeny α : \underline{F}′ → \underline{F} over S giving rise to α̂. If we require that α is an isomorphism over C′ then (\underline{F}′, α) is unique up to canonical isomorphism. In this case we denote \underline{F}′ by α^{∗}\underline{F}.

Proof. We denote by β : S → Spf 𝔽_{q^{ℓ}}[[ζ]] the structure morphism of S. For each i = 0, …, ℓ − 1 we extract from the sheaf F̂′ of O_{S}[[z]]-modules underlying \underline{F̂}′ the locally free subsheaf F̂′_{i} ⊂ F̂′ of rank r on which λ ∈ 𝔽_{q^{ℓ}} acts through the character λ ↦ β^{∗}λ^{q^{i}}. The quasi-isogeny α̂ gives an inclusion F̂′_{i} ↪ F_{i} ⊗_{O_{C′_{S}}} O_{S}[[z]][\frac{1}{z}]. This

permits us to glue F̂′_{i} with the corresponding sheaf F_{i}|_{C′_{S}} to obtain a locally free sheaf F′_{i} of rank r on C_{S}. For arbitrary i ∈ ℤ we take n ∈ ℤ such that 0 ≤ i−nℓ < ℓ and define

F′_{i} := F′_{i−nℓ} ⊗ O_{C_{S}}(nk · ∞) .

From the morphisms Π′ and F′ of \underline{F̂}′ and Π_{i} and τ_{i} of \underline{F} we obtain morphisms

Π′_{i} : F′_{i} → F′_{i+1} and τ′_{i} : σ^{∗}F′_{i} → F′_{i+1} .

The morphism Π′_{ℓ−1} : F′_{ℓ−1} → F′_{ℓ} is induced from z^{−k}Π′ : F̂′_{ℓ−1} → F̂′_{ℓ}(k · ∞). The same applies to τ′_{ℓ−1}. These morphisms make α^{∗}\underline{F} := (F′_{i}, Π′_{i}, τ′_{i}) into an abelian sheaf of rank r and dimension d over S whose associated formal abelian sheaf is \underline{F̂}′. By construction there is a quasi-isogeny α : α^{∗}\underline{F} → \underline{F} which is an isomorphism over C′ and induces the quasi-isogeny α̂ on formal abelian sheaves. ⊓ ⊔

Proposition 8.2. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} and let j : S̄ ↪ S be a closed subscheme defined by a sheaf of ideals which is locally nilpotent. Let \underline{F} and \underline{F}′ be two abelian sheaves of rank r and dimension d over S. Then every quasi-isogeny from j^{∗}\underline{F} to j^{∗}\underline{F}′ lifts in a unique way to a quasi-isogeny from \underline{F} to \underline{F}′.

Proof. Let ᾱ : j^{∗}\underline{F} → j^{∗}\underline{F}′ be a quasi-isogeny. It suffices to treat the case where the q-th power of the ideal sheaf defining S̄ is zero. In this case the morphisms σ_{S} and σS̄ factor through j

σ_{S} = j ∘ σ̄ : S → S̄ → S  and  σS̄ = σ̄ ∘ j : S̄ → S → S̄ .

Consider the quasi-isogeny σ̄^{∗}ᾱ[1] : σ̄^{∗}j^{∗}\underline{F}[1] → σ̄^{∗}j^{∗}\underline{F}′[1] where [1] denotes the shift by 1 (cf. 1.7). We view the morphisms τ_{i} as an isogeny (τ_{i}) : σ^{∗}\underline{F}[1] → \underline{F} and obtain a commutative diagram

σ̄^{∗}j^{∗}\underline{F}[1] ⟶^{σ̄^{∗}ᾱ[1]} σ̄^{∗}j^{∗}\underline{F}′[1]
(τ_{i})↓  ↓^{(τ′_{i})}
\underline{F}  ⟶^{α}  \underline{F}′

which defines a quasi-isogeny α. Pulling back this diagram under j we see that j^{∗}α = ᾱ. Moreover the diagram shows that α is uniquely determined by ᾱ. This proves the proposition. ⊓ ⊔

From the proof we even see the following.

Corollary 8.3. Keep the situation of the proposition.

1. If ᾱ : j^{∗}\underline{F} → j^{∗}\underline{F}′ is an isogeny then the lift is an isogeny α : \underline{F} → \underline{F}′(n · ∞) for some integer n ≥ 0.

2. If ᾱ is an isomorphism over C′ then the same holds for α.

Next we come to the analogue of the Serre-Tate-Theorem. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} and let j : S′ ↪ S be a closed subscheme defined by a sheaf of ideals which is locally nilpotent. Let \underline{F}′ be an abelian sheaf of rank r and dimension d over S′ and let \underline{F̂}′ be the associated formal abelian sheaf. The category of lifts of \underline{F}′ to S has as objects: all pairs (\underline{F} , α : j^{∗}\underline{F} ⟶^{∼} \underline{F}′) where \underline{F} is an abelian sheaf over S and

α an isomorphism of abelian sheaves over S′, as morphisms: isomorphisms between the \underline{F}’s that are compatible with the α’s.

Similarly we define the category of lifts of \underline{F̂}′ to S. By Propositions 8.2 and 6.4 all Hom-sets in these categories contain at most one element.

Theorem 8.4 (Analogue of the Serre-Tate-Theorem). The category of lifts of \underline{F}′ to S and the category of lifts of \underline{F̂}′ to S are equivalent.

Proof. Let be the functor that assigns to a lift of \underline{F}′ the corresponding lift of \underline{F̂}′. Full faithfulness of follows from Corollary 8.3. It remains to show that is essentially surjective. So let (\underline{F̂} , α̂ : j^{∗}\underline{F̂} ⟶^{∼} \underline{F̂}′) be a lift of \underline{F̂}′ to S.

It suffices to treat the case where the q-th power of the ideal sheaf defining S′ is zero. In this case the morphism σ_{S} factors through j

σ_{S} = j ∘ σ′ : S → S′ → S .

Consider the abelian sheaf \underline{F̃} := (σ′^{∗}\underline{F}′)[1] over S. I.e.

(F̃_{i}, Π̃_{i}, τ̃_{i}) = (σ′^{∗}F′_{i−1}, σ′^{∗}Π′_{i−1}, σ′^{∗}τ′_{i−1}) .

The morphisms τ′_{i} constitute an isogeny τ′ := (τ′_{i}) : j^{∗}\underline{F̃} → \underline{F}′ which is an isomorphism over C′. We let

γ′ = τ′^{−1} ∘ α̂ : j^{∗}\underline{F̂} → \underline{F̂}′ → j^{∗}\underline{ˆF̃}

be the resulting quasi-isogeny of formal abelian sheaves. By Proposition 6.4 it lifts

to a quasi-isogeny γ̂ : \underline{F̂} → \underline{ˆF̃}. We put \underline{F} := γ^{∗}\underline{F̃} (Prop. 8.1) and we let γ : \underline{F} → \underline{F̃} be the induced quasi-isogeny of abelian sheaves. Then (\underline{F}, τ′ ∘ j^{∗}γ) is the desired lift of \underline{F}. ⊓ ⊔

In the remainder of this section we give an example for an abelian sheaf and we compute the associated formal abelian sheaf. This example will be crucial for the uniformization of Ab-Sh_{H}^{r,d} in Part Three.

Example 8.5. On the scheme S̄ = Spec 𝔽_{q} we set for i = 0, …, ℓ

M_{i} = O_{CS̄}(k · ∞)^{⊕i} ⊕ O_{CS̄}^{⊕ℓ−i} .

We let Π_{i} : M_{i} → M_{i+1} be the morphism coming from the natural inclusion O_{CS̄} ⊂ O_{CS̄}(k · ∞) in the (i + 1)-st summand. We define the σ-linear morphism

τ_{i} = ( \begin{matrix}0 & … & 1 \\ 1 & ... & ⋮ \\ ... & ... \\ & 1 & 0 \end{matrix} )· σ^{∗} : σ^{∗}M_{i} → M_{i+1} .

For arbitrary i ∈ ℤ we take n ∈ ℤ such that 0 ≤ i − nℓ < ℓ and set

M_{i} := M_{i−nℓ} ⊗ O_{CS̄}(nk · ∞) , Π_{i} := Π_{i−nℓ} ⊗ id , τ_{i} := τ_{i−nℓ} ⊗ id .

Then \underline{M} := (M_{i}, Π_{i}, τ_{i}) is an abelian sheaf of rank ℓ and dimension k over Spec 𝔽_{q}. We let e = \frac{r}{ℓ} and set \overline{𝕄} = \underline{M}^{⊕e}. This is an abelian sheaf of rank r and dimension

d. We compute the formal abelian sheaf ˆ \overline{𝕄} associated to \overline{𝕄}.

In order to do this we have to extend the base scheme S̄ to S̄′ = Spec 𝔽_{q^{ℓ}}. Since

we afterwards want to lift ˆ \overline{𝕄} to S′ = Spf 𝔽_{q^{ℓ}}[[ζ]], we right away describe this lift. For i = 0, …, ℓ − 1 we consider the ℓ × ℓ-matrices

(8.1)  Π_{i} = ( \begin{matrix}1 \\ & . & . & . \\ & & & & z^{k} \\ & & & & & . & . & . \\ & & & & & & & & 1 \end{matrix} ) and T = ( \begin{matrix}0 & … & (z − ζ)^{k} \\ 1 & ... & ⋮ \\ ... & ... \\ 1 & 0 \end{matrix} ) ,

where the z^{k} in Π_{i} sits in the (i + 1)-st row. We let the formal abelian sheaf M̂ of rank ℓ and dimension k be the O_{S′}[[z]]-module M̂ = O_{S′}[[z]]^{ℓ^{2}} together with the morphisms

Π = ( \begin{matrix}0 & … & Π_{ℓ−1} \\ Π^{0} & ... & ⋮ \\ & ... & ... \\ Π_{ℓ−2} & 0 \end{matrix} ) , λ = ( \begin{matrix}λ Id_{r} \\ & λ^{q} Id_{r} \\ & & . & . & . \\ & & & & & λ^{q^{ℓ−1}} Id_{r} \end{matrix} ) : M̂ → M̂

and F = ( \begin{matrix}0 & … & & T \\ T & ... & & ⋮ \\ & ... & ... \\ & & T & 0 \end{matrix} )· σ^{∗} : σ^{∗}M̂ → M̂.

We set 𝕄̂ = M̂^{⊕e}. If j : S̄′ ↪ S′ denotes the inclusion, then j^{∗}𝕄̂ is the formal abelian sheaf associated to \overline{𝕄}. We denote it by ˆ \overline{𝕄}. Via Theorem 7.12 we obtain from ˆ \overline{𝕄} a special z-divisible O_{Δ}-module \overline{𝔼} over Spec 𝔽_{q^{ℓ}} whose Dieudonné 𝔽_{q}[[z]]-

module is ˆ \overline{𝕄}. One easily checks F^{ℓ} = z^{k} · σ^{ℓ∗} on ˆ \overline{𝕄}. In the terminology of [21] this means that \overline{𝔼} is descent. The Newton-polygon of ˆ \overline{𝕄} is a straight line between the end points (0, 0) and (rℓ, dℓ).

Via the Serre-Tate-Theorem 8.4 we obtain from the lift 𝕄̂ of the formal abelian sheaf of \overline{𝕄} an abelian sheaf 𝕄 over Spf 𝔽_{q^{ℓ}} which lifts \overline{𝕄}.

Let us compute the group of quasi-isogenies of these (formal) abelian sheaves.

Proposition 8.6. Fix a generator λ ∈ 𝔽_{q^{ℓ}} of the extension 𝔽_{q^{ℓ}}/𝔽_{q}. Let V be the Vandermonde matrix

(8.2) V = ( \begin{matrix}1 & λ & λ^{2} & … & λ^{ℓ−1} \\ 1 & λ^{q} & λ^{2q} & … & λ^{(ℓ−1)q} \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 1 & λ^{q^{ℓ−1}} & λ^{2q^{ℓ−1}} & … & λ^{(ℓ−1)q^{ℓ−1}} \end{matrix} ) and V_{e} = ( \begin{matrix}V \\ & . & . & . \\ & & & & V \end{matrix} )

be the block diagonal matrix of dimension r. Then the group of quasi-isogenies of \overline{𝕄} over 𝔽_{q}^{alg} is the group of Q-valued points of the algebraic group J = V_{e} GL_{r} V_{e}^{−1} over Q.

Proof. Let g ∈ J(Q) and let D ⊂ C be an effective divisor satisfying div(g_{μν}) ≥ −D for all entries of g. Then multiplication with g defines morphisms ι^{∗}M_{i}^{⊕e} → ι^{∗}M_{i}^{⊕e}(D) which form a quasi-isogeny of \overline{𝕄} (defined over Spec 𝔽_{q^{ℓ}}). Conversely one computes that every quasi-isogeny of \overline{𝕄} arises in this way. ⊓ ⊔

If T ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} is a scheme we let β : T → Spf 𝔽_{q^{ℓ}}[[ζ]] be its structure morphism and β̄ : T̄ → Spec 𝔽_{q^{ℓ}} be the reduction modulo ζ.

Proposition 8.7. Let T ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} be connected. Then there is a canonical isomorphism g ↦ g_{T} of J(Q) to the group QIsog_{T} (β^{∗}𝕄) of quasi-isogenies of β^{∗}𝕄 over T.

Proof. The map g ↦ g_{T} is defined as

J(Q) ⟶ QIsogT̄(β̄^{∗}\overline{𝕄}) ⟶^{∼} QIsog_{T} (β^{∗}𝕄)
g ↦  β̄^{∗}g  ↦  g_{T} ,

the last isomorphism coming from Proposition 8.2. Clearly this defines a monomorphism of groups.

We show that g ↦ β̄^{∗}g is surjective. Let α : \overline{𝕄}T̄ → \overline{𝕄}T̄(D) be an isogeny for some effective divisor D ⊂ C. There exists an a ∈ A whose divisor is ≥ D on C′. We view α|_{C′ ; T̄} as a matrix U ∈ M_{r}(OT̄ ⊗ A[\frac{1}{a}]). Then the matrix V_{e}^{−1}UV_{e} satisfies

σ(V_{e}^{−1}UV_{e}) = V_{e}^{−1}UV_{e}

and hence lies in GL_{r}(Q). We conclude that U ∈ J(Q) and β̄^{∗}U = α. ⊓ ⊔

Proposition 8.8. The group of quasi-isogenies of the formal abelian sheaf 𝕄̂ is isomorphic to J(Q_{∞}).

Proof. A straightforward calculation shows that this isomorphism can be described as follows. Let

W_{i} = ( \begin{matrix}z^{k} Id_{i} & 0 \\ 0 & Id_{ℓ−i} \end{matrix} )  and  W = ( \begin{matrix}g \\ & W_{1}gW_{1}^{−1} \\ & & . & . & . \\ & & & & & W_{ℓ−1}gW_{ℓ−1}^{−1} \end{matrix} ) .

Then an element ∈ (Q_{∞}) is mapped to the quasi-isogeny W^{⊕e} of M̂. ⊓ ⊔

9 Moduli Spaces for z-Divisible Groups

Consider the formal abelian sheaf 𝕄̂ from Example 8.5. We will define a moduli problem for formal abelian sheaves which are quasi-isogenous to 𝕄̂. This is a higher dimensional variant of a moduli problem studied by Drinfeld [11]. At the same time it is a close analogue of a moduli problem for p-divisible groups considered by Rapoport–Zink [34]. Like these two problems our moduli problem too will be solved by a formal scheme over Spf 𝔽_{q^{ℓ}}[[ζ]]. Following [11, 34] we will use this formal moduli scheme in Section 12 to (partly) uniformize the stacks of abelian sheaves.

For a scheme S in Nilp_{𝔽_{q}[[ζ]]} we denote by S̄ the closed subscheme defined by the sheaf of ideals ζO_{S}. We call S̄ the special fiber of S. If E is a z-divisible group over S we denote by ES̄ = E ×_{S} S̄ the base change to the special fiber. A similar notation will be applied to Dieudonné 𝔽_{q}[[z]]-modules, etc. If β : S → Spf 𝔽_{q^{ℓ}}[[ζ]] is a morphism of formal schemes we denote by β̄ : S̄ → Spec 𝔽_{q^{ℓ}} its restriction to special fibers. We define the following moduli problem for formal abelian sheaves.

Definition 9.1. Let G be the contravariant functor Nilp_{𝔽_{q}[[ζ]]} ⟶ Sets

S ⟼ \{ Isomorphism classes of triples (β, \underline{F̂}, α̂) where

• β : S → Spf 𝔽_{q^{ℓ}}[[ζ]] is a morphism of formal schemes,

• \underline{F̂} is a formal abelian sheaf of rank r and dimension d over S ,

• α̂ : \underline{F̂}S̄ → β̄^{∗} ˆ\overline{𝕄} is a quasi-isogeny of formal abelian sheaves. \}

Thereby two triples (β, \underline{F̂}, α̂) and (β′, \underline{F̂}′, α′) are isomorphic if β = β′ and if there is an isomorphism between \underline{F̂} and \underline{F̂}′ over S which is compatible with α̂ and α′.

Using the equivalence between special z-divisible O_{Δ}-modules and formal abelian sheaves from Theorem 7.12 we see that

Proposition 9.2. G can be described as the functor Nilp_{𝔽_{q}[[ζ]]} ⟶ Sets

S ⟼ { Isomorphism classes of triples (β, E, ρ) where

• β : S → Spf 𝔽_{q^{ℓ}}[[ζ]] is a morphism of formal schemes,

• E is a special z-divisible O_{Δ}-module of height rℓ and dimension dℓ over S ,

• ρ : β̄^{∗}\overline{𝔼} → ES̄ is a quasi-isogeny of z-divisible O_{Δ}-modules. \}

Sending (β, \underline{F̂}, α̂) to β gives a morphism G → Spf 𝔽_{q^{ℓ}}[[ζ]]. We define an action of the Galois group Gal(𝔽_{q^{ℓ}}/𝔽_{q}) on G over Spf 𝔽_{q^{ℓ}}[[ζ]]. Namely for each π^{∗} ∈ Gal(𝔽_{q^{ℓ}}/𝔽_{q}) = Gal(𝔽_{q^{ℓ}}[[ζ]]/𝔽_{q}[[ζ]]) consider the cartesian square

S^{π}  ⟶^{π_{S}}  S
↓^{π^{∗}β}  ↓^{β}
Spf 𝔽_{q^{ℓ}}[[ζ]] ⟶^{π} Spf 𝔽_{q^{ℓ}}[[ζ]] .

Then we let π^{∗} act by mapping the element (β, \underline{F̂}, α̂) ∈ G(S) to

(π^{∗}β, π_{S}^{∗}\underline{F̂}, π_{S}^{∗}α̂) ∈ G(S^{π}) .

Following the arguments given in Rapoport–Zink [34] in the case of p-divisible groups, one can prove the following representability theorem. Recall that an adic formal scheme G over Spf 𝔽_{q^{ℓ}}[[ζ]] is called locally formally of finite type if G_{red} is locally of finite type over Spec 𝔽_{q^{ℓ}}.

Theorem 9.3. The functor that assigns to a scheme S ∈ Nilp_{𝔽_{q}[[ζ]]} the set of isomorphism classes of triples (β, E, ρ) where

• β : S → Spf 𝔽_{q^{ℓ}}[[ζ]] is a morphism of formal schemes,

• E is a z-divisible group of dimension dℓ and height rℓ over S,

• ρ : β̄^{∗}\overline{𝔼} → ES̄ is a quasi-isogeny of z-divisible groups,

is representable by a quasi-separated, locally noetherian, adic formal scheme which is locally formally of finite type over Spf 𝔽_{q^{ℓ}}[[ζ]].

The proof of this theorem is given in [21]. It makes use of Dieudonné 𝔽_{q}[[z]]- modules which replace the crystals of the p-divisible groups in [34].

Corollary 9.4. The functor G is representable by a quasi-separated, locally noetherian, adic formal scheme which is locally formally of finite type over Spf 𝔽_{q^{ℓ}}[[ζ]].

Proof. Let G̃ be the formal scheme whose existence is stated in Theorem 9.3. Let E be the universal z-divisible group over G̃. We transport the O_{Δ}-action from 𝔼 to E via ρ. Then G is the closed formal subscheme of G̃ on which E is special and O_{Δ} acts through isogenies (Propositions 6.5 and 6.8). ⊓ ⊔ 9.5. We define an action of the group J(Q_{∞}) on G. By Proposition 8.8 there is an isomorphism ε_{∞} from the group J(Q_{∞}) to the group of quasi-isogenies of ˆ \overline{𝕄}. We let g ∈ J(Q_{∞}) act on G through ε_{∞}

(β, \underline{F̂}, α̂) ↦ (β, \underline{F̂}, β̄^{∗}ε_{∞}(g) ∘ α̂) .

This action commutes with the Galois action on G.

Let now Γ ⊂ J(Q_{∞}) be a discrete subgroup. We say that Γ is separated if it is separated in the profininte topology. This means that for every g ∈ Γ there is a normal subgroup Γ′ ⊂ Γ of finite index that does not contain g.

Again following the arguments of Rapoport–Zink we prove in [21]

Theorem 9.6. Let Γ ⊂ J(Q_{∞}) be a separated discrete subgroup. Then the quotient Γ\G is a locally noetherian, adic formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stack locally formally of finite type over Spf 𝔽_{q^{ℓ}}. Moreover, the 1-morphism G → Γ\G is adic.

See the appendix A.5 – A.9 for an explanation of this statement.

Part Three Uniformization

We now turn towards the uniformization of the stacks Ab-Sh_{H}^{r,d} at ∞. Therefore we again consider their base change

Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q}[[ζ]] .

As remarked in Part Two these are formal algebraic Spf 𝔽_{q}[[ζ]]-stacks (A.5). In analogy with the work of Rapoport–Zink [34] and Drinfeld [11] the space used to uniformize these stacks will be the formal scheme G. We view it as a formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stack (A.10). The uniformization we obtain will only be partial. To be precise, we find a closed subset Z in Ab-Sh_{H}^{r,d} ×_{C} ∞ and we consider the formal completion Ab-Sh_{H}^{r,d}_{/Z} of Ab-Sh_{H}^{r,d} along Z. It is a formal algebraic Spf 𝔽_{q}[[ζ]]-stack (A.14). We will uniformize Ab-Sh_{H}^{r,d}_{/Z}. This uniformization therefore takes place in the 2-category of formal algebraic Spf 𝔽_{q}[[ζ]]-stacks.

10 Algebraizations

We first give still another interpretation of the moduli space G. Namely since the formal abelian sheaf ˆ \overline{𝕄} comes from the abelian sheaf \overline{𝕄}, the universal formal abelian sheaf on G and its quasi-isogeny α̂ to ˆ \overline{𝕄} can be algebraized. I.e. they too come from an abelian sheaf, namely from α^{∗}\overline{𝕄}. Now consider a scheme S ∈ Nilp_{𝔽_{q}[[ζ]]} and denote by S̄ the closed subscheme of S defined by ζ = 0.

Definition 10.1. Let G′ be the contravariant functor Nilp_{𝔽_{q}[[ζ]]} ⟶ Sets

S ⟼ \{ Isomorphism classes of pairs (\underline{F}, α) where

• \underline{F} is an abelian sheaf of rank r and dimension d over S

• α : \underline{F} S̄ → \overline{𝕄}S̄ is a quasi-isogeny which is an isomorphism over C′ \}

Thereby two such pairs (\underline{F}, α) and (\underline{F}′, α′) are isomorphic if there is an isomorphism between \underline{F} and \underline{F}′ over S which is compatible with α and α′.

Theorem 10.2. The functors G and G′ ×_{Spf 𝔽_{q}[[ζ]]} Spf 𝔽_{q^{ℓ}}[[ζ]] are canonically isomorphic as Gal(𝔽_{q^{ℓ}}/𝔽_{q})-modules (where Gal(𝔽_{q^{ℓ}}/𝔽_{q}) acts trivially on G′).

Proof. We will exhibit two mutually inverse maps between these two functors. We start by describing the morphism G′ ×_{Spf 𝔽_{q}[[ζ]]} Spf 𝔽_{q^{ℓ}}[[ζ]] → G.

So let (\underline{F}, α) ∈ G′(S) and β : S → Spf 𝔽_{q^{ℓ}}[[ζ]]. By Construction 7.13 we obtain from \underline{F} and β a formal abelian sheaf \underline{F̂} of rank r and dimension d over S. The quasi-isogeny α induces a quasi-isogeny of formal abelian sheaves α̂ : \underline{F̂}S̄ → β̄^{∗} ˆ\overline{𝕄}. The triple (β, \underline{F̂}, α̂) defines an S-valued point of the functor G. One easily checks that the morphism just constructed is Gal(𝔽_{q^{ℓ}}/𝔽_{q})-equivariant.

Conversely let (β, \underline{F̂}, ˆ \overline{α}) ∈ G(S). By Proposition 6.4 there is a unique lift of ˆ \overline{α} to a quasi-isogeny α̂ : \underline{F̂} → β^{∗}𝕄̂. From Proposition 8.1 we obtain an abelian sheaf \underline{F} := α^{∗}(β^{∗}𝕄) whose formal abelian sheaf is \underline{F̂}, and a quasi-isogeny α : \underline{F} → β^{∗}𝕄 which is an isomorphism over C′ and which induces α̂ on the formal abelian sheaves. We have (\underline{F}, αS̄) ∈ G′(S). Thus we have constructed a morphism G → G′. Again one checks that this morphism is Gal(𝔽_{q^{ℓ}}/𝔽_{q})-invariant. The two morphisms just described are mutually inverse and yield the desired isomorphism between G and G′ ×_{Spf 𝔽_{q}[[ζ]]} Spf 𝔽_{q^{ℓ}}[[ζ]]. ⊓ ⊔

The action of J(Q_{∞}) on G from 9.5 induces an action of J(Q_{∞}) on G′, since it is compatible with the Galois action on G.

Definition 10.3. The pair (\underline{F}, α) ∈ G′(S) which is associated to an element (β, \underline{F̂}, α̂) ∈ G(S) by Theorem 10.2 will be called the algebraization of (β, \underline{F̂}, α̂).

Example 10.4. We want to explain how a quasi-isogeny α : \underline{F} S̄ → \overline{𝕄}S̄ which is an isomorphism over C′ induces H-level structures on \underline{F} for compact open subgroups H ⊂ GL_{r}(𝔸_{f}).

Consider the abelian sheaf \overline{𝕄} = (M_{i}^{⊕e}, Π_{i}^{⊕e}, τ_{i}^{⊕e}) from Example 8.5 pulled back to S̄ = Spec 𝔽_{q^{ℓ}}. The restrictions M_{i}^{⊕e}|_{C′ ; S̄} are all isomorphic via the morphisms Π_{i}. We denote this restriction by \overline{𝕄}|_{C′ ; S̄}. The same holds for the morphisms τ_{i}. So we obtain a morphism

τ|_{C′ ; S̄} : σ^{∗}\overline{𝕄}|_{C′ ; S̄} → \overline{𝕄}|_{C′ ; S̄} .

Via the canonical identification \overline{𝕄}|_{C′ ; S̄} = O_{C′ ; S̄}^{r} this morphism is expressed by the block diagonal matrix

τ|_{C′ ; S̄} = ( \begin{matrix}T \\ & . & . & . \\ & & & & T \end{matrix} )· σ^{∗}  where  T = ( \begin{matrix}0 & … & 1 \\ 1 & ... & ⋮ \\ ... & ... \\ & 1 & 0 \end{matrix} ) ∈ GL_{ℓ}(𝔽_{q}) .

Therefore multiplication with the matrix V_{e} from (8.2) defines an isomorphism

ψ : (\overline{𝕄}|_{C′ ; S̄}, τ|_{C′ ; S̄}) ⟶^{∼} (O_{C′ ; S̄}^{r}, Id_{r} ·σ^{∗})

that commutes with the σ-linear endomorphisms τ|_{C′ ; S̄} on one and Id_{r} ·σ^{∗} on the other side. This ψ induces an isomorphism

γ = (ψ|_{𝔸_{f}})^{τ} : (\overline{𝕄}|_{𝔸_{f}})^{τ} ⟶^{∼} 𝔸_{f}^{r}

which gives rise to an H-level structure on \overline{𝕄}.

Now let S ∈ Nilp_{𝔽_{q}[[ζ]]} be an arbitrary scheme equipped with a fixed morphism β : S → Spf 𝔽_{q^{ℓ}}[[ζ]]. Let \underline{F} be an abelian sheaf over S and let α : \underline{F} S̄ → \overline{𝕄}S̄ be a quasi-isogeny which is an isomorphism over C′. For an algebraically closed base point ι : s → S the composition of α with β^{∗}ψ gives rise to an isomorphism

γ ∘ (α|_{𝔸_{f}})^{τ} := ((β^{∗}ψ ∘ α)|_{𝔸_{f}} )^{τ} : (ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s) ⟶^{∼} 𝔸_{f}^{r}

which is fixed under the action of π_{1}(S, s) on the source. This induces an H-level structure on \underline{F}.

11 Closedness of the Uniformizable Locus

In this section we show that an abelian sheaf of rank r and dimension d over an algebraically closed field in Nilp_{𝔽_{q^{ℓ}} [[ζ]]} is isogenous to 𝕄 if and only if its formal

abelian sheaf is isoclinic. The locus of these points inside Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]] will be the one uniformized by G. We prove that this locus is formally closed. A couple of lemmas are needed beforehand.

Lemma 11.1. Let \underline{F} be an abelian sheaf of rank r and dimension d over a finite field in Nilp_{𝔽_{q^{ℓ}} [[ζ]]}. Assume that the formal abelian sheaf associated to \underline{F} is isoclinic. Then for some integer n > 0 divisible by r there is an isomorphism (σ^{n})^{∗}\underline{F} ≅ \underline{F} which maps the isogeny

T := (τ_{i}) ∘ σ^{∗}(τ_{i}) ∘ … ∘ (σ^{n−1})^{∗}(τ_{i}) : (σ^{n})^{∗}\underline{F} [n] → \underline{F}

to the isogeny (Π_{i})^{n} : \underline{F} [n] → \underline{F}.

Proof. It suffices to prove the assertion for an abelian sheaf isogenous to \underline{F}. By Proposition 7.9 the formal abelian sheaf associated to \underline{F} is isogenous to a formal abelian sheaf (F̂, F) which satisfies im F^{r} = z^{d}F̂. Pulling back \underline{F} along this isogeny (Proposition 8.1) we can assume that (F̂, F) is the formal abelian sheaf associated to \underline{F}. Since \underline{F} is defined over a finite field we certainly obtain (σ^{n})^{∗}\underline{F} ≅ \underline{F} for a suitable n > 0. We may even assume that r divides n. The isogeny T is an isomorphism over C′. Now the equation im F^{r} = z^{d}F̂ implies that the image of T is \underline{F}(−\frac{nk}{ℓ} · ∞). Hence T factors as T = α ∘ (Π_{i})^{n} for an automorphism α of \underline{F}. Since the automorphism group of \underline{F} is finite by Proposition 1.5 the lemma follows. ⊓ ⊔

Definition 11.2. Let \underline{F} and \underline{F}′ be abelian sheaves of rank r and dimension d over a field K in Nilp_{𝔽_{q^{ℓ}} [[ζ]]}. We define the Q-vector space

Hom_{K}^{0}(\underline{F}, \underline{F}′) = lim_{⟶_{D}} Hom_{K}(\underline{F}, \underline{F}′(D))

where the limit is taken over all effective divisors D ⊂ C. Furthermore for formal abelian sheaves \underline{F̂} and \underline{F̂}′ over K we define the Q_{∞}-vector space

Hom_{K}^{0}(\underline{F̂}, \underline{F̂}′) = Hom_{K}(\underline{F̂}, \underline{F̂}′) ⊗_{𝔽_{q}[[z]]} Q_{∞} .

Note that each time the invertible elements in Hom^{0} are precisely the quasiisogenies.

Corollary 11.3. Let \underline{F} and \underline{F}′ be abelian sheaves of rank r and dimension d over a finite field K in Nilp_{𝔽_{q^{ℓ}} [[ζ]]}. Assume that their associated formal abelian sheaves

\underline{F̂} and \underline{F̂}′ are isoclinic. Then for a suitable finite extension K′/K we have an isomorphism

Hom_{K′}^{0}(\underline{F}, \underline{F}′) ⊗_{Q} Q_{∞} ⟶^{∼} Hom_{K^{alg}}^{0}(\underline{F̂}, \underline{F̂}′) .

Proof. We first assume that C = ℙ_{𝔽_{q}}^{1}. Let Spec K[z] ⊂ ℙ_{K}^{1} be a neighborhood of ∞ = V(z). Restricted to this neighborhood all the sheaves F_{i} and F′_{i} are free of rank r. We fix bases for i = 0, …, ℓ − 1 and consider for the other i the bases induced by the periodicity F_{i} = F_{i−ℓ}(k · ∞). Then the morphism

⨁_{i=0}^{ℓ−1} τ_{i} : ⨁_{i=0}^{ℓ−1} σ^{∗}F_{i} ⟶ ⨁_{i=1}^{ℓ} F_{i}

is represented by a matrix U ∈ M_{rℓ}(K[z]). We endow the formal abelian sheaf (F̂, F) of \underline{F} with the induced basis. With respect to this basis F is of the form F = Û · σ^{∗} for the matrix

Û = ( \begin{matrix}0 & … & Id_{r} \\ ^{Id^{r}} & ... & ⋮ \\ ... & ... \\ Id_{r} & 0 \end{matrix} )· U = ∑_{ν=0}^{N} Û_{ν}z^{ν} ∈ M_{rℓ}(K[z]) .

The same holds for \underline{F}′ where we denote the corresponding matrix by Û′. Let us for a moment forget the structure of the (formal) abelian sheaves that is given by the Π’s. Let n = rm be the integer from Lemma 11.1 and let K′ be the compositum of 𝔽_{q^{n}} and K inside K^{alg}. We claim that there is an isomorphism of Q_{∞}-vector spaces

{ Φ ∈ M_{rℓ}(Q ⊗_{𝔽_{q}} K′) : Φ Û = Û′ ^{σ}Φ } ⊗_{Q} Q_{∞} ⟶^{∼} (11.1)

{ Φ ∈ M_{rℓ}(K^{alg}((z))) : Φ Û = Û′ ^{σ}Φ }

where the superscript ^{σ}Φ denotes the application of σ^{∗} to the entries of the matrix Φ. The injectivity is obvious. We have to prove the surjectivity. So let an element of the right hand side be given. After multiplying it with a power of z it is represented by a matrix

Φ = ∑_{μ=0}^{∞} Φ_{μ}z^{μ} ∈ M_{rℓ}(K^{alg}[[z]]) .

We expand the equation Φ Û = Û′ ^{σ}Φ into powers of z and get for all μ

(11.2)  Φ_{μ} Û_{0} − Û′_{0} ^{σ}Φ_{μ} = ∑_{ν=1}^{N}(Φ_{μ−ν} Û_{ν} − Û′_{ν} ^{σ}Φ_{μ−ν}) .

Now by Lemma 11.1 we have

Û^{σ}Û ⋯ ^{σ^{rm−1}}Û = z^{dm} · Id_{rℓ}  and  Û′ ^{σ}Û′ ⋯ ^{σ^{rm−1}}Û′ = z^{dm} · Id_{rℓ} .

This implies the equation z^{dm} · Φ = z^{dm} · ^{σ^{rm}}Φ, whence Φ_{μ} = ^{σ^{rm}}Φ_{μ} for all μ. We find Φ_{μ} ∈ M_{rℓ}(K′). Now we consider for an integer i the sequence of matrices (Φ_{i}, …, Φ_{i+N}). As i varies, these sequences run through a finite set. Therefore there are infinitely many i giving rise to the same sequence. Let j be the difference of two such i and consider the matrix

Φ − z^{j}Φ =: ∑_{μ=0}^{∞} Φ̃_{μ}z^{μ} .

In this matrix we find a sequence with Φ̃_{i} = … = Φ̃_{i+N} = 0. Looking at equation (11.2) we see that we may set all Φ̃_{μ} = 0 for μ > i + N to obtain a matrix

Φ̃ = ∑_{μ=0}^{i−1} Φ̃_{μ}z^{μ} ∈ M_{rℓ}(K′[z])

which satisfies Φ̃ Û = Û′ ^{σ}Φ̃ and is congruent to Φ modulo z^{j}. As j can be chosen arbitrarily large the surjectivity of (11.1) is established.

Now note that the Π’s on the (formal) abelian sheaves induce endomorphisms of the Q_{∞}-vector spaces in (11.1). So the compatibility with the Π’s is a condition that cuts out isomorphic linear subspaces on both sides of (11.1). From this the corollary follows in the case C = ℙ^{1}. For arbitrary C consider a finite flat morphism π : C → ℙ^{1} mapping ∞_{C} to ∞. We have just proved the assertion for π_{∗}\underline{F} and π_{∗}\underline{F}′. Since again the elements of O_{C} induce endomorphisms of the Q_{∞}-vector spaces in (11.1) we may deduce the assertion for \underline{F} and \underline{F}′. ⊓ ⊔

Recall the abelian sheaf 𝕄 over Spf 𝔽_{q^{ℓ}}[[ζ]] and the level structures on 𝕄 from Example 10.4.

Proposition 11.4. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} be locally of finite type over 𝔽_{q^{ℓ}}[[ζ]] and let H ⊂ GL_{r}(𝔸_{f}) be a compact open subgroup. Let \underline{F} be an abelian sheaf of rank r and dimension d with a rational H-level structure over S. Then for a point s ∈ S the following assertions are equivalent:

1. The formal abelian sheaf associated to \underline{F}_{s} is isoclinic.

2. Over a finite extension of the residue field κ(s) of s there is a quasi-isogeny between \underline{F}_{s} and 𝕄_{s} which is compatible with the H-level structures on both sides.

3. There is an abelian sheaf \underline{F}′ over a finite field 𝔽 ⊂ 𝔽_{q^{ℓ}}, a finite quasi-isogeny α : \overline{𝕄}_{𝔽} → \underline{F}′ over 𝔽 compatible with the H-level structures, and a quasi-isogeny φ_{s} : \underline{F}′_{s} → \underline{F}_{s} which is an isomorphism over C′ and which is defined over a finite extension of κ(s).

Proof. Note that the formal abelian sheaf of \underline{F}_{s} is isoclinic if and only if it is isogenous over an algebraically closed extension of κ(s) to the formal abelian sheaf 𝕄̂ of 𝕄. Therefore 2 implies 1. Clearly 2 follows from 3.

To prove that 1 implies 3 we proceed by induction on the transcendence degree of the residue field κ(s) of s over 𝔽_{q}. Observe that κ(s) is finitely generated since S is locally of finite type over 𝔽_{q^{ℓ}}[[ζ]]. If κ(s) ⊂ 𝔽_{q}^{alg} we obtain from Corollary 11.3 a quasi-isogeny α : 𝕄_{𝔽} → \underline{F}_{𝔽} over a finite field 𝔽 ⊃ κ(s). Altering α by a quasiisogeny of 𝕄_{𝔽} we can achieve that α is compatible with the H-level structures. Let α̂ be the induced quasi-isogeny on formal abelian sheaves. We set \underline{F}′ = α^{∗}\underline{F}_{𝔽}. Then α factors as

\overline{𝕄}_{𝔽} ⟶^{α′} \underline{F}′ ⟶^{α′′} \underline{F}_{𝔽} ,

where α′ is a finite quasi-isogeny compatible with the H-level structures, and α′′ is a quasi-isogeny which is an isomorphism over C′.

If κ(s) ⊄ 𝔽_{q}^{alg} we choose a point s′ of codimension 1 in the closure of s inside S. By Theorem 7.7 the formal abelian sheaf of \underline{F}_{s′} is also isoclinic. So the induction hypothesis asserts that there is an abelian sheaf \underline{F}′ over a finite field 𝔽 ⊂ 𝔽_{q^{ℓ}}, a finite quasi-isogeny α : \overline{𝕄}_{𝔽} → \underline{F}′ over 𝔽 compatible with the H-level structures, and a quasi-isogeny φ_{K} : \underline{F}′_{K} → \underline{F}_{K} over an algebraic closure K of κ(s′), which is an isomorphism over C′. A suitable extension of O_{\overline{\{s\}},s′} is a power series ring in one variable K[[π]] over K.

Now consider the abelian sheaf \underline{F}_{S′} over S′ := Spec K[[π]] which is the pullback of \underline{F} under the morphism S′ → S. Let \underline{F̂}_{S′} be the formal abelian sheaf associated to \underline{F}_{S′}. The Newton polygon of \underline{F̂}_{S′} is constant over S′. So by Theorem 7.10 there is a quasi-isogeny of formal abelian sheaves φ̂ : \underline{F̂}′_{S′} → \underline{F̂}_{S′}. After changing φ̂ by a quasi-isogeny of \underline{F̂}′ we may assume that φ_{K} gives rise to φ̂|_{K}. Due to the Serre-Tate-Theorem 8.4 φ̂ induces for all n a lift of φ_{K} to a quasi-isogeny φ_{n} over S′_{n} := Spec K[[π]]/(π^{n+1}). By 8.3 φ_{n} is an isomorphism over C′. Let D be the divisor of the pole of φ̂ at ∞. Then φ_{n} : \underline{F}′_{S′_{n}} → \underline{F}(D)_{S′_{n}} is a true isogeny. So in the limit we obtain a quasi-isogeny between \underline{F}′_{S′} and \underline{F}_{S′} over Spf K[[π]] which is an isomorphism over C′. By Grothendieck’s existence theorem it comes from a quasi-isogeny over Spec K[[π]] which gives us a quasi-isogeny φ_{s} : \underline{F}′_{s} → \underline{F}_{s} over K((π)). From the following lemma we obtain the desired quasi-isogeny over a finite extension of κ(s). ⊓ ⊔

Lemma 11.5. Let K ⊃ 𝔽_{q} be an arbitrary field. Consider two abelian sheaves \underline{F} and \underline{F}′ over K and a quasi-isogeny φ : \underline{F}′_{L} → \underline{F}_{L} defined over some extension L of K. Then there is a quasi-isogeny φ′ : \underline{F}′ → \underline{F} defined over a finite extension of K. If φ is an isomorphism over C′ we can find a φ′ which is also an isomorphism over C′.

Proof. Let D ⊂ C be an effective divisor such that φ : \underline{F}′_{L} → \underline{F}_{L}(D) is an isogeny. In the description of the morphisms φ_{i} : F′_{i} → F_{i}(D) there are only finitely many coefficients from L involved due to the periodicity. Let R be the K-subalgebra of L generated by these coefficients. Now consider the locally closed subscheme S of Spec R defined by the conditions that the φ_{i} give an isogeny. These are the equations Π_{i} ∘φ_{i} = φ_{i+1} ∘Π′_{i}, and τ_{i} ∘σ^{∗}φ_{i} = φ_{i+1} ∘τ′_{i}, and the conditions that φ_{i} is injective, and that coker φ_{i} is locally free of finite rank and supported on D̃ × S for an effective divisor D̃ ⊂ C. As S is of finite type over K we find a K′-rational point on it for a finite extension K′/K. The data over this point defines a quasi-isogeny φ′ : \underline{F}′_{K′} → \underline{F}_{K′}. If φ is an isomorphism over C′ it is clear that we can achieve the same for φ′. ⊓ ⊔

Corollary 11.6. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} and let H ⊂ GL_{r}(𝔸_{f}) be a compact open subgroup. Let \underline{F} be an abelian sheaf of rank r and dimension d with a rational H-level structure over S. Then the set of points in S over which there is a quasi-isogeny between \underline{F} and 𝕄 which is compatible with the H-level structures, is closed.

Proof. By Theorem 7.7 the set of points over which the formal abelian sheaf associated to \underline{F} is isoclinic is closed in S. If S is locally of finite type over 𝔽_{q^{ℓ}}[[ζ]] then the corollary follows from Proposition 11.4.

Let S be arbitrary. We only need to treat the case where S is reduced. Then the abelian sheaf \underline{F} induces a 1-morphism f : S → Ab-Sh_{H}^{r,d} ×_{C} ∞ of algebraic Spec 𝔽_{q^{ℓ}}-stacks. As the question is local on S we can assume that S is quasi-compact. Since Ab-Sh_{H}^{r,d} is locally of finite type over C we may further assume that f factors through a presentation X → Ab-Sh_{H}^{r,d} ×_{C} ∞, where X is a scheme of finite type over 𝔽_{q^{ℓ}}[[ζ]]. Then the closedness of the set on X implies the closedness of the set on S. ⊓ ⊔

Example 11.7. Consider the universal abelian sheaf over M_{I}^{2,2} from the example in Section 4. Let S = M_{I}^{2,2} ×_{C} ∞. Pink has computed that the closed set from Corollary 11.6 is the proper subset

⋃ g V(ζ, a_{11}, a_{22}, a_{21}) ⊂ S ,

where the union runs over all g ∈ GL_{2}(𝔽_{q}) which act on the points (a_{μν}) ∈ S by conjugation (a_{μν}) ↦ g(a_{μν})g^{−1} ; cf. [3, §7]. (Note that Ab-Sh_{H_{I}}^{2,2} is a GL_{2}(𝔽_{q})-torsor over Ab-Sh^{2,2}.)

Remark 11.8. There is an interesting consequence for the uniformizability of tmotives. Namely by work of Gardeyn [17] the t-motive associated to an abelian sheaf over a complete extension K of 𝔽_{q}((ζ)) is uniformizable in the sense of Anderson [1], if and only if firstly the abelian sheaf extends to an abelian sheaf over the valuation ring R of a finite extension of K, and secondly its reduction modulo the maximal ideal of R is isogenous to 𝕄.

Let X be an admissible formal scheme in the sense of Raynaud [35, 4] and let (\underline{F}, η̄) be an abelian sheaf with H-level structure over X. Then we deduce that the set of points on the associated rigid-analytic space X^{rig} over which the abelian sheaf \underline{F} is uniformizable, is formally closed. In particular if X^{rig} is quasi-compact then the complement of this set is also quasi-compact. For more elaboration on this issue see Böckle–Hartl [3].

In the remainder of this section we prove a weak result on the uniform existence of the quasi-isogeny between \underline{F} and 𝕄 which above has been studied point-wise. It will suffice for our purposes in this article. Undoubtedly there should be much stronger results in this direction.

Lemma 11.9. Let 𝔽 ⊃ 𝔽_{q} be a finite field and let S = Spec R ∈ Nilp_{𝔽_{q}[[ζ]]} be a reduced noetherian affine scheme. Consider abelian sheaves \underline{F} and \underline{F}′ of rank r and dimension d over S and 𝔽 respectively. Let s : Spec L → S be an algebraically closed point over which a quasi-isogeny φ_{s} : \underline{F}′_{s} → \underline{F}_{s} is given. Assume that φ_{s} is an isomorphism over C′. Then there exists a quasi-compact reduced scheme S′ of finite type over S containing a lift of s, such that φ_{s} extends over all of S′ to a quasi-isogeny φ : \underline{F}′_{S′} → \underline{F}_{S′} which is an isomorphism over C′.

Proof. We first treat the case where C = ℙ_{𝔽_{q}}^{1} and C′ = Spec 𝔽_{q}[t]. Then the projective R[t]-module F|_{C′_{S}} underlying \underline{F} is a direct summand of a free R[t]-module

F|_{C′_{S}} ⊕ F^{nil} = F̃

of rank r̃. We extend τ\underline{F} by zero to the endomorphism τ̃ := τ\underline{F}|_{C′_{S}} ⊕ 0 : σ^{∗}F̃ → F̃. After choosing bases of F̃ and \underline{F}′|_{C′_{S}} the coherent sheaves on C′_{S} with σ-linear endomorphism (F̃, τ̃) and \underline{F}′|_{C′_{S}} are isomorphic to

(R[t]r̃, U · σ^{∗})  and  (R[t]^{r}, U′ · σ^{∗})

for suitable matrices U = ∑_{ν} U_{ν}t^{ν} ∈ Mr̃(R[t]) and U′ = ∑_{ν} U′_{ν}t^{ν} ∈ GL_{r}(𝔽[t]). The quasi-isogeny φ : \underline{F}′_{S′} → \underline{F}_{S′} we are looking for, then corresponds to a matrix Φ = ∑_{μ} Φ_{μ} t^{μ} ∈ M_{r̃×r}(O_{S′}[t]) satisfying the condition Φ U′ = U ^{σ}Φ. We expand this condition according to powers of t to get

(11.3)  Φ_{μ} U′_{0} − U_{0} ^{σ}Φ_{μ} = ∑_{ν=1}^{μ}(U_{ν} ^{σ}Φ_{μ−ν} − Φ_{μ−ν} U′_{ν}) .

The quasi-isogeny φ_{s} over Spec L corresponds to a matrix for some integer N

\overline{Φ} = ∑_{μ=0}^{N} \overline{Φ}_{μ}t^{μ} ∈ M_{r̃×r}(L[t])

satisfying (11.3). In order to extend \overline{Φ} to a neighborhood of s we simply adjoin the entries of indeterminant matrices Φ_{0}, …, Φ_{N} to the ring R and divide out the relations (11.3) for μ = 0, …, N. Thus we obtain an R-algebra R′′ and an étale morphism S′′ = Spec R′′ → S. The point s lifts to a point s′′ : Spec L → S′′ by mapping Φ_{μ} to \overline{Φ}_{μ}.

We let S̃ ⊂ S′′ be the closed subset on which the right hand side of (11.3) and the matrices Φ_{μ} are zero for all μ > N. Clearly s′′ lies in S̃. Over S̃ the matrix Φ defines a morphism f : \underline{F}′|_{C′ ; _{S}} → (F̃, τ̃)S̃. Since τ\underline{F′}|_{C′} is an isomorphism and τ̃|_{F^{nil}} = 0 we see that f factors through a morphism φ : \underline{F}′|_{C′ ; _{S}} → \underline{F}|_{C′ ; _{S}}. The coherent sheaves ker φ and coker φ are supported on closed subschemes of C′ ; S̃. Observe that S̃ is noetherian. So the projection of these closed subschemes to S̃ are constructible subsets of S̃ by Chevalley’s theorem [EGA, Corollaire IV.1.8.5]. Their complement S′ is also constructible. Hence S′ is a finite union of locally close reduced subschemes of S̃. We replace S′ with the disjoint union of these subschemes. Then S′ is a quasi-compact reduced scheme of finite type over S. Now φ defines a quasi-isogeny \underline{F}′_{S′} → \underline{F}_{S′} which is an isomorphism over C′_{S′}. Moreover s′′ ∈ S′ and φ specializes to φ_{s} at s′′.

If C is arbitrary we consider a finite flat morphism π : C → ℙ_{𝔽_{q}}^{1} mapping ∞_{C} to ∞. Then we have just proved the existence of a quasi-isogeny π_{∗}\underline{F}_{S′} → π_{∗}\underline{F}′_{S′} over S′. In order for this to give a quasi-isogeny \underline{F}_{S′} → \underline{F}′_{S′} it must commute with the elements of O_{C}. Clearly this condition cuts out a reduced closed subscheme of S′ which contains s′′. We replace S′ by this subset. This concludes the proof of the lemma. ⊓ ⊔

Proposition 11.10. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} be a quasi-compact and reduced scheme and let H ⊂ GL_{r}(𝔸_{f}) be a compact open subgroup. Consider an abelian sheaf (\underline{F}, γ̄) of rank r and dimension d with rational H-level structure over S. Assume that for every point s ∈ S the formal abelian sheaf associated to \underline{F}_{s} is isoclinic. Then there exists a surjective morphism of 𝔽_{q^{ℓ}}-schemes S′ → S with S′ quasi-compact and reduced, and a quasi-isogeny over S′ between \underline{F}_{S′} and 𝕄_{S′} compatible with the H-level structures.

Proof. Since the data (\underline{F}, γ̄) involves only finitely many coefficients from O_{S} we may assume that S is of finite type over 𝔽_{q^{ℓ}}. In particular S is noetherian. (We could also use the fact that Ab-Sh_{H}^{r,d} is locally of finite type over C.)

Let s ∈ S be a point. From Proposition 11.4 we obtain an abelian sheaf \underline{F}′ over a finite field 𝔽, a finite quasi-isogeny α : \overline{𝕄}_{𝔽} → \underline{F}′ over 𝔽 compatible with the H-level structures, and a quasi-isogeny φ_{s} : \underline{F}′_{s} → \underline{F}_{s} which is an isomorphism over C′. The quasi-isogeny φ_{s} is defined over an algebraically closed extension of κ(s). So by the previous lemma there is a morphism S′_{s} → S of finite type from a quasicompact reduced scheme S′_{s}, such that s lifts to a point of S′_{s}, and φ_{s} extends to a quasi-isogeny φ over all of S′_{s} which is an isomorphism over C′. Clearly the quasiisogeny φ ∘ α : \overline{𝕄}_{S′_{s}} → \underline{F}_{S′_{s}} over S′_{s} is compatible with the H-level structures. By Chevalley’s theorem the image of the morphism S′_{s} → S is a constructible subset S_{s} of S containing s. Since S is of finite type over 𝔽_{q^{ℓ}} the S_{s} form a countable covering of S by constructible subsets. By [EGA, Corollaire 0.9.2.4] finitely many of the S_{s} suffice to cover S. We let S′ be the finite disjoint union of the corresponding S′_{s}. ⊓ ⊔

12 The Uniformization Theorem

Let H ⊂ GL_{r}(𝔸_{f}) be a compact open subgroup. We will define 1-morphisms of formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stacks from G to the stacks Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]] of abelian sheaves with rational H-level structure (Definition 5.7). For this purpose recall the H-level structures on \overline{𝕄} constructed in Example 10.4.

12.1. Let S be in Nilp_{𝔽_{q}[[ζ]]} and denote by S̄ its special fiber. The morphism G → G′ from Theorem 10.2 associates to an element (β, \underline{F̂}, α̂) ∈ G(S) an abelian sheaf \underline{F} and a quasi-isogeny α : \underline{F} S̄ → β̄^{∗}\overline{𝕄} which is an isomorphism over C′. For an algebraically closed base point ι : s → S the quasi-isogeny α has lead in 10.4 to an isomorphism

γ ∘ (α|_{𝔸_{f}})^{τ} : (ι^{∗}\underline{F}|_{𝔸_{f}})^{τ} (s) ⟶^{∼} 𝔸_{f}^{r}

which is fixed under the action of π_{1}(S, s) on the source. Then we define a 1morphism Θ of formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stacks by the following map on S-valued points

(12.1)  Θ : G × GL_{r}(𝔸_{f})/H ⟶ Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]]
(β, \underline{F̂}, α̂) × hH ↦ (\underline{F}, h^{−1}γ (α|_{𝔸_{f}})^{τ} ) × β .

It is equivariant with respect to the right GL_{r}(𝔸_{f})-action on the projective systems on both sides of (12.1).

12.2. There is an action of J(Q) on the source which we describe next. Recall that we have defined in 9.5 an action of J(Q_{∞}) on G through the isomorphism ε_{∞} from J(Q_{∞}) to the group of quasi-isogenies of ˆ \overline{𝕄}. We let J(Q) act on G via the inclusion J(Q) ⊂ J(Q_{∞}).

On the other hand we have a morphism

ε^{∞} : J(Q) ↪ GL_{r}(𝔸_{f})

which is defined by the commutative diagram

(\overline{𝕄}|_{𝔸_{f}})^{τ} ⟶^{(g|_{𝔸_{f}})^{τ}} (\overline{𝕄}|_{𝔸_{f}})^{τ}
γ↓  ↓^{γ}
𝔸_{f}^{r}  ⟶^{ε^{∞}(g)} 𝔸_{f}^{r}

for g ∈ J(Q). Note that ε^{∞} identifies J(Q) with the diagonal embedding of GL_{r}(Q) into GL_{r}(𝔸_{f}).

We define a left action of the group J(Q) on G × GL_{r}(𝔸_{f})/H

(β, \underline{F̂}, α̂) × hH ⟼ (β, \underline{F̂}, β̄^{∗}ε_{∞}(g) ∘ α̂) × ε^{∞}(g)hH .

Proposition 12.3. We abbreviate Y = G × GL_{r}(𝔸_{f})/H. The action of J(Q) induces a 1-isomorphism of formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stacks

Y × J(Q) ⟶^{∼} Y ×_{Ab-Sh_{H}^{r,d} × Spf 𝔽_{q^{ℓ}} [[ζ]]} Y .

Proof. We have to show that the functor between the categories of S-valued points on both sides is an equivalence. Note that in the stack Y the only morphisms are the identities. From this full faithfulness follows. For essential surjectivity we have to show that two S-valued points of Y lie in the same orbit if and only if they are mapped to the same point by Θ. So let (β, \underline{F̂}, α̂, hH) and (β, \underline{F̂}′, α′, h′H) be in Y (S) and let (\underline{F}, α, hH) and (\underline{F}′, α, h′H) be their algebraizations.

First we assume that

( β, \underline{F̂}′, α′, h′H ) = ( β, \underline{F̂}, β̄^{∗}ε_{∞}(g) ∘ α̂, ε^{∞}(g)hH )

in Y (S) for a g ∈ J(Q). Consider the following diagram of quasi-isogenies over S̄

(12.2)  \begin{matrix}\underline{F} S̄ & ⟶^{α} & \overline{𝕄}S̄ \\ φ̄↓ & & ↓^{gS̄} \\ \underline{F} ′ ; S̄ & ⟶^{α′} & \overline{𝕄}S̄ , \end{matrix}

where gS̄ is the quasi-isogeny obtained from g by Proposition 8.7. The quasi-isogeny φ̄ defined by this diagram is finite by construction. By Proposition 8.2 it lifts uniquely to a quasi-isogeny φ : \underline{F} → \underline{F}′. We claim that φ is finite. Namely the induced quasi-isogeny φ̂ on formal abelian sheaves satisfies φ̂S̄ = id_{\underline{F̂}S̄}. Hence we find φ̂ = id\underline{F̂} by the uniqueness of the lift. This shows that φ is a finite quasi-isogeny.

Now φ̄ induces on \underline{F} the H-level structure

h^{−1} ε^{∞}(g)^{−1} γ (α′|_{𝔸_{f}})^{τ} (φ̄|_{𝔸_{f}})^{τ} = h^{−1} γ (α|_{𝔸_{f}})^{τ} .

Therefore the two points have the same image under Θ.

Conversely let φ : \underline{F} → \underline{F′} be a finite quasi-isogeny inducing an equality of H-level structures on \underline{F}

h′^{−1} γ (α′|_{𝔸_{f}})^{τ} (φ|_{𝔸_{f}})^{τ} = h^{−1} γ (α|_{𝔸_{f}})^{τ}

This time diagram (12.2) defines a quasi-isogeny gS̄ from \overline{𝕄}S̄ to itself. By Proposition 8.7 it comes from an element g ∈ J(Q). Then we have

( β, \underline{F̂}′, α′, h′H ) = ( β, \underline{F̂}, β̄^{∗}ε_{∞}(g) ∘ α̂, ε^{∞}(g)hH ) .
⊓ ⊔

Hence the map Θ factors through a 1-morphism of formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]- stacks

J(Q)\G × GL_{r}(𝔸_{f})/H ⟶ Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]] .

Note that the quotient J(Q)\G × GL_{r}(𝔸_{f})/H is a formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stack due to Theorem 9.6. Indeed, the subgroup J(Q) ↪ J(Q_{∞}) × GL_{r}(𝔸_{f}) is discrete. Hence

J(Q)\G × GL_{r}(𝔸_{f})/H ≅ ∐_{Γ} Γ\G

where Γ runs through a countable set of subgroups of J(Q_{∞}) of the form

(J(Q_{∞}) × gHg^{−1}) ∩ J(Q) ⊂ J(Q_{∞}) .

These are separated discrete subgroups.

Proposition 12.4. The 1-morphism Θ defines a 1-morphism of formal algebraic Spf 𝔽_{q}[[ζ]]-stacks

Θ′ : G′ × GL_{r}(𝔸_{f})/H ⟶ Ab-Sh_{H}^{r,d} ×_{C} Spf 𝔽_{q}[[ζ]]

which is invariant with respect to the action of J(Q) on the source defined in Section 10.

Proof. We have to show that Θ and the action of J(Q) commute with the Galoisdescent data on the source and the target of the 1-morphism Θ. For the J(Q)-action this was already observed in 9.5. For Θ we must check that

(\underline{F}, h^{−1}γ (α|_{𝔸_{f}})^{τ} ) = (\underline{F}, h^{−1}^{σ}γ (α|_{𝔸_{f}})^{τ} )

in Ab-Sh_{H}^{r,d}, where ^{σ}γ is obtained from γ by the action of the generator F rob_{q} of Gal(𝔽_{q^{ℓ}}/𝔽_{q}). Now^{σ}γ^{−1}∘γ comes in fact from an automorphism of \overline{𝕄} (cf. 10.4). This induces an automorphism of \underline{F} which carries the H-level structure h^{−1}γ (α|_{𝔸_{f}})^{τ} to h^{−1}^{σ}γ (α|_{𝔸_{f}})^{τ}. ⊓ ⊔

Let Z be the set of points s : Spec L → Ab-Sh_{H}^{r,d} ×_{C} ∞ such that the universal abelian sheaf over s is isogenous to \overline{𝕄} over an algebraic closure of L. Consider the preimage Z′ ⊂ Ab-Sh_{H}^{r,d} ×_{𝔽_{q}}𝔽_{q^{ℓ}} of Z under the base change morphism coming from 𝔽_{q} ⊂ 𝔽_{q^{ℓ}}.

Lemma 12.5. The set Z′ can also be described as the set of points s over which the associated formal abelian sheaf is isoclinic. In particular Z and Z′ are closed subsets.

Proof. The formal abelian sheaf ˆ \overline{𝕄} of \overline{𝕄} is isoclinic. Therefore Z′ is contained in the set of the lemma. Conversely let s belong to this later set. Since Ab-Sh_{H}^{r,d} is locally of finite type over C we can assume that s comes from a point on a local presentation X → Ab-Sh_{H}^{r,d} ×_{C} ∞ where X is a scheme of finite type over 𝔽_{q}. From Proposition 11.4 we obtain a quasi-isogeny α : \underline{F}_{s} → 𝕄_{s} over an algebraic closure of L. Hence s belongs to Z′.

Now Theorem 7.7 implies that the subset Z′ is closed. Namely Z′ is the complement of the open substack on which the associated Newton polygon lies strictly below the Newton polygon of ˆ \overline{𝕄}. Therefore also the image Z of Z′ is closed. ⊓ ⊔

We denote by Ab-Sh_{H}^{r,d}_{/Z} the formal completion of Ab-Sh_{H}^{r,d} along Z (A.12). It is a formal algebraic Spf 𝔽_{q}[[ζ]]-stack. By its definition the 1-morphism Θ′ factors through Ab-Sh_{H}^{r,d}_{/Z}. Indeed, if a point s ∈ Ab-Sh_{H}^{r,d} ×_{C} ∞ lies in the image of Θ′ the formal abelian sheaf associated to s is isogenous to 𝕄̂ by definition and hence isoclinic. We can now formulate our

Uniformization Theorem 12.6. There are GL_{r}(𝔸_{f})-equivariant 1-isomorphisms of formal algebraic Spf 𝔽_{q}[[ζ]]-stacks

Θ̄ : J(Q)\G × GL_{r}(𝔸_{f})/H ⟶^{∼} Ab-Sh_{H}^{r,d}_{/Z} ×_{𝔽_{q}} 𝔽_{q^{ℓ}} ,
Θ̄′ : J(Q)\G′ × GL_{r}(𝔸_{f})/H ⟶^{∼} Ab-Sh_{H}^{r,d}_{/Z} .

Example 12.7. In the case of elliptic sheaves

Ab-Sh_{H}^{r,1}_{/Z} ×_{𝔽_{q}} 𝔽_{q^{ℓ}} = Ab-Sh_{H}^{r,1} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]] .

This follows from the fact that there is only one polygon between the points (0, 0) and (r, 1) with non-negative slopes and integral break points, namely the straight line. So all formal abelian sheaves are isoclinic and Z is all of Ab-Sh_{H}^{r,d} ×_{C} ∞. In this case the open and closed subscheme of G on which the universal quasiisogeny has height zero is the formal scheme Ω^{(r)} used by Drinfeld [11]. A detailed account on this can be found in Genestier [18]. Therefore we have an isomorphism of formal schemes G ≅ ℤ × Ω^{(r)}. This decomposition of G is compatible with the decomposition of Ab-Sh_{H_{I}}^{r,1} ≅ ℤ × Dr-Mod_{I}^{r} from Example 1.8. So Drinfeld’s uniformization theorem which announces an isomorphism of formal schemes

GL_{r}(Q)\Ω^{(r)} × GL_{r}(𝔸_{f})/H_{I} ⟶^{∼} Dr-Mod_{I}^{r} ×_{C} Spf 𝔽_{q^{ℓ}}[[ζ]]

is equivalent to ours.

Example 12.8. Consider the algebraic stack Ab-Sh_{H}^{2,2} from Section 4. We describe Z(0) := Z ∩ Ab-Sh_{H}^{2,2}(0). In 11.7 we have remarked that

Z ∩ M_{I}^{2,2} ×_{C} ∞ = ⋃ g V(ζ, a_{11}, a_{22}, a_{21}) .

Now we claim that

Z(0) = (Z ∩ M_{I}^{2,2} ×_{C} ∞) ∪ (Ab-Sh_{H}^{2,2}(0) ∖ M_{I}^{2,2}) ×_{C} ∞ .

Indeed let s : Spec L → (Ab-Sh_{H}^{2,2}(0) ∖ M_{I}^{2,2}) ×_{C} ∞ be a point. We must show that the abelian sheaf \underline{F} over L is isogenous to \overline{𝕄}. By what was said in Section 4 we have F_{0} = O_{ℙ_{L}^{1}}(m · ∞) ⊕ O_{ℙ_{L}^{1}}(−m · ∞) for an integer m ≥ 1. With respect to this basis, τ_{0} is described by a matrix

τ_{0} = ( \begin{matrix}a_{0} + a_{1}t & b_{0} + … + b_{2m+1} t^{2m+1} \\ 0 & d_{0} + d_{1}t \end{matrix} ) · σ^{∗} .

Due to the presence of the I-level structure we may assume a_{0} = d_{0} = 1 and b_{0} = 0. Since coker τ_{0} is supported at ∞ we must have a_{1} = d_{1} = 0. Now let u_{i} ∈ L^{alg} be solutions of the equations u_{i}^{q} − u_{i} + b_{i} = 0 for i = 1, …, 2m + 1. Then the isomorphism

\overline{𝕄}|_{C′} → \underline{F}|_{C′} , ( \begin{matrix}x \\ y \end{matrix} ) ↦ ( \begin{matrix}1 & u_{0}t + … + u_{2m+1} t^{2m+1} \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}x \\ y \end{matrix} )

extends to a quasi-isogeny \overline{𝕄} → \underline{F} over L^{alg}. Hence s belongs to Z(0).

Remark 12.9. In [34] Rapoport–Zink study the uniformization of Shimura varieties of EL- and PEL-type. One of their theorems yields the uniformization of the formal completion of a Shimura variety along the most supersingular isogeny class [34, 6.30]. The Newton polygon in this isogeny class is maximal. In this sense our Uniformization Theorem is closely analogous to theirs. Beyond this, Rapoport–Zink also obtain uniformization theorems of other isogeny classes. There is no doubt that these theorems too have counterparts for abelian sheaves.

The remainder of this article is devoted to the proof of the Uniformization Theorem.

13 Proof of the Uniformization Theorem

13.1. By Proposition 12.4 it suffices to prove the assertion for the 1-morphism Θ̄. We fix the following notation. On the formal scheme G we let J be the largest ideal of definition of G; cf. [EGA, I_{new}, 10.5.4]. For an integer n ≥ 0 we denote by G_{n} the scheme (G, O_{G}/J^{n+1}). We set

Y := G × GL_{r}(𝔸_{f})/H ,
Y_{n} := G_{n} × GL_{r}(𝔸_{f})/H ,
Y := J(Q)\G × GL_{r}(𝔸_{f})/H ,
X := Ab-Sh_{H}^{r,d}_{/Z} ×_{Spf 𝔽_{q}[[ζ]]} Spf 𝔽_{q^{ℓ}}[[ζ]] .

Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let (\underline{F}, γ̄, β) ∈ X (S) which we consider as a 1-morphism S → X. We have to show that the stack

Y ×_{X} S

is a scheme mapping isomorphically to S. The proof relies on several intermediate lemmas.

Lemma 13.2. The 1-morphism Θ̄ : Y → X is a 1-monomorphism of formal algebraic stacks, i.e. for every S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} the functor Θ̄(S) : Y(S) → X (S) is fully faithful.

Proof. We can view the formal algebraic Spf 𝔽_{q^{ℓ}}[[ζ]]-stacks X and Y as Spec 𝔽_{q^{ℓ}}[[ζ]]- stacks X̃ and Ỹ by setting for an Spec 𝔽_{q^{ℓ}}[[ζ]]-scheme S

X̃(S) = \{ \begin{matrix}X (S) & if S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} \\ ∅ & if S ∉ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} . \end{matrix}

Then the assertion follows from Proposition 12.3, Lemma A.13 and [30, Proposition 3.8]. ⊓ ⊔

Lemma 13.3. The 1-morphism of algebraic Spec 𝔽_{q^{ℓ}}-stacks Θ̄_{red} : Y_{red} → X_{red} (A.6) is representable by a morphism of schemes.

Proof. This follows from the fact that every 1-monomorphism of algebraic stacks is representable by a morphism of schemes; cf. [30, Théorème A.2 and Corollaire 8.1.3]. ⊓ ⊔

Lemma 13.4. Θ̄_{red} : Y_{red} → X_{red} is surjective.

Proof. Note that X_{red} is the closed substack Z ⊂ Ab-Sh_{H}^{r,d} ×_{C} ∞ with its induced reduced structure. Let s ∈ X_{red} be a point. By definition of Z there is a quasi-isogeny α : \underline{F}_{s} → 𝕄_{s}. We can multiply it with a quasi-isogeny of 𝕄 and thus assume that α is compatible with the H-level structures. The induced quasi-isogeny \underline{F}_{s} → α^{∗}𝕄_{s} is finite and also compatible with the H-level structures. Therefore s lies in the image of Θ̄_{red}. ⊓ ⊔

Lemma 13.5. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} be quasi-compact and reduced, and consider a 1-morphism S → X_{red}. Then there exists a surjective morphism of 𝔽_{q^{ℓ}}-schemes S′ → S with S′ quasi-compact and reduced, and a 2-commutative diagram

S′ ⟶ Y_{red}
↓  ↓
S ⟶ X_{red} .

Proof. This is just a reformulation of Proposition 11.10. ⊓ ⊔

Lemma 13.6. Θ̄_{red} : Y_{red} → X_{red} is quasi-compact.

Proof. Let S ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} be quasi-compact and reduced, and let S → X_{red} be a 1-morphism. Let S′ → S be the surjective morphism from Lemma 13.5. It gives rise to a surjective morphism of schemes over 𝔽_{q^{ℓ}}

Y_{red} ×_{X_{red}} S ← ⟵ Y_{red} ×_{X_{red}} S′ ← ⟵ Y_{red} ×_{X_{red}} S′ .

Since Y_{red} ×_{X_{red}} S′ ≅ S′ × J(Q) by Proposition 12.3, we obtain an epimorphism S′ ⟶ Y_{red} ×_{X_{red}} S. Now the lemma follows from the quasi-compactness of S′. ⊓ ⊔

Lemma 13.7. Θ̄_{red} : Y_{red} → X_{red} is proper.

Proof. Since Y_{red} and X_{red} are locally of finite type and Θ̄_{red} is quasi-compact, we see that Θ̄_{red} is of finite type. Being a 1-monomorphism it is also separated. So it remains to prove that it is universally closed. For this we use the valuative criterion [30, Théorème 7.3]. Let R be a valuation ring with Spec R ∈ Nilp_{𝔽_{q^{ℓ}} [[ζ]]} and let K be its field of fractions. We have to show that for every 2-commutative diagram

Spec K ⟶ Y_{red}
↓  ↓
Spec R ⟶ X_{red}

there exists a finite extension R′/R of valuation rings and a 1-morphism of Spec 𝔽_{q^{ℓ}}- stacks Spec R′ → Y_{red} which 2-commutes with the above diagram. However, the existence of this data follows from Lemma 13.5. ⊓ ⊔

Lemma 13.8. The 1-morphism Θ̄ : Y → X is adic (A.9).

Proof. We will show that Θ̄_{red} is a 1-isomorphism. Since both X and Y are adic (A.7) this suffices.

Let P : X → X_{red} be a presentation. Then P is an epimorphism since X_{red} is an algebraic Spec 𝔽_{q^{ℓ}}-stack. Thus it suffices to show that Y_{red} ×_{X_{red}} X → X is a 1isomorphism. From the previous lemmas we know that it is a proper monomorphism of schemes, hence a closed immersion. Since it is also surjective and X is reduced it is an isomorphism as desired. ⊓ ⊔

Lemma 13.9. The 1-morphism Θ̄ : Y → X is étale.

Proof. Since quasi-isogenies of z-divisible groups lift to infinitesimal neighborhoods we first see that Y → X is étale.

Now let J be an ideal of definition of X and let Z be the closed substack defined by J. Since Θ̄ and also the presentation Y → Y are adic (Theorem 9.6), we obtain 1-morphisms

Y ×_{X} Z → Y ×_{X} Z → Z .

of algebraic Spec 𝔽_{q^{ℓ}}[[ζ]]-stacks. Since these 1-morphisms are representable by morphisms of schemes, we can apply [EGA] to see that Y ×_{X} Z → Z is étale. This proves the lemma. ⊓ ⊔

We can now finish the

Proof (of the Uniformization Theorem 12.6). Keep the notation of the proof of Lemma 13.9. We have to show that

Y ×_{X} Z → Z

is a 1-isomorphism of algebraic Spec 𝔽_{q}[[ζ]]-stacks. Let S be a Spec 𝔽_{q}[[ζ]]-scheme and let S → Z be a 1-morphism. Then from the previous lemmas we conclude that

Y ×_{X} S → S

is an étale monomorphism of schemes, hence an open immersion. Being also surjective, it is indeed an isomorphism. This completes the proof of the Uniformization Theorem. ⊓ ⊔

A Background on Formal Algebraic Stacks

For a general introduction to the theory of algebraic stacks we refer to Laumon– Moret-Bailly [30] or Deligne–Mumford [9]. In this appendix we propose the notion of formal algebraic stacks which generalizes the notion of algebraic stacks in the same way as formal schemes are a generalization of usual schemes. In fact much of the theory of algebraic stacks can be developed also for formal algebraic stacks. See Hartl [22] for details.

For a scheme S let Sch_{S} be the category of S-schemes equipped with the étale topology. An algebraic stack X over S is defined as a category fibered in groupoids over Sch_{S} satisfying further conditions [30, Definition 4.1]. We transfer this concept to the formal category.

For the rest of this appendix we let S be a formal scheme. We denote by Nilp_{S} the category of schemes over S on which an ideal of definition of S is locally nilpotent. We remind the reader that every scheme may be considered as a formal scheme having (0) as an ideal of definition. In this sense every U ∈ Nilp_{S} is itself a formal scheme. We equip Nilp_{S} with the étale topology. We make the following definitions (compare [29, 26]).

Definition A.1. A formal S-space is a sheaf of sets on the site Nilp_{S}.

Definition A.2. A (quasi-separated) formal algebraic S-space is a formal S-space X such that

1. the diagonal morphism X → X ×_{S} X is relatively representable by a quasicompact morphism of formal schemes and

2. there is a formal scheme X′ over S and a morphism of formal S-spaces X′ → X which is representable (automatic because of 1) by an étale surjective morphism of formal schemes.

Definition A.3. A formal S-stack X is a category X fibered in groupoids over Nilp_{S} such that

1. for every U ∈ Nilp_{S} and every x, y ∈ X (U) the presheaf

Isom(x, y) : Nilp_{U} → Sets
(V → U) ↦ Hom_{X (V)}(x_{V} , y_{V})

is in fact a sheaf on Nilp_{U},

2. for every covering U_{i} → U in Nilp_{S} all descent data for this covering are effective.

Definition A.4. A formal S-stack is called representable if it is 1-isomorphic to a formal algebraic space.

A 1-morphism X → Y of formal S-stacks is called representable if for every U ∈ Nilp_{S} and every y ∈ Y(U) viewed as a 1-morphism U → Y of formal S-stacks the fiber product X ×_{Y} U (in the 2-category of formal S-stacks) is representable.

Definition A.5. A (quasi-separated) formal algebraic S-stack is a formal S-stack X such that

1. the diagonal 1-morphism of formal S-stacks

X → X ×_{S} X

is representable, separated, and quasi-compact,

2. there exists a formal algebraic S-space X and a 1-morphism of formal S-stacks

P : X → X

which is representable (automatic because of 1) by a smooth and surjective morphism of formal algebraic S-spaces.

The 1-morphism P is called a presentation of X. We say that X is of DM-type if the presentation P in 2 can be chosen étale.

A.6. Let X be a formal algebraic S-stack and let P : X → X be a presentation. We define the underlying reduced stack X_{red} as follows: For every U ∈ Nilp_{S} we let X_{red}(U) be the full subcategory of X (U) whose objects are the x ∈ X (U) such that there is a covering U′ → U in Nilp_{S}, an element x′ ∈ X_{red}(U′), and an isomorphism in X (U′) between x_{U′} and P (x′). Then X_{red} is an algebraic S_{red}-stack. If moreover X is of DM-type then X_{red} is an algebraic S_{red}-stack in the sense of Deligne–Mumford. In this way we obtain from every 1-morphism f : Y → X of formal algebraic Sstacks a 1-morphism f_{red} : Y_{red} → X_{red} of algebraic S_{red}-stacks. We say that f is locally formally of finite type if f_{red} is locally of finite type.

A.7. For an algebraic stack X one can define its structure sheaf O_{X} which is a sheaf on the lisse-étale site of X; cf. [30, § 12]. Then one has the usual bijection between closed substacks of X and quasi-coherent sheaves of ideals of O_{X}.

The same can be done for formal algebraic S-stacks X. In this setting we say that a sheaf of ideals J of O_{X} is an ideal of definition of X if for some (any) presentation P : X → X of X the ideal sheaf P^{∗}J is an ideal of definition of X. The formal algebraic S-stack is called J -adic if J^{n} is an ideal of definition for every n. It is called adic if it is J -adic for some J. If X (i.e. X) is locally noetherian then there exists a unique largest ideal of definition K of X, namely the one defining the closed substack X_{red} of X. Note that if X is J -adic for some J then it is also K-adic. One easily verifies the following proposition which generalizes A.6.

Proposition A.8. Let I be an ideal of definition of S and let J be an ideal of definition of a formal algebraic S-stack X with I ·O_{X} ⊂ J. Then the closed substack of X which is defined by the ideal J is an algebraic (S, O_{S}/I)-stack.

Definition A.9. A 1-morphism f : Y → X of locally noetherian formal algebraic S-stacks is called adic if for some (any) ideal of definition J of X the ideal f^{∗}J is an ideal of definition of Y.

We discuss some examples.

Example A.10. Every formal scheme G over S can be viewed as a sheaf of sets on the category Nilp_{S}. Moreover, every sheaf on Nilp_{S} is a formal S-stack. Therefore we can view every quasi-separated formal scheme G over S as a formal algebraic S-stack of DM-type.

Example A.11. Let S^{alg} be a scheme and let S_{0} be a closed subscheme. (We allow the case S_{0} = S^{alg}.) We let the formal scheme S be the formal completion of S^{alg} along S_{0}. If X is an (algebraic) S^{alg}-stack (in the sense of Deligne–Mumford) then X ×_{S^{alg}} S is an adic formal (algebraic) S-stack (of DM-type).

We generalize this example as follows.

Definition A.12. Let S^{alg} be a scheme and let S_{0} be a closed subscheme. We let the formal scheme S be the formal completion of S^{alg} along S_{0}. Let X be an algebraic S^{alg}-stack and let Z ⊂ X be a closed substack, contained in X ×_{S^{alg}} S_{0}. We view the objects of X as 1-morphisms U → X for varying U ∈ Sch_{S^{alg}}. We define the formal completion X̂_{Z} of X along Z as the full subcategory of X consisting of those objects U → X such that U_{red} → X factors through Z, i.e. such that there exists a 2-commutative diagram of 1-morphisms

U ⟶ X
↑  ↑
U_{red} ⟶ Z .

Note that if X is an S^{alg}-scheme this definition coincides with the usual definition of formal completion along a closed subscheme.

One verifies directly that X̂_{Z} is an S^{alg}-stack. The embedding X̂_{Z} → X is a 1-morphism of S^{alg}-stacks which is a 1-monomorphism, i.e. the functors X̂_{Z}(U) → X (U) are fully faithful. This implies that the diagonal 1-morphism

X̂_{Z} → X̂_{Z} ×_{X} X̂_{Z}

is a 1-isomorphism of S^{alg}-stacks [30, 2.3]. As a consequence we obtain

Lemma A.13. Let Y → X̂_{Z} and Y′ → X̂_{Z} be two 1-morphisms of S^{alg}-stacks. Then there is a 1-isomorphism of S^{alg}-stacks

Y ×X̂_{Z} Y′ ⟶^{∼} Y ×_{X} Y′ .

The fibration X̂_{Z} → Sch_{S^{alg}} factors through Nilp_{S} ↪ Sch_{S^{alg}}. Note the fact that for an étale covering U_{i} → U in Sch_{S^{alg}} we have U ∈ Nilp_{S} if and only if U_{i} ∈ Nilp_{S} for all i. This implies that X̂_{Z} is a formal S-stack. We show that X̂_{Z} is even a formal algebraic S-stack. Namely, condition 1 of Definition A.5 can be read off from the following 2-cartesian diagram of S^{alg}-stacks

X̂_{Z} ⟶^{∼} X̂_{Z} ×_{X} X̂_{Z} ⟶ X̂_{Z} ×_{S^{alg}} X̂_{Z} ⟵^{∼} X̂_{Z} ×_{S} X̂_{Z}
↓  ↓
X  ⟶ X ×_{S^{alg}} X .

Condition 2 results from the fact that every presentation X → X of X induces a presentation X̂_{Z} → X̂_{Z} of X̂_{Z} by the formal completion X̂_{Z} of X along the closed subscheme Z = X ×_{X} Z. Thus we have proved

Proposition A.14. Keep the notation of Definition A.12. Then the formal completion X̂_{Z} of X along Z is a formal algebraic S-stack. If X is an algebraic S^{alg}-stack in the sense of Deligne–Mumford then X̂_{Z} is of DM-type. If J is the ideal sheaf on X defining the closed substack Z then X̂_{Z} is J · OX̂_{Z} -adic.

A.15. Let S^{alg} be a scheme and let X be an algebraic S^{alg}-stack. There is a notion of points of X. Namely a point of X is given by a 1-morphisms Spec K → X for an S^{alg}-field K. The set |X | of points of X forms a topological space; cf. [30, § 5]. Let Z ⊂ |X | be a closed subset, i.e. there is an open substack U ⊂ X such that Z = |X | ∖ |U|. We can equip Z in a unique way with a structure of reduced closed substack [30, 4.10]. By A.12 we can consider the formal completion of X along Z.

Example A.16 (Quotients). Let U ∈ Nilp_{S} and let G be a formal U-group space (i.e. a group object in the category of formal U-spaces). A (left) G-torsor is a formal U-space P with an action of G (from the left) such that there is a covering U′ → U in Nilp_{S} for which P ×_{U} U′ is G ×_{U} U′-isomorph to G ×_{U} U′ which acts on itself by left translation.

Let X be a formal S-space, Y an X-space (i.e. a formal S-space equipped with a morphism Y → X) and G an X-group space which acts on Y from the left. We define the quotient stack G\Y as the following category fibered in groupoids over Nilp_{S}: For every U ∈ Nilp_{S} the category (G\Y)(U) consists of all triples (x, P, α) where x ∈ X(U), P is a G ×_{X,x} U-torsor and α : P → Y_{X,x}U is a G ×_{X,x} U-equivariant morphism of formal U-spaces. One easily verifies that the quotient G\Y is a formal S-stack.

In particular, if X is an adic formal algebraic S-space and G a finite étale Sgroup scheme then the quotient G\X is even an adic formal algebraic S-stack of DM-type. In this case, the canonical projection X → G\X is an étale presentation of G\X.

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