Local ℙ-shtukas and their relation to global 𝔊-shtukas

Esmail Arasteh Rad and Urs Hartl^{∗}

December 29, 2015

Abstract

This is the first in a sequence of two articles investigating moduli stacks of global 𝔊-shtukas, which are function field analogs for Shimura varieties. Here 𝔊 is a flat affine group scheme of finite type over a smooth projective curve, and global 𝔊-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. Our moduli stacks generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields.

In the present article we explain the relation between global 𝔊-shtukas and local ℙ-shtukas, which are the function field analogs of p-divisible groups with additional structure. We prove the analog of a theorem of Serre and Tate stating the equivalence between the deformations of a global 𝔊-shtuka and its associated local ℙ-shtukas. We also investigate local ℙ-shtukas alone and explain their relation with Galois representations through their Tate modules. And if ℙ is a smooth affine group scheme with connected reductive generic fiber we prove the existence of Rapoport–Zink spaces for bounded local ℙ-shtukas as formal schemes locally formally of finite type. In the sequel to this article we use these Rapoport–Zink spaces to uniformize the moduli stacks of global 𝔊-shtukas. Mathematics Subject Classification (2000): 11G09, (11G18, 14L05, 14M15)

Contents

1 Introduction 2 1.1 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Local ℙ-Shtukas and Global 𝔊-Shtukas 4 2.1 Loop Groups and Local ℙ-Shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Global 𝔊-Shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Tate Modules for Local ℙ-Shtukas 8

4 The Rapoport–Zink Spaces for Local ℙ-Shtukas 11 4.1 Unbounded Rapoport–Zink Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Bounded Local ℙ-Shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Representability of the Bounded Rapoport–Zink Functor . . . . . . . . . . . . . . . . . . . 20

5 The Relation Between Global 𝔊-Shtukas and Local ℙ-Shtukas 28 5.1 Preliminaries on 𝔊-Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 The Global-Local Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 The Analog of the Serre–Tate Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

References 35

∗Both authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) in form of the research grant HA3002/2-1 and the SFB’s 478 and 878.

1 Introduction

Let 𝔽_{q} be a finite field with q elements, let C be a smooth projective geometrically irreducible curve over 𝔽_{q}, and let 𝔊 be a flat affine group scheme of finite type over C. A global 𝔊-shtuka \underline{G} over an 𝔽_{q}-scheme S is a tuple (G, s_{1}, …, s_{n}, τ) consisting of a 𝔊-torsor G over C_{S} := C ×_{𝔽_{q}} S, an n-tuple of (characteristic) sections (s_{1}, …, s_{n}) ∈ C^{n}(S) and a Frobenius connection τ defined outside the graphs of the sections s_{i}, that is, an isomorphism τ: σ^{∗}G|_{C_{S}∖∪_{i}Γ_{s_{i}}} ^{∼} ⟶ G|_{C_{S}∖∪_{i}Γ_{s_{i}}} where σ^{∗} = ( id_{C} × Frob_{q,S})^{∗}.

In [AH13] we will show that the moduli stack ∇_{n}ℋ ^{1}(C, 𝔊) of global 𝔊-shtukas, after imposing suitable boundedness conditions and level structures, is an algebraic Deligne–Mumford stack over C^{n}. One can hope that ∇_{n}ℋ ^{1}(C, 𝔊) may play the same role that Shimura varieties play for number fields. More specifically one can hope that the Langlands correspondence for function fields is realized on its cohomology. Note that in particular our moduli stack generalizes the space F Sh_{D,r} of F -sheaves (also called “Drinfeld-shtukas”) which was considered by Drinfeld [Dri87] and Laurent Lafforgue [Laf02] in their proof of the Langlands correspondence for 𝔊 = GL_{2} (resp. 𝔊 = GL_{r}), and which in turn was generalized by Varshavsky’s [Var04] moduli stacks F Bun to the case where 𝔊 is a constant split reductive group. Varshavsky’s moduli stack and our generalization are used by Vincent Lafforgue [Laf12] to prove Langlands parameterization over function fields. Strictly speaking Drinfeld and L. Lafforgue did not use the language of GL_{r}-torsors but rather the equivalent one of locally free sheaves. Our space ∇_{n}ℋ ^{1}(C, 𝔊) likewise generalizes the moduli stacks Cht\underline{λ} of Ngô and Ngô Dac [NN08] who explain a simple method to count 𝔊-shtukas over finite fields, the stacks Eℓℓ_{C,𝒟,I} of Laumon, Rapoport and Stuhler [LRS93] who used them to prove the local Langlands correspondence for GL_{r}, and the stacks Ab-Sh_{H}^{r,d} of the second author [Har05]; see [AH13, Remark 3.19] for a detailed comparison between these moduli stacks.

In [AH13] we also prove that ∇_{n}ℋ ^{1}(C, 𝔊) has a Rapoport–Zink uniformization by Rapoport–Zink spaces for local ℙ-shtukas. More precisely, let A_{ν} ≅ 𝔽_{ν}[[ζ]] be the completion of the local ring O_{C,ν} at a closed point ν ∈ C, let Q_{ν} be its fraction field, and consider the group schemes ℙ = ℙ_{ν} := 𝔊 ×_{C} Spec A_{ν} and P_{ν} = 𝔊 ×_{C} Spec Q_{ν}. Let Nilp_{A_{ν}} denote the category of A_{ν}-schemes on which the uniformizer ζ of A_{ν} is locally nilpotent. A local ℙ_{ν}-shtuka over a scheme S ∈ Nilp_{A_{ν}} is a pair \underline{L} = (L_{+}, τ̂) consisting of an L^{+}ℙ_{ν}-torsor L_{+} on S and an isomorphism of the LP_{ν}-torsors τ̂: σ̂^{∗}L ^{∼} ⟶ L. Here LP_{ν} (resp. L^{+}ℙ_{ν}) denotes the group of loops (resp. positive loops) of ℙ_{ν} (see Section 2.1), L denotes the LP_{ν}- torsor associated with L_{+} and σ̂^{∗}L the pullback of L under the absolute 𝔽_{ν}-Frobenius endomorphism Frob_{(#𝔽_{ν}),S}: S → S. Building on earlier work of Anderson [And93], Drinfeld [Dri76], Genestier [Gen96], Laumon [Lau96], Rosen [Ros03] and Taguchi [Tag93], local GL_{r}-shtukas were studied by the second author in [Har11] as function field analogs of p-divisible groups and F -crystals. Local ℙ_{ν}-shtukas, which can be viewed as function field analogs of p-divisible groups with extra structure by the group scheme ℙ_{ν}, were introduced by Viehmann and the second author in [HV11, HV12] in the case where ℙ_{ν} is a constant split reductive group. Our definition is a generalization to flat affine group schemes ℙ_{ν} of finite type.

As a preparation to [AH13] we show in this article that for ℙ_{ν} smooth over A_{ν} and for a fixed local ℙ_{ν}-shtuka \underline{𝕃}_{0} over a field k, the unbounded Rapoport–Zink functor

\underline{M}_{\underline{𝕃}_{0}} : (Nilp_{k[[ζ]]})^{o} ⟶ Sets
S ⟼ \{Isomorphism classes of (\underline{L}, δ̄): where \underline{L} is a local ℙ_{ν}-shtuka
over S and δ̄: \underline{L}S̄ → \underline{𝕃}_{0,S̄} is a quasi-isogeny over S̄\},

where S̄ = V(ζ) ⊂ S, is representable by an ind-scheme, ind-quasi-projective over Spf k[[ζ]]; see Theorem 4.4. More precisely, if the L^{+}ℙ_{ν}-torsor underlying \underline{𝕃}_{0} is trivial then \underline{M}_{\underline{𝕃}_{0}} ≅ Fℓ_{ℙ_{ν}} ˆ×_{𝔽_{ν}} Spf 𝔽_{ν}[[ζ]], where Fℓ_{ℙ_{ν}} is the affine flag variety of ℙ_{ν}; see Remark 4.3. To obtain a formal scheme locally formally of finite type, as in the analog for p-divisible groups, one has to assume that ℙ_{ν} has connected reductive generic fiber, and one has to bound the Hodge polygon, that is the relative position of σ̂^{∗}L_{+} and L_{+} under τ̂. We give an axiomatic treatment of bounds in Section 4.2 and prove the representability of the bounded Rapoport–Zink functor by a formal scheme locally formally of finite type over Spf k[[ζ]] in Theorem 4.18.

Our proof, which generalizes [HV11, Theorem 6.3], is inspired by Rapoport’s and Zink’s original result [RZ96, Theorem 2.16] for p-divisible groups.

In addition, in Chapter 3 we discuss the relation between local ℙ-shtukas and Galois representations which is given by the associated Tate module. This chapter is largely independent of the rest of this article and is only used in Remark 5.6. In Chapter 5 we consider the formal stack ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}, which is obtained by taking the formal completion of the stack ∇_{n}ℋ ^{1}(C, 𝔊) at a fixed n-tuple of pairwise different characteristic places \underline{ν} = (ν_{1}, …, ν_{n}). This means we let A\underline{ν} be the completion of the local ring O_{C^{n},\underline{ν}}, and we consider global 𝔊-shtukas only over schemes S whose characteristic morphism S → C^{n} factors through Nilp_{A\underline{ν}}. Recall that with an abelian variety over a scheme in Nilp_{ℤ_{p}} one can associate its p-divisible group. In the analogous situation for global 𝔊-shtukas one can associate a tuple (Γ̂_{ν_{i}}(\underline{G}))_{i} of local ℙ_{ν_{i}}-shtukas Γ̂_{ν_{i}}(\underline{G}) with a global 𝔊-shtuka \underline{G} in ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S). We construct this global-local functor in Section 5.2 by first generalizing the glueing lemma of Beauville and Laszlo [BL95] in Lemma 5.1. In analogy with a theorem of Serre and Tate relating the deformation theory of abelian varieties over schemes in Nilp_{ℤ_{p}} and their associated p-divisible groups, we prove in Theorem 5.10 the equivalence between the infinitesimal deformations of a global 𝔊-shtuka and the infinitesimal deformations of its associated n-tuple of local ℙ_{ν_{i}}-shtukas. Note that unlike abelian varieties, 𝔊-shtukas posses more than one characteristic and we must keep track of the deformations of the local ℙ_{ν_{i}}-shtukas at each of these characteristic places ν_{i}. This theorem for abelian τ-sheaves (corresponding to the case 𝔊 = GL_{r}) and their associated z-divisible groups was first stated and proved by the second author in [Har05].

Acknowledgements. We would like to thank E. Viehmann and L. Kramer for helpful discussions and the anonymous referee for his careful reading and valuable remarks.

1.1 Notation and Conventions

Throughout this article we denote by

𝔽_{q} a finite field with q elements and characteristic p, C a smooth projective geometrically irreducible curve over 𝔽_{q}, Q := 𝔽_{q}(C) the function field of C, ν a closed point of C, also called a place of C, 𝔽_{ν} the residue field at the place ν on C, A_{ν} the completion of the stalk O_{C,ν} at ν, Q_{ν} := Frac(A_{ν}) its fraction field, 𝔽 a finite field containing 𝔽_{q},

𝔻_{R} := Spec R[[z]]  the spectrum of the ring of formal power series in z with coefficients in an 𝔽- algebra R,

𝔻̂_{R} := Spf R[[z]] the formal spectrum of R[[z]] with respect to the z-adic topology.

When R = 𝔽 we drop the subscript R from the notation of 𝔻_{R} and 𝔻̂_{R}.

For a formal scheme Ŝ we denote by NilpŜ the category of schemes over Ŝ on which an ideal of definition of Ŝ is locally nilpotent. We equip NilpŜ with the fppf -topology. We also denote by

n ∈ ℕ_{>0} a positive integer, \underline{ν} := (ν_{i})_{i=1...n} an n-tuple of closed points of C, A\underline{ν} the completion of the local ring O_{C^{n},\underline{ν}} of C^{n} at the closed point \underline{ν} = (ν_{i}), Nilp_{A\underline{ν}} := Nilp_{Spf A\underline{ν}} the category of schemes over C^{n} on which the ideal defining the closed point \underline{ν} ∈ C^{n} is locally nilpotent,

2 LOCAL ℙ-SHTUKAS AND GLOBAL 𝔊-SHTUKAS 4

Nilp_{𝔽[[ζ]]} := Nilp𝔻̂  the category of 𝔻-schemes S for which the image of z in O_{S} is locally nilpotent. We denote the image of z by ζ since we need to distinguish it from z ∈ O_{𝔻}.

𝔊 a flat affine group scheme of finite type over C, ℙ_{ν} := 𝔊 ×_{C} Spec A_{ν} the base change of 𝔊 to Spec A_{ν}, P_{ν} := 𝔊 ×_{C} Spec Q_{ν} the generic fiber of ℙ_{ν} over Spec Q_{ν}, ℙ a flat affine group scheme of finite type over 𝔻 = Spec 𝔽[[z]], P := ℙ ×_{𝔻} Spec 𝔽((z)) the generic fiber of ℙ over Spec 𝔽((z)).

Let S be an 𝔽_{q}-scheme. We denote by σ_{S}: S → S its 𝔽_{q}-Frobenius endomorphism which acts as the identity on the points of S and as the q-power map on the structure sheaf. Likewise we let σ̂_{S}: S → S be the 𝔽-Frobenius endomorphism of an 𝔽-scheme S. We set

C_{S} := C ×_{Spec 𝔽_{q}} S, and σ := id_{C} ×σ_{S}.

Let H be a sheaf of groups (for the fppf -topology) on a scheme X. In this article a (right) H-torsor (also called an H-bundle) on X is a sheaf G for the fppf -topology on X together with a (right) action of the sheaf H such that G is isomorphic to H on an fppf -covering of X. Here H is viewed as an H-torsor by right multiplication.

2 Local ℙ-Shtukas and Global 𝔊-Shtukas

Global 𝔊-shtukas are function field analogs of abelian varieties. They were introduced by Drinfeld [Dri87] in the case where 𝔊 = GL_{r} and used by him and by L. Lafforgue [Laf02] to establish the Langlands correspondence for GL_{r} over global function fields. As we mentioned in the introduction, they were generalized by Laumon, Rapoport and Stuhler [LRS93], Varshavsky [Var04], Ngô and Ngô Dac [NN08], and in [Har05]. We further generalize all these variants in Definition 2.12. Varshavsky’s and our generalization are used by Vincent Lafforgue [Laf12] to prove Langlands parameterization over function fields. The local p-adic properties of abelian varieties are largely captured by their associated p-divisible groups. In the theory of global 𝔊-shtukas the latter correspond to local shtukas; see [Har09, Chapter 3].

2.1 Loop Groups and Local ℙ-Shtukas

Since we want to develop the theory of local ℙ-shtukas partly independently of global 𝔊-shtukas we let 𝔽 be a finite field and 𝔽[[z]] be the power series ring over 𝔽 in the variable z. We let ℙ be a flat affine group scheme of finite type over 𝔻 := Spec 𝔽[[z]], and we let P := ℙ ×_{𝔻} 𝔻̇ be the generic fiber of ℙ over 𝔻̇ := Spec 𝔽((z)). We are mainly interested in the situation where we have an isomorphism 𝔻 ≅ Spec A_{ν} for a place ν of C and where ℙ = ℙ_{ν} := 𝔊 ×_{C} Spec A_{ν}. We recall the following

Definition 2.1. The group of positive loops associated with ℙ is the infinite dimensional affine group scheme L^{+}ℙ over 𝔽 whose R-valued points for an 𝔽-algebra R are

L^{+}ℙ(R) := ℙ(R[[z]]) := ℙ(𝔻_{R}) := Hom_{𝔻}(𝔻_{R}, ℙ) .

The group of loops associated with P is the fpqc-sheaf of groups LP over 𝔽 whose R-valued points for an 𝔽-algebra R are

LP (R) := P (R((z))) := P (𝔻̇_{R}) := Hom𝔻̇(𝔻̇_{R}, P) ,

where we write R((z)) := R[[z]][\frac{1}{z}] and 𝔻̇_{R} := Spec R((z)). It is representable by an ind-scheme of ind-finite type over 𝔽; see [PR08, § 1.a], or [BD, §4.5], [NP01], [Fal03] when ℙ is constant. Let ℋ ^{1}(Spec 𝔽, L^{+}ℙ) := [Spec 𝔽/L^{+}ℙ] (respectively ℋ ^{1}(Spec 𝔽, LP) := [Spec 𝔽/LP]) denote the classifying space of L^{+}ℙ-torsors (respectively LP -torsors). It is a stack fibered in groupoids over the category of 𝔽-schemes S whose category ℋ ^{1}(Spec 𝔽, L^{+}ℙ)(S) consists of all L^{+}ℙ-torsors (resp. LP -torsors) on S. The inclusion of sheaves L^{+}ℙ ⊂ LP gives rise to the natural 1-morphism

L: ℋ ^{1}(Spec 𝔽, L^{+}ℙ) ⟶ ℋ ^{1}(Spec 𝔽, LP), L_{+} ↦ L .  (2.1)

Definition 2.2. Let ℙ̂ be the formal group scheme over 𝔻̂ := Spf 𝔽[[z]], obtained by the formal completion of ℙ along V (z). A formal ℙ̂-torsor over an 𝔽-scheme S is a z-adic formal scheme P̂ over 𝔻̂_{S} := 𝔻̂ ˆ×_{𝔽} S together with an action ℙ̂ ˆ×𝔻̂ P̂ → P̂ of ℙ̂ on P̂ such that there is a covering 𝔻̂_{S′} → 𝔻̂_{S} where S′ → S is an fpqc-covering and a ℙ̂-equivariant isomorphism

P̂ ˆ×𝔻̂_{S} 𝔻̂_{S′} ^{∼} ⟶ ℙ̂ ˆ×𝔻̂ 𝔻̂_{S′}.

Here ℙ̂ acts on itself by right multiplication. Let ℋ ^{1}(𝔻̂, ℙ̂) be the category fibered in groupoids that assigns to each 𝔽-scheme S the groupoid consisting of all formal ℙ̂-torsors over 𝔻̂_{S}.

Remark 2.3. If ℙ is smooth over 𝔻 then for any P̂ in ℋ ^{1}(𝔻̂, ℙ̂)(Spec R) one can find an étale covering R → R′ such that P̂ ˆ×𝔻̂_{R} 𝔻̂_{R′} is isomorphic to ℙ̂_{R′} in ℋ ^{1}(𝔻̂, ℙ̂)(R′). Indeed, since P̂ → 𝔻̂_{R} is smooth, the restriction P̂_{0} of P̂ to V (z) ⊆ 𝔻̂_{R} is likewise smooth over R. Therefore P̂_{0} has a section over an étale covering R → R′. Then by smoothness this section extends over 𝔻̂_{R}.

In [HV11, Proposition 2.2.(a)] Viehmann and the second author proved that for a split reductive group G, there is a bijection of (pointed) sets between the Čech cohomology Ȟ^{1}(S_{fpqc}, L^{+}G) and the set of isomorphism classes of z-adic formal schemes over 𝔻̂_{S}. By the same arguments one can even see that there is a canonical equivalence between the corresponding categories.

Proposition 2.4. There is a natural isomorphism

ℋ ^{1}(𝔻̂, ℙ̂) ^{∼} ⟶ ℋ ^{1}(Spec 𝔽, L^{+}ℙ)

of groupoids. In particular, if ℙ is smooth over 𝔻 then all L^{+}ℙ-torsors for the fpqc-topology on S are already trivial étale locally on S.

Proof. With a given element P̂ of ℋ ^{1}(𝔻̂, ℙ̂)(S) one can associate the following sheaf

\underline{K}: S_{fpqc} ⟶ Sets
T ⟼ Hom𝔻̂_{S}(𝔻̂_{T} , P̂),

where S_{fpqc} denotes the big fpqc-site on S. This sheaf is a torsor under the action of L^{+}ℙ(T) = Hom𝔻̂(𝔻̂_{T} , ℙ̂).

Conversely let K be an L^{+}ℙ-torsor. Let S′ → S be an fpqc-covering that trivializes K and fix a trivialization K_{S′} ^{∼} ⟶ (L^{+}ℙ)_{S′}. This gives a 1-cocycle g ∈ L^{+}ℙ(S′′), where S′′ = S′ ×_{S} S′. Now ḡ = g(mod z^{n}) can be viewed as a descent data on ℙ̂ ˆ×_{𝔻} 𝔻_{n,S′} = ℙ×_{𝔻}𝔻_{n,S′} where 𝔻_{n,S′} := Spec 𝔽[[z]]/(z^{n})×_{𝔽}S′. Since 𝔻_{n,S′} → 𝔻_{n,S} is an fpqc-covering and ℙ is affine, the descent data is effective by [BLR90, § 6.1, Theorem 6] and gives an affine finitely presented scheme Ĝ_{n} over 𝔻_{n,S} by [EGA, IV_{2}, Proposition 2.7.1], which is moreover smooth by [EGA, IV_{4}, Corollaire 17.7.3] if ℙ is smooth over 𝔻. These schemes form an inductive system {Ĝ_{n}}_{n∈ℕ}. Now set Ĝ := lim_{⟶} Ĝ_{n}, the existence of this limit (in the category of z-adic

formal schemes over 𝔻̂_{S}) follows from [EGA, I_{new}, Corollary 10.6.4]. This shows that the functor is essentially surjective. By the above construction we see that the functor is also fully faithful.

The last statement now follows from Remark 2.3.

Definition 2.5. Assume that we have two morphisms f, g: X → Y of schemes or stacks. We denote by equi(f, g: X ⇉ Y) the pull back of the diagonal under the morphism (f, g): X → Y ×_{ℤ} Y, that is, we let

equi(f, g: X ⇉ Y) := X ×_{(f,g),Y ×Y,Δ} Y

where Δ = Δ_{Y/ℤ}: Y → Y ×_{ℤ} Y is the diagonal morphism.

Generalizing [HV11, Definition 3.1] we define the space of local ℙ-shtukas as follows.

Definition 2.6. Let X be the fiber product

ℋ ^{1}(Spec 𝔽, L^{+}ℙ) ×_{ℋ ^{1}(Spec 𝔽,LP)} ℋ ^{1}(Spec 𝔽, L^{+}ℙ)

of groupoids. Let pr_{i} denote the projection onto the i-th factor. We define the groupoid of local ℙ-shtukas Sht_{ℙ}^{𝔻} to be

Sht_{ℙ}^{𝔻} := equi (σ̂ ∘ pr_{1}, pr_{2} : X ⇉ ℋ ^{1}(Spec 𝔽, L^{+}ℙ)) ˆ×_{Spec 𝔽} Spf 𝔽[[ζ]].

(see Definition 2.5) where σ̂ := σ̂_{ℋ ^{1}(Spec 𝔽,L^{+}ℙ)} is the absolute 𝔽-Frobenius of ℋ ^{1}(Spec 𝔽, L^{+}ℙ). The category Sht_{ℙ}^{𝔻} is fibered in groupoids over the category Nilp_{𝔽[[ζ]]} of 𝔽[[ζ]]-schemes on which ζ is locally nilpotent. We call an object of the category Sht_{ℙ}^{𝔻}(S) a local ℙ-shtuka over S.

More explicitly a local ℙ-shtuka over S ∈ Nilp_{𝔽[[ζ]]} is a pair \underline{L} = (L_{+}, τ̂) consisting of an L^{+}ℙ-torsor L_{+} on S and an isomorphism of the associated loop group torsors τ̂: σ̂^{∗}L → L from (2.1).

Definition 2.7. A local ℙ-shtuka (L_{+}, τ̂) is called étale if τ̂ comes from an isomorphism of L^{+}ℙ-torsors σ̂^{∗}L_{+} ^{∼} ⟶ L_{+}. We denote by ÉtSht_{ℙ}^{𝔻}(S) the category of étale local ℙ-shtukas over S.

Lemma 2.8. Let k be a separably closed field extension of 𝔽. If ℙ is smooth over 𝔻 with connected special fiber, then for any b ∈ L^{+}ℙ(k) there exists some c ∈ L^{+}ℙ(k) such that b σ̂^{∗}(c) = c.

Proof. Let ℙ̂ be as in Definition 2.2. Then L^{+}ℙ(k) = ℙ̂(k[[z]]). We view ℙ̂ as the inductive limit lim_{⟶} ℙ_{n},

where ℙ_{n} = ℙ ×_{𝔻} 𝔻_{n} with 𝔻_{n} := Spec 𝔽[[z]]/(z^{n}). Let G̃_{n} denote the linear algebraic group over 𝔽 given by the Weil restriction Res_{𝔻_{n}/ Spec 𝔽}(ℙ_{n}). The reduction of b mod z^{n} gives an element b_{n} ∈ G̃_{n}(k). Since ℙ is smooth with connected special fiber, G̃_{n} is connected by [CGP10, Proposition A.5.9]. Thus by Lang’s theorem [Lan56, Corollary on p. 557] there exists a c_{n} ∈ G̃_{n}(k) such that b_{n}σ̂^{∗}(c_{n}) = c_{n}. Here σ̂ is the 𝔽-Frobenius on G̃_{n} which coincides with the Frobenius σ̂ induced from ℙ. Now consider the reduction map α_{n}: G̃_{n+1}(k) → G̃_{n}(k) and the element d̄_{n} := α_{n}(c_{n+1})^{−1}c_{n} which satisfies σ̂^{∗}(d̄_{n}) = d̄_{n} and hence lies in G̃_{n}(𝔽) = ℙ(𝔻_{n+1}). Since ℙ is smooth d̄_{n} lifts to an element d_{n} ∈ ℙ(𝔻_{n+1}) = G̃_{n+1}(𝔽). Replacing c_{n+1} by c_{n+1}d_{n} we may assume that α_{n}(c_{n+1}) = c_{n} and then take c := lim_{⟵} c_{n}.

Corollary 2.9. If ℙ is smooth over 𝔻 with connected special fiber, then every étale local ℙ-shtuka over a separably closed field k is isomorphic to ((L^{+}ℙ)_{k}, 1·σ̂^{∗}).

Proof. Let \underline{L} = (L_{+}, τ̂) be an étale local ℙ-shtuka over k. By Proposition 2.4 there is a trivialization of the L^{+}ℙ-torsor L_{+} giving rise to an isomorphism \underline{L} ≅ ((L^{+}ℙ)_{k}, b · σ̂^{∗}) for some b ∈ L^{+}ℙ(k). By Lemma 2.8 there is an element c ∈ L^{+}ℙ(k) with b σ̂^{∗}(c) = c and multiplication with c is an isomorphism ((L^{+}ℙ)_{k}, 1·σ̂^{∗}) ^{∼} ⟶ ((L^{+}ℙ)_{k}, b·σ̂^{∗}).

Local ℙ-shtukas can be viewed as function field analogs of p-divisible groups. This inspires the following notions of quasi-isogenies; compare [HV11, Definition 3.8].

Definition 2.10. A quasi-isogeny f: \underline{L} → \underline{L}′ between two local ℙ-shtukas \underline{L} := (L_{+}, τ̂) and \underline{L}′ := (L′_{+}, τ̂′) over S is an isomorphism of the associated LP -torsors f: L → L′ satisfying f ∘τ̂ = τ̂′ ∘σ̂^{∗}f. We denote by QIsog_{S}(\underline{L}, \underline{L}′) the set of quasi-isogenies between \underline{L} and \underline{L}′ over S, and we write QIsog_{S}(\underline{L}) := QIsog_{S}(\underline{L}, \underline{L}) for the quasi-isogeny group of \underline{L}.

As in the theory of p-divisible groups, also our quasi-isogenies are rigid. Here we prove the case of local ℙ-shtukas which is analogous to p-divisible groups. Like for abelian varieties, the case of global 𝔊shtukas only holds in fixed finite characteristics. We will define quasi-isogenies between global 𝔊-shtukas in Section 2.2 and prove rigidity for them in Proposition 5.9.

Proposition 2.11 (Rigidity of quasi-isogenies for local ℙ-shtukas). Let S be a scheme in Nilp_{𝔽[[ζ]]} and let j: S̄ → S be a closed immersion defined by a sheaf of ideals I which is locally nilpotent. Let \underline{L} and \underline{L}′ be two local ℙ-shtukas over S. Then

QIsog_{S}(\underline{L}, \underline{L}′) ⟶ QIsogS̄(j^{∗}\underline{L}, j^{∗}\underline{L}′), f ↦ j^{∗}f

is a bijection of sets.

Proof. This was proved in [HV11, Proposition 3.9] when ℙ = G ×_{𝔽} 𝔻 for a constant split reductive group G over 𝔽. The proof carries over literally. Compare also Proposition 5.9.

2.2 Global 𝔊-Shtukas

Let 𝔽_{q} be a finite field with q elements, let C be a smooth projective geometrically irreducible curve over 𝔽_{q}, and let 𝔊 be a flat affine group scheme of finite type over C. The relation to Section 2.1 is as follows. We are mainly interested in the case where 𝔽[[z]] ≅ A_{ν} and ℙ = ℙ_{ν}.

Definition 2.12. A global 𝔊-shtuka \underline{G} = (G, s_{1}, …, s_{n}, τ) over an 𝔽_{q}-scheme S is a tuple where

• G is a 𝔊-torsor over C_{S},

• s_{1}, …, s_{n} ∈ C(S) are 𝔽_{q}-morphisms called the characteristic sections of \underline{G}, and

• τ: σ^{∗}G_{|_{C_{S} ∖Γs_{1} ∪...∪Γs_{n}}} ^{∼} ⟶ G_{|_{C_{S} ∖Γs_{1} ∪...∪Γs_{n}}} is an isomorphism of 𝔊-torsors. Here Γ_{s_{i}} ⊂ C_{S} denotes the graph of the morphism s_{i}.

We write σ = id_{C} ×σ_{S} for the 𝔽_{q}-Frobenius endomorphism σ_{S}: S → S which acts as the identity on the points of S and as the q-power map on the structure sheaf. We denote the moduli stack of global 𝔊-shtukas by ∇_{n}ℋ ^{1}(C, 𝔊). It is a stack fibered in groupoids over the category of 𝔽_{q}-schemes. Sometimes we will fix the sections (s_{1}, …, s_{n}) ∈ C^{n}(S) and simply call \underline{G} = (G, τ) a global 𝔊-shtuka over S.

In [AH13, Theorem 3.15] we prove that ∇_{n}ℋ ^{1}(C, 𝔊) is an ind-algebraic stack over C^{n} (in the sense of [AH13, Definition 3.14]) which is ind-separated and locally of ind-finite type. However, we will not use this result in the present article, as we will mainly focus on local ℙ-shtukas, and the relation between individual global 𝔊-shtukas and local ℙ-shtukas. For a thorough discussion how our global 𝔊-shtukas and their moduli spaces generalize similar concepts in the literature, we refer to the introduction and to [AH13, Remark 3.19].

There is also a notion of quasi-isogenies for global 𝔊-shtukas.

Definition 2.13. Consider a scheme S together with characteristic morphisms s_{i}: S → C for i = 1, …, n and let \underline{G} = (G, τ) and \underline{G}′ = (G′, τ′) be two global 𝔊-shtukas over S with the same characteristics s_{i}. A quasi-isogeny from \underline{G} to \underline{G}′ is an isomorphism f: G|_{C_{S}∖D_{S}} ^{∼} ⟶ G′|_{C_{S}∖D_{S}} satisfying τ′σ^{∗}(f) = fτ, where D is some effective divisor on C. We denote the group of quasi-isogenies of \underline{G} to itself by QIsog_{S}(\underline{G}).

Like for abelian varieties, rigidity of global 𝔊-shtukas only holds in fixed finite characteristics; see Proposition 5.9.

3 TATE MODULES FOR LOCAL ℙ-SHTUKAS 8

3 Tate Modules for Local ℙ-Shtukas

In this chapter we assume that ℙ is a flat affine group scheme of finite type over 𝔻. For a scheme S ∈ Nilp_{𝔽[[ζ]]} let O_{S}[[z]] be the sheaf of O_{S}-algebras on S for the fpqc-topology whose ring of sections on an S-scheme Y is the ring of power series O_{S}[[z]](Y) := Γ(Y, O_{Y})[[z]]. Let O_{S}((z)) be the fpqc-sheaf of O_{S}-algebras on S associated with the presheaf Y ↦ Γ(Y, O_{Y})[[z]][\frac{1}{z}]. A sheaf M of O_{S}[[z]]-modules on S which is finite free fpqc-locally on S is already finite free Zariski-locally on S by [HV11, Proposition 2.3]. We call those modules finite locally free sheaves of O_{S}[[z]]-modules. We denote by σ̂^{∗} the endomorphism of O_{S}[[z]] and O_{S}((z)) that acts as the identity on the variable z, and is the 𝔽-Frobenius b ↦ (b)^{#𝔽} on local sections b ∈ O_{S}. For a sheaf M of O_{S}[[z]]-modules on S we set σ̂^{∗}M := M ⊗_{O_{S}[[z]],σ̂^{∗}} O_{S}[[z]]. We recall the definition of local shtukas and their quasi-isogenies from [HV11, Definition 4.1] and [Har11, Definition 2.1.1].

Definition 3.1. (a) A local shtuka over S is a pair (M, τ̂) consisting of a locally free sheaf M of O_{S}[[z]]- modules of finite rank on S and an isomorphism τ̂: σ̂^{∗}M ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M ⊗_{O_{S}[[z]]} O_{S}((z)).

(b) A local shtuka \underline{M} := (M, τ̂) is called étale if τ̂ comes from an isomorphism of O_{S}[[z]]-modules σ̂^{∗}M ^{∼} ⟶ M.

(c) A morphism f: (M, τ̂) → (M′, τ̂′) of local shtukas over S is a morphism f: M → M′ of O_{S}[[z]]- modules which satisfies τ̂′ ∘σ̂^{∗}f = f ∘τ̂. We do not require that f is an isomorphism. We denote by Sht_{𝔻}(S) the category of local shtukas over S and by ÉtSht_{𝔻}(S) the category of étale local shtukas over S.

Remark 3.2. There is an equivalence of categories between the category ℋ ^{1}(Spec 𝔽, L^{+} GL_{r})(S) and the category of locally free sheaves of O_{S}[[z]]-modules of rank r; see [HV11, §4]. It induces an equivalence between the category of local GL_{r}-shtukas over S and the category consisting of local shtukas over S of rank r with isomorphisms as the only morphisms; see [HV11, Lemma 4.2].

Definition 3.3. A quasi-isogeny between two local shtukas (M, τ̂) → (M′, τ̂′) is an isomorphism of O_{S}((z))-modules

f: M ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M′ ⊗_{O_{S}[[z]]} O_{S}((z))

with τ̂′ ∘ σ̂^{∗}f = f ∘ τ̂.

In analogy with p-divisible groups and abelian varieties, one can also assign a Galois representation to a given étale local shtuka as follows. Assume that S is connected. Let s̄ be a geometric point of S and let π_{1}^{ét}(S, s̄) denote the algebraic fundamental group of S at s̄. We define the (dual) Tate functor from the category of étale local shtukas ÉtSht_{𝔻}(S) over S to the category 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]} of finite free 𝔽[[z]]-modules equipped with a continuous action of π_{1}^{ét}(S, s̄) as follows

Ť_{−}: ÉtSht_{𝔻}(S) ⟶ 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]},
\underline{M} := (M, τ̂) ⟼ Ť_{\underline{M}} := (M ⊗_{O_{S}[[z]]} κ(s̄)[[z]])τ̂.

Here the superscript τ̂ denotes τ̂-invariants. Sometimes also the notation H_{ét}^{1}(\underline{M}, 𝔽[[z]]) := Ť_{\underline{M}} is used. Inverting z we also consider the rational (dual) Tate functor

V̌_{−}: ÉtSht_{𝔻}(S) ⟶ 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]},
\underline{M} := (M, τ̂) ⟼ V̌_{\underline{M}} := (M ⊗_{O_{S}[[z]]} κ(s̄)[[z]])τ̂ ⊗_{𝔽[[z]]} 𝔽((z)).

where 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]} denotes the category of finite 𝔽((z))-vector spaces equipped with a continuous action of π_{1}^{ét}(S, s̄). The functor V̌_{−} transforms quasi-isogenies into isomorphisms.

Proposition 3.4. Let S ∈ Nilp_{𝔽[[ζ]]} be connected. Then the functor Ť_{−} is an equivalence between the categories ÉtSht_{𝔻}(S) and 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]}. The functor V̌_{−} is an equivalence between the category of étale local shtukas over S with quasi-isogenies as morphisms and the category 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]} with isomorphisms as the only morphisms. There is a canonical isomorphism Ť_{\underline{M}} ⊗_{𝔽[[z]]} κ(s̄)[[z]] ^{∼} ⟶ \underline{M} ⊗_{O_{S}[[z]]} κ(s̄)[[z]] of κ(s̄)[[z]]-modules which is equivariant for the action of π_{1}^{ét}(S, s̄) and τ̂, where π_{1}^{ét}(S, s̄) acts trivially on \underline{M} and τ̂ acts trivially on Ť_{\underline{M}}.

Proof. The statement for Ť_{−} follows by the same arguments as [Har11, Proposition 1.3.7]. It is analogous to [Kat73, Proposition 4.1.1] and can be thought of as a positive characteristic analog of the Riemann-Hilbert correspondence. We describe the quasi-inverse functor. Consider an 𝔽[[z]][π_{1}^{ét}(S, s̄)]-module of rank r and the corresponding representation π: π_{1}^{ét}(S, s̄) → GL_{r}(𝔽[[z]]). For each m ∈ ℕ let S_{m} → S be the finite Galois covering corresponding to the kernel of π_{1}^{ét}(S, s̄) ⟶^{π} GL_{r}(𝔽[[z]]) ⟶^{mod z^{m}} GL_{r}(𝔽[[z]]/(z^{m})). Let M̃_{m} be the free module of rank r over O_{S_{m}}[[z]]/(z^{m}) = O_{S_{m}} ⊗_{𝔽} 𝔽[[z]]/(z^{m}) and equip it with the Frobenius τ̂ := σ̂ ⊗ id and the action of γ ∈ Gal(S_{m}/S) by γ(b ⊗ f) = γ^{∗}(b) ⊗ π(γ^{−1})(f) for b ∈ O_{S_{m}} and f ∈ 𝔽[[z]]/(z^{m})^{⊕r}. Then M̃_{m} descends to a locally free O_{S}[[z]]/(z^{m})-module M_{m} of rank r and τ̂ descends to an isomorphism τ̂: σ̂^{∗}M_{m} ^{∼} ⟶ M_{m}. This makes M := lim_{⟵} M_{m} into an étale local shtuka over S and

yields the quasi-inverse to Ť_{−}.

That V̌_{−} is essentially surjective follows from the fact that π_{1}^{ét}(S, s̄) is compact which implies that every 𝔽((z))[π_{1}^{ét}(S, s̄)]-module arises by inverting z from an 𝔽[[z]][π_{1}^{ét}(S, s̄)]-module. To see that V̌_{−} is fully faithful consider two local shtukas \underline{M} and \underline{M}′ over S and an isomorphism f: V̌_{\underline{M}} ^{∼} ⟶ V̌_{\underline{M}}′. There are powers z^{N} and z^{N′} of z such that z^{N}f and z^{N′}f^{−1} come from morphisms Ť_{\underline{M}} → Ť_{\underline{M}}′ respectively Ť_{\underline{M}}′ → Ť_{\underline{M}}. Under the equivalence Ť_{−} these in turn come from morphisms g: \underline{M} → \underline{M}′ and g′: \underline{M}′ → \underline{M}. Then gg′ = z^{N+N′} and this implies that g and g′ are quasi-isogenies. Clearly V̌_{z^{−N}g} = f and V̌_{z^{−N′}g′} = f^{−1}. This proves that V̌_{−} is an equivalence of categories.

Let Vect_{𝔻} be the groupoid over Nilp_{𝔽[[ζ]]} whose S-valued points is the category of locally free sheaves of O_{S}[[z]]-modules with isomorphisms as the only morphisms. Let Rep_{𝔽[[z]]} ℙ be the category of representations ρ: ℙ → GL(V) of ℙ in finite free 𝔽[[z]]-modules V, that is, ρ is a morphism of algebraic groups over 𝔽[[z]]. Any such representation ρ gives a functor

ρ_{∗}: ℋ ^{1}(Spec 𝔽, L^{+}ℙ) → Vect_{𝔻}

which sends an L^{+}ℙ-torsor L_{+} ∈ ℋ ^{1}(Spec 𝔽, L^{+}ℙ)(S) to the sheaf of O_{S}[[z]]-modules associated with the following presheaf

Y ⟼ (L_{+}(Y) × (V ⊗_{𝔽[[z]]} O_{S}[[z]](Y)))/L^{+}ℙ(Y) .  (3.1)

The functor ρ_{∗}: ℋ ^{1}(Spec 𝔽, L^{+}ℙ) → Vect_{𝔻} induces a functor from the category of local ℙ-shtukas to the category of local shtukas which we likewise denote ρ_{∗}. This functor is also compatible with quasi-isogenies.

Definition 3.5. Let F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]}), resp. F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]}), denote the category whose objects are tensor functors from Rep_{𝔽[[z]]} ℙ to 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]}, respectively to 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]}, and whose morphisms are isomorphisms of functors. We define the (dual) Tate functor Ť_{−}, respectively the rational (dual) Tate functor V̌_{−} as the functors

Ť_{−}: ÉtSht_{ℙ}^{𝔻}(S) ⟶ F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]})
\underline{L} ⟼ Ť\underline{L}: ρ ↦ Ť_{ρ_{∗}\underline{L}},
V̌_{−}: ÉtSht_{ℙ}^{𝔻}(S) ⟶ F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]})
\underline{L} ⟼ V̌\underline{L}: ρ ↦ V̌_{ρ_{∗}\underline{L}}.

That Ť_{−} and V̌_{−} are indeed tensor functors, follows from the fact that \underline{L} ↦ ρ_{∗}\underline{L} is a tensor functor and from the equivariant isomorphism Ť_{ρ_{∗}\underline{L}} ⊗_{𝔽[[z]]} κ(s̄)[[z]] ^{∼} ⟶ ρ_{∗}\underline{L} ⊗_{O_{S}[[z]]} κ(s̄)[[z]] from Proposition 3.4. If ℙ is smooth with connected special fiber and \underline{L} is an étale local ℙ-shtuka then the composition of the tensor functor Ť\underline{L} followed by the forgetful functor F: 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]} → 𝔉𝔐od_{𝔽[[z]]} is isomorphic to the forgetful fiber functor ω^{∘}: Rep_{𝔽[[z]]} ℙ → 𝔉𝔐od_{𝔽[[z]]} by Corollary 2.9. Indeed, the base change \underline{L}s̄ of \underline{L} to s̄ = Spec κ(s̄) is isomorphic to \underline{𝕃}_{0} := ((L^{+}ℙ)s̄, 1·σ̂^{∗}) and the functor F ∘ Ť_{\underline{𝕃}_{0}} is isomorphic to ω^{∘}. This yields a conjugacy class of isomorphisms Aut^{⊗}(F ∘ Ť\underline{L}) ≅ Aut^{⊗}(ω^{∘}) = ℙ. Since every γ ∈ π_{1}^{ét}(S, s̄) acts as a tensor automorphism of Ť\underline{L}, the tensor functor Ť\underline{L} corresponds to a conjugacy class of Galois representations π: π_{1}^{ét}(S, s̄) → ℙ(𝔽[[z]]). Now Proposition 3.4 generalizes as follows.

Proposition 3.6. Let ℙ be smooth over 𝔻 with connected special fiber and let S ∈ Nilp_{𝔽[[ζ]]} be a connected scheme. Then the functor Ť_{−} is an equivalence between the category ÉtSht_{ℙ}^{𝔻}(S) and the category F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]}). The functor V̌_{−} from the category of étale local ℙ-shtukas over S with quasi-isogenies as morphisms to the category F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]}) is fully faithful.

Proof. To construct the functor which is quasi-inverse to Ť_{−} we fix a tensor functor F in F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔉𝔐od_{𝔽[[z]][π_{1}^{ét}(S,s̄)]}). The difference of the two 𝔽[[z]]-rational fiber functors ω^{∘} and F ∘F on Rep_{𝔽[[z]]} ℙ is given by the torsor Isom^{⊗}(ω^{∘}, F ∘ F) over ℙ = Aut^{⊗}(ω^{∘}); use [Wed04, Corollary 5.20]. Since the special fiber of ℙ is connected, this torsor has an 𝔽-valued point by Lang’s theorem [Lan56, Theorem 2]. Since ℙ and hence Isom^{⊗}(ω^{∘}, F ∘F) is smooth over 𝔻 this point lifts to an 𝔽[[z]]-valued point of Isom^{⊗}(ω^{∘}, F ∘ F), that is to a tensor isomorphism α: ω^{∘} ^{∼} ⟶ F ∘ F over 𝔽[[z]] inducing an isomorphism α_{∗}: ℙ ^{∼} ⟶ Aut^{⊗}(F ∘ F). Since π_{1}^{ét}(S, s̄) acts as automorphisms of the fiber functor F ∘ F, the functor F corresponds to a representation π: π_{1}^{ét}(S, s̄) → ℙ(𝔽[[z]]) which depends on α up to conjugation in ℙ. For each m ∈ ℕ we let S_{m} → S be the finite étale Galois covering corresponding to the kernel of π_{1}^{ét}(S, s̄) ⟶^{π} ℙ(𝔽[[z]]) ⟶^{mod z^{m}} ℙ_{m}(𝔽[[z]]/(z^{m})) where ℙ_{m} := ℙ ×_{𝔻} Spec 𝔽[[z]]/(z^{m}). Let G̃_{m} be the trivial ℙ_{m}-torsor over S_{m} ×_{𝔽} Spec 𝔽[[z]]/(z^{m}) and equip it with the Frobenius τ̂ := id: σ̂^{∗}G̃_{m} ^{∼} ⟶ G̃_{m}. Via the action of Gal(S_{m}/S) through π on G̃_{m} the latter descends to a ℙ_{m}-torsor G_{m} over S ×_{𝔽} Spec 𝔽[[z]]/(z^{m}) and τ̂ descends to an isomorphism τ̂: σ̂^{∗}G_{m} → G_{m}. This makes Ĝ := lim_{⟵} G_{m} into a formal ℙ̂-torsor over

S together with an isomorphism τ̂: σ̂^{∗}Ĝ ^{∼} ⟶ Ĝ; see Definition 2.2. By Proposition 2.4 it corresponds to an L^{+}ℙ-torsor L_{+} together with an isomorphism τ̂: σ̂^{∗}L_{+} ^{∼} ⟶ L_{+}, that is, to the étale local ℙ-shtuka \underline{L} = (L_{+}, τ̂). It satisfies Ť\underline{L} ≅ F. A different isomorphism α gives a different local ℙ-shtuka which is canonically isomorphic to \underline{L}. This yields the quasi-inverse to Ť_{−}.

To prove that V̌_{−} is fully faithful let \underline{L} = (L_{+}, τ̂) and \underline{L}′ = (L′_{+}, τ̂′) be two étale local ℙ-shtukas over S and let δ: V̌\underline{L} ^{∼} ⟶ V̌_{\underline{L}}′ be an isomorphism of tensor functors. We consider the following functor

M̂_{−}: ℋ ^{1}(Spec 𝔽, L^{+}ℙ)(S) → F unct^{⊗}(Rep_{𝔽[[z]]} ℙ, 𝔐od_{O_{S}[[z]]}),

which sends an L^{+}ℙ-torsor L_{+} to the tensor functor mapping the representation ρ to the O_{S}[[z]]-module ρ_{∗}L_{+} from (3.1). By Proposition 3.4 the isomorphism δ|_{ρ} between V̌\underline{L}(ρ) = V̌_{ρ_{∗}\underline{L}} and V̌_{\underline{L}}′(ρ) = V̌_{ρ_{∗}\underline{L}′} comes from a quasi-isogeny between ρ_{∗}\underline{L} = (M̂_{L_{+}}(ρ), ρ_{∗}τ̂) and ρ_{∗}\underline{L}′ = (M̂_{L′_{+}}(ρ), ρ_{∗}τ̂′). Therefore the

isomorphism δ induces an isomorphism M̂_{L_{+}} ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M̂_{L′_{+}} ⊗_{O_{S}[[z]]} O_{S}((z)). Take an fppf -cover S′ → S trivializing L_{+} and L′_{+} and fix trivializations L_{+} ≅ (L^{+}ℙ)_{S′} ≅ L′_{+}. Then we have

Isom^{⊗}(M̂_{(L_{+})_{S′}} ⊗_{O_{S′}[[z]]} O_{S′}((z)) , M̂_{(L_{+})_{S′}} ⊗_{O_{S′}[[z]]} O_{S′}((z))) ≅ Aut^{⊗}(ω^{∘})(O_{S′}((z))) = LP (S′),

because M̂_{(L^{+}ℙ)_{S′}} = ω^{∘} ⊗_{𝔽[[z]]} O_{S′}((z)) and ℙ = Aut^{⊗}(ω^{∘}) by [Wed04, Corollary 5.20]. Therefore δ gives an isomorphism h_{S′}: (LP)_{S′} → (LP)_{S′}. The morphism h_{S′} inherits the descent datum coming from the fact that δ is defined over S, and hence it defines an isomorphism h: L ^{∼} ⟶ L′, where L and L′ denote the LP -torsors associated with L_{+} and L′_{+}. One easily checks that h satisfies τ̂′ ∘ σ̂^{∗}h = h ∘ τ̂ and gives a quasi-isogeny h: \underline{L} → \underline{L}′.

4 THE RAPOPORT–ZINK SPACES FOR LOCAL ℙ-SHTUKAS 11

Remark 3.7. In general the functor V̌_{−} does not need to be an equivalence, not even onto the category of those tensor functors F: Rep_{𝔽[[z]]} ℙ → 𝔉𝔐od_{𝔽((z))[π_{1}^{ét}(S,s̄)]} for which F ∘F ≅ ω^{∘} ⊗_{𝔽[[z]]} 𝔽((z)). For example let ℙ be the Iwahori subgroup of GL_{2}, that is,

ℙ(𝔽[[z]]) = \{ A ∈ GL_{2}(𝔽[[z]]): A ≡ \binom{∗ ∗}{0 ∗} mod z \} .

Let x be transcendental over 𝔽 and let S = Spec 𝔽(x). Set G := π_{1}^{ét}(S, s̄) = Gal(𝔽(x)^{sep}/𝔽(x)) and consider a representation π: G → GL_{2}(𝔽[[z]]) such that the residual representation \overline{π}: G → GL_{2}(𝔽) is irreducible. This implies that π(G) ⊄ ℙ(𝔽[[z]]) and hence the tensor functor F: Rep_{𝔽[[z]]} ℙ → 𝔉𝔐od_{𝔽((z))[G]} given by

ρ ⟼ [G ⟶^{π} GL_{2}(𝔽[[z]]) ↪ P (𝔽((z))) ⟶^{ρ} GL(ω^{∘}(ρ))(𝔽((z)))]

cannot come from a local ℙ-shtuka over 𝔽(x).

Note that such a representation π exists. For example if φ: 𝔽_{q}[t] → End 𝔾_{a,𝔽(x)} is a Drinfeld-𝔽_{q}[t]- module of rank 2 over 𝔽(x) without potential complex multiplication, then for almost all primes ν of 𝔽_{q}[t] the Galois representation π_{φ,ν}: G → GL_{2}(A_{ν}) on the ν-adic Tate module of φ has this property by [PT06, Theorem A] or [PR09, Theorem 0.1]. For a concrete example let φ_{t} = 1 − xτ + τ^{2} and ν = (t). Then φ[t] = {y ∈ 𝔽(x)^{sep} : y^{q^{2}} − xy^{q} + y = 0}. If q = 2 it is easy to see that y^{3} − xy + 1 is irreducible in 𝔽(x)[y] and has splitting field of degree 6 over 𝔽(x). This implies that \overline{π}_{φ,(t)}(G) = GL_{2}(𝔽_{2}). The reason for the failure of V̌_{−} to be an equivalence of course lies in the fact that the Drinfeld-module φ does not carry a level structure over 𝔽(x) whereas any étale local ℙ-shtuka for the above Iwahori group ℙ carries a Γ_{0}(ν)-level structure.

Even if we assume that ℙ is a maximal parahoric subgroup of P as in Remark 4.3, we expect that it depends on the group P whether V̌_{−} is an equivalence. Namely, in the proof of Proposition 3.6, when we try to extend the construction of the quasi-inverse of Ť_{−} to V̌_{−} we obtain a representation π_{1}^{ét}(S, s̄) → P (𝔽((z))). For V̌_{−} to be an equivalence we need that up to conjugation this representation factors through ℙ(𝔽[[z]]). We know that π_{1}^{ét}(S, s̄) is a profinite group and hence compact. Therefore the image of the representation is contained in a maximal compact subgroup. So the question arises whether every maximal compact subgroup of P (𝔽((z))) is conjugate to ℙ(𝔽[[z]]). This is true when ℙ = GL_{r} or SL_{r} and in this case V̌_{−} is an equivalence.

But in general the answer may be negative for two reasons. First of all, although every maximal parahoric subgroup is maximally compact, the converse may fail. For example for P = PGL_{2} the subgroup generated by the Iwahori subgroup \{ A ∈ PGL_{2}(𝔽[[z]]): A ≡ \binom{∗ ∗}{0 ∗} mod z \} and by \binom{0 1}{z 0} is maximally compact but not parahoric, because it is the stabilizer of the midpoint of an edge in the Bruhat–Tits tree of PGL_{2}. This group contains the Iwahori subgroup with index 2. Secondly, not all maximal parahoric subgroups need to be conjugate, because they are the stabilizers of 0-simplices in the Bruhat– Tits building, but not all 0-simplices are conjugate in general. This occurs for example when P = Sp_{2r}.

4 The Rapoport–Zink Spaces for Local ℙ-Shtukas

In this chapter we assume that ℙ is a smooth affine group scheme over 𝔻.

Rapoport and Zink constructed a moduli space for p-divisible groups together with a quasi-isogeny to a fixed one (and with some extra structure such as a polarization, endomorphisms, or a level structure). They proved that this moduli space is ind-representable by a formal scheme locally formally of finite type over ℤ_{p}.

We already mentioned that local ℙ-shtukas behave analogously to p-divisible groups. However, this analogy is not perfect, unless we restrict to “bounded” local ℙ-shtukas as the analogous objects corresponding to p-divisible groups. More precisely we bound the Hodge polygon of a local ℙ-shtuka (L_{+}, τ̂), that is, the relative position of σ̂^{∗}L_{+} and L_{+} under the isomorphism τ̂; see Definition 4.8(b) below. This is motivated by the fact that F -isocrystals and Dieudonné-modules also have bounded Hodge slopes. We will show that Rapoport–Zink spaces for bounded local ℙ-shtukas are formal schemes locally formally of finite type over Spf 𝔽[[ζ]]. When ℙ := G_{0} ×_{𝔽} 𝔻 for a connected split reductive group G_{0} over 𝔽 this was proved in [HV11, Theorem 6.3]. In the non-constant case for a smooth affine group ℙ over 𝔻 we will give an axiomatic definition of the boundedness condition in Section 4.2. We start with the unbounded situation.

4.1 Unbounded Rapoport–Zink Spaces

For a scheme S in Nilp_{𝔽[[ζ]]} let S̄ denote the closed subscheme V_{S}(ζ) ⊆ S. On the other hand for a scheme T̄ over 𝔽 we set T̂ := T̄ ˆ×_{Spec 𝔽} Spf 𝔽[[ζ]]. Then T̂ is a ζ-adic formal scheme with T̄ = VT̂(ζ). So the underlying topological spaces of T̄ and T̂ coincide. We let NilpT̂ be the category of T̂-schemes on which ζ is locally nilpotent.

Definition 4.1. With a given local ℙ-shtuka \underline{L}_{0} over an 𝔽-scheme T̄ we associate the functor

\underline{M}_{\underline{L}_{0}} : (NilpT̂)^{o} ⟶ Sets
S ⟼ \{Isomorphism classes of (\underline{L}, δ̄): where \underline{L} is a local ℙ-shtuka
over S and δ̄: \underline{L}S̄ → \underline{L}_{0,S̄} is a quasi-isogeny over S̄\}.

Here we say that (\underline{L}, δ̄) and (\underline{L}′, δ̄′) are isomorphic if δ̄^{−1} ∘ δ̄′ lifts via Proposition 2.11 to an isomorphism \underline{L}′ ^{∼} ⟶ \underline{L}. The group QIsogT̄(\underline{L}_{0}) of quasi-isogenies of \underline{L}_{0} acts on the functor \underline{M}_{\underline{L}_{0}} via g: (\underline{L}, δ̄) ↦ (\underline{L}, g∘δ̄) for g ∈ QIsogT̄(\underline{L}_{0}). We will show that \underline{M}_{\underline{L}_{0}} is representable by an ind-scheme which we call an unbounded Rapoport–Zink space for local ℙ-shtukas.

Remark 4.2. Note that by rigidity of quasi-isogenies (Proposition 2.11) the functor \underline{M}_{\underline{L}_{0}} is naturally isomorphic to the functor

S ↦ \{Isomorphism classes of (\underline{L}, δ): where \underline{L} is a local ℙ-shtuka
over S and δ: \underline{L}_{S} → \underline{L}_{0,S} is a quasi-isogeny over S\}.

This also shows that id\underline{L} is the only automorphism of (\underline{L}, δ), and for this reason we do not need to consider \underline{M}_{\underline{L}_{0}} as a stack.

Remark 4.3. In order to show that \underline{M}_{\underline{L}_{0}} is representable by an ind-scheme we recall the definition of the affine flag variety Fℓ_{ℙ}. It is defined to be the fpqc-sheaf associated with the presheaf

R ⟼ LP (R)/L^{+}ℙ(R) = ℙ (R((z))) /ℙ (R[[z]])

on the category of 𝔽-algebras; compare Definition 2.1. Pappas and Rapoport [PR08, Theorem 1.4] show that Fℓ_{ℙ} is represented by an ind-scheme which is ind-quasi-projective over 𝔽, and hence ind-separated and of ind-finite type over 𝔽. Moreover, they show that the quotient morphism LP → Fℓ_{ℙ} admits sections locally for the étale topology. They proceed as follows. When ℙ = SL_{r,𝔻}, the fpqc-sheaf Fℓ_{ℙ} is called the affine Grassmanian. It is an inductive limit of projective schemes over 𝔽, that is, ind-projective over 𝔽; see [BD, Theorem 4.5.1] or [Fal03, NP01]. By [PR08, Proposition 1.3] and [AH13, Proposition 2.1] there is a faithful representation ℙ ↪ SL_{r} with quasi-affine quotient. Pappas and Rapoport show in the proof of [PR08, Theorem 1.4] that Fℓ_{ℙ} → Fℓ_{SL_{r}} is a locally closed embedding, and moreover, if SL_{r} /ℙ is affine, then Fℓ_{ℙ} → Fℓ_{SL_{r}} is even a closed embedding and Fℓ_{ℙ} is ind-projective. More generally, if the fibers of ℙ over 𝔻 are geometrically connected, it was proved by Richarz [Ric13b, Theorem A] that Fℓ_{ℙ} is ind-projective if and only if ℙ is a parahoric group scheme in the sense of Bruhat and Tits [BT72, Définition 5.2.6]; see also [HR08]. Note that, in particular, a parahoric group scheme is smooth with connected fibers and reductive generic fiber.

Let us view the formal scheme T̂ = T̄ ˆ×_{𝔽} Spf 𝔽[[ζ]] as the ind-scheme lim_{⟶} T̄ ×_{𝔽} Spec 𝔽[ζ]/(ζ^{m}). We may

form the fiber product F̂ℓ_{ℙ,T̂} := Fℓ_{ℙ} ˆ×_{𝔽} T̂ in the category of ind-schemes (see [BD, 7.11.1]). Note that this fiber product can be either viewed as the restriction of the sheaf Fℓ_{ℙ} to the fpqc-site of schemes in NilpT̂ or also as the formal completion of Fℓ_{ℙ} ×_{𝔽} T̄ ×_{𝔽} Spec 𝔽[[ζ]] along the special fiber V(ζ).

Theorem 4.4. If ℙ is a smooth affine group scheme over 𝔻 the functor \underline{M}_{\underline{L}_{0}} from Definition 4.1 is represented by an ind-scheme, ind-quasi-projective over T̂ = T̄ ˆ×_{𝔽} Spf 𝔽[[ζ]], hence ind-separated and of ind-finite type over T̂. If the fibers of ℙ over 𝔻 are connected then \underline{M}_{\underline{L}_{0}} is ind-projective if and only if ℙ is parahoric in the sense of Bruhat and Tits [BT72, Définition 5.2.6] and [HR08].

If \underline{L}_{0} is trivialized by an isomorphism α: \underline{L}_{0} ^{∼} ⟶ ((L^{+}ℙ)T̄, bσ̂^{∗}) over T̄ with b ∈ LP (T̄) then \underline{M}_{\underline{L}_{0}} is represented by the ind-scheme F̂ℓ_{ℙ,T̂} := Fℓ_{ℙ} ˆ×_{𝔽} T̂.

Proof. We first assume that \underline{L}_{0} is trivialized by an isomorphism α. We regard \underline{M}_{\underline{L}_{0}} in the equivalent form mentioned in Remark 4.2. Consider a pair (\underline{L}, δ) = ((L_{+}, τ̂), δ) ∈ \underline{M}_{\underline{L}_{0}}(S). Choose an fppf -covering S′ → S which trivializes \underline{L}, then the quasi-isogeny α_{S′} ∘ δ is given by an element g′ ∈ LP (S′). The image of the element g′ ∈ LP (S′) in F̂ℓ_{ℙ,T̂}(S′) is independent of the choice of the trivialization, and since (\underline{L}, δ)

is defined over S, it descends to a point x ∈ F̂ℓ_{ℙ,T̂}(S). Note in particular that τ̂_{S′} is determined by b and g′ through the diagram

σ̂^{∗}(LP)_{S′} ⟶^{τ̂_{S′}} (LP)_{S′}
σ̂^{∗}(g′)↓  ↓^{g′}
σ̂^{∗}(LP)_{S′} ⟶^{b} (LP)_{S′}.

Conversely let x ∈ F̂ℓ_{ℙ,T̂}(S) for a scheme S in NilpT̂. The projection morphism LP → Fℓ_{ℙ} admits local sections for the étale topology by [PR08, Theorem 1.4]. Consequently there is an étale covering S′ → S such that x is represented by an element g′ ∈ LP (S′). We set S′′ := S′×_{S} S′ and define (L′_{+}, τ̂′, δ′) over S′ as follows. Let L′_{+} := (L^{+}ℙ)_{S′}, let the quasi-isogeny δ′: (L′_{+}, τ̂′) → ((L^{+}ℙ)_{S′}, bσ̂^{∗}) be given by y ↦ g′y, and the Frobenius by τ̂′ := (g′)^{−1}bσ̂^{∗}(g′)σ̂^{∗}. We descend (L′_{+}, τ̂′, δ′) to S. For an S-scheme Y let Y′ = Y ×_{S} S′ and Y′′ = Y′ ×_{Y} Y′ = Y ×_{S} S′′, and let p_{i}: Y′′ → Y′ be the projection onto the i-th factor. Since g′ comes from an element x ∈ F̂ℓ_{ℙ,T̂}(S) there is an h ∈ L^{+}ℙ(S′′) with p_{1}^{∗}(g′) = p_{2}^{∗}(g′) · h. Consider the fpqc-sheaf L_{+} on S whose sections over an S-scheme Y are given by

L_{+}(Y) := { y′ ∈ L^{+}ℙ(Y′): p_{1}^{∗}(y′) = h^{−1} · p_{2}^{∗}(y′) in L^{+}ℙ(Y′′) }

on which L^{+}ℙ(Y) acts by right multiplication. Then \underline{L} is an L^{+}ℙ-torsor on S because over Y = S′ there is a trivialization

(L^{+}ℙ)_{S′} ^{∼} ⟶ (L_{+})_{S′} , f ↦ h p_{1}^{∗}(f) ∈ (L^{+}ℙ)(S′′)

due to the cocycle condition for h. Moreover, τ̂′ descends to an isomorphism

τ̂: σ̂^{∗}L(Y) ^{∼} ⟶ L(Y) , σ̂^{∗}(y′) ↦ (g′)^{−1}bσ̂^{∗}(g′)σ̂^{∗}(y′)

making (L_{+}, τ̂) into a local ℙ-shtuka over S. Also δ′ descends to a quasi-isogeny of local ℙ-shtukas

δ: L(Y) ^{∼} ⟶ LP (Y) = { f′ ∈ LP (Y′): p_{1}^{∗}(f′) = p_{2}^{∗}(f′) in LP (Y′′) } ,
y′ ↦ g′y′ .

Note that this is well defined. Namely, if g′ is replaced by g̃′ with u′ = (g̃′)^{−1}g′ ∈ L^{+}ℙ(S′) then left multiplication with u′ defines an isomorphism

((L^{+}ℙ)_{S′}, (g′)^{−1}bσ̂^{∗}(g′)σ̂^{∗}, g′) ^{∼} ⟶ ((L^{+}ℙ)_{S′}, (g̃′)^{−1}bσ̂^{∗}(g̃′)σ̂^{∗}, g̃′).

Also h̃ = p_{2}^{∗}(u′) h p_{1}^{∗}(u′)^{−1} and hence left multiplication with u′ descends to an isomorphism \underline{L} ^{∼} ⟶ \underline{L̃} over S. This establishes the last statement of the theorem.

To prove the first assertion we may choose an fppf -covering T̄′ → T̄ and a trivialization α: \underline{L}_{0,T̄′} ^{∼} ⟶ ((L^{+}ℙ)T̄′, bσ̂^{∗}) over T̄′. We set T̂′ := T̄′ ˆ×_{𝔽} Spf 𝔽[[ζ]] and T̂′′ := T̂′ ˆ×T̂ T̂′, and let pr_{i}: T̂′′ → T̂′ be the projection onto the i-th factor. By what we have proved above, we obtain an isomorphism

\underline{M}_{\underline{L}_{0}} ˆ×T̂ T̂′ ^{∼} ⟶ F̂ℓ_{ℙ,T̂′}, δ ↦ α_{S′} ∘ δ = g′ · L^{+}ℙ(S′) .

Note that F̂ℓ_{ℙ,T̂′} is an ind-scheme which is ind-quasi-projective over T̂′ by Remark 4.3 and, when ℙ has connected fibers, even ind-projective if and only if ℙ is parahoric. Over T̂′′ there is an isomorphism

\xymatrix{pr_{1}^{∗}(F̂ℓ_{ℙ,T̂′}) \ar[r]^{≅} & \underline{M}_{\underline{L}_{0}} ˆ×T̂ T̂′′ \ar[r]^{≅} & pr_{2}^{∗}(F̂ℓ_{ℙ,T̂′}) \\ g′ · L^{+}ℙ(S′) \ar@{|->}[r] & pr_{1}^{∗}α_{S′}^{−1} ∘ g′ \ar@{|->}[rr] &  & pr_{2}^{∗}α_{S′} ∘ pr_{1}^{∗}α_{S′}^{−1} ∘ g′ · L^{+}ℙ(S′)}  (4.1)

It is given by left multiplication on pr_{1}^{∗}(F̂ℓ_{ℙ,T̂′}) = F̂ℓ_{ℙ,T̂′′} = pr_{2}^{∗}(F̂ℓ_{ℙ,T̂′}) with the element pr_{2}^{∗}α_{S′}∘pr_{1}^{∗}α_{S′}^{−1} ∈ L^{+}ℙ(T̂′′).

We write Fℓ_{ℙ} = lim_{⟶} Fℓ_{ℙ}^{(N)} for quasi-projective 𝔽-schemes Fℓ_{ℙ}^{(N)}. There is a line bundle F on Fℓ_{ℙ}

which is “ample” in the sense that its restriction to any Fℓ_{ℙ}^{(N)} is ample, and which is equivariant for the L^{+}ℙ-action by left multiplication, i.e. “L^{+}ℙ-linearized” in the sense of [MFK94, Definition 1.6]. For example one can take a faithful representation ℙ ↪ SL_{r,𝔻} with quasi-affine quotient (Remark 4.3), take the fundamental line bundles L_{i} on Fℓ_{SL_{r}} from [Fal03, p. 46], take F as the pullback of a tensor product of strictly positive powers of the L_{i}, see [Fal03, p. 54], and take the Fℓ_{ℙ}^{(N)} as the preimages in Fℓ_{ℙ} of the Schubert varieties in Fℓ_{SL_{r}}. Then (4.1) defines a descent datum on the pair (Fℓ_{ℙ}^{(N)} ×_{𝔽}T̄′×_{𝔽}𝔽[[ζ]]/(ζ^{m}), F) for all N and m. This descent datum is effective by [SGA 1, VIII, Proposition 7.8], see also [BLR90, § 6.1, Theorem 7]. Therefore there is a quasi-projective scheme \underline{M}_{\underline{L}_{0}}^{(N,m)} over T̄ ×_{𝔽} 𝔽[[ζ]]/(ζ^{m}) with \underline{M}_{\underline{L}_{0}}^{(N,m)} ×T̄

T̄′ ≅ Fℓ_{ℙ}^{(N)} ×_{𝔽}T̄′ ×_{𝔽} 𝔽[[ζ]]/(ζ^{m}). Moreover, if the fibers of ℙ are connected then \underline{M}_{\underline{L}_{0}}^{(N,m)} is projective if

and only if ℙ is parahoric by Remark 4.3. It follows that \underline{M}_{\underline{L}_{0}} = lim_{⟶_{N,m}} \underline{M}_{\underline{L}_{0}}^{(N,m)} is an ind-(quasi-)projective

ind-scheme over T̂.

4.2 Bounded Local ℙ-Shtukas

We want to introduce boundedness conditions for local ℙ-shtukas where ℙ is again a smooth affine group scheme over 𝔻. Due to the problem discussed in Example 4.14 below we will base our boundedness conditions on an axiomatic definition of “bounds”. For this purpose we fix an algebraic closure 𝔽((ζ))^{alg} of 𝔽((ζ)). Since its ring of integers is not complete we prefer to work with finite extensions of discrete valuation rings R/𝔽[[ζ]] such that R ⊂ 𝔽((ζ))^{alg}. For such a ring R we denote by κ_{R} its residue field, and we let Nilp_{R} be the category of R-schemes on which ζ is locally nilpotent. We also set F̂ℓ_{ℙ,R} := Fℓ_{ℙ} ˆ×_{𝔽} Spf R and F̂ℓ_{ℙ} := F̂ℓ_{ℙ,𝔽[[ζ]]}. Before we can define “bounds” we need to make the following observations.

Definition 4.5. (a) For a finite extension 𝔽[[ζ]] ⊂ R ⊂ 𝔽((ζ))^{alg} of discrete valuation rings we consider closed ind-subschemes Ẑ_{R} ⊂ F̂ℓ_{ℙ,R}. We call two closed ind-subschemes Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} and Ẑ′_{R′} ⊂ F̂ℓ_{ℙ,R′} equivalent if there is a finite extension of discrete valuation rings 𝔽[[ζ]] ⊂ R̃ ⊂ 𝔽((ζ))^{alg} containing R and R′ such that Ẑ_{R} ˆ×_{Spf R} Spf R̃ = Ẑ′_{R′} ˆ×_{Spf R′} Spf R̃ as closed ind-subschemes of F̂ℓ_{ℙ,R̃}.

(b) Let Ẑ = [Ẑ_{R}] be an equivalence class of closed ind-subschemes Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} and consider the group GẐ := {γ ∈ Aut_{𝔽[[ζ]]}(𝔽((ζ))^{alg}): γ(Ẑ) = Ẑ }. We define the ring of definition RẐ of Ẑ as the intersection of the fixed field of GẐ in 𝔽((ζ))^{alg} with all the finite extensions R ⊂ 𝔽((ζ))^{alg} of 𝔽[[ζ]] over which a representative Ẑ_{R} of Ẑ exists.

Let us give some explanations for this definition.

Remark 4.6 (about Definition 4.5(a)). Consider an ind-scheme structure on Fℓ_{ℙ} given as an inductive limit Fℓ_{ℙ} = lim_{⟶} Fℓ_{ℙ}^{(m)} of quasi-compact 𝔽-schemes Fℓ_{ℙ}^{(m)} indexed by m ∈ ℕ. Then F̂ℓ_{ℙ,R} =

lim_{⟶}(Fℓ_{ℙ}^{(m)} ×_{𝔽} Spec R/(ζ^{m})) is an ind-scheme structure on F̂ℓ_{ℙ,R}. A closed ind-subscheme Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} is

of the form Ẑ_{R} = lim_{⟶} Ẑ_{R,m} for an inductive system of closed subschemes Ẑ_{R,m} ⊂ Fℓ_{ℙ}^{(m)} ×_{𝔽} Spec R/(ζ^{m}). The latter correspond to sheaves of ideals I_{m} ⊂ O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R/(ζ^{m}) for all m.

If R ⊂ R̃ and Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} is a closed ind-subscheme then ẐR̃ := Ẑ_{R} ˆ×_{Spf R} Spf R̃ ⊂ F̂ℓ_{ℙ,R̃} is a closed

ind-subscheme, such that Ẑ_{R} is the ind-scheme theoretic image of ẐR̃ in F̂ℓ_{ℙ,R}. Indeed, in terms of the ideal sheaves I_{m} ⊂ O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R/(ζ^{m}) of Ẑ_{R} and Ĩ_{m} ⊂ O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R̃/(ζ^{m}) of ẐR̃ this means Ĩ_{m} = I_{m} ⊗_{R} R̃

and I_{m} = Ĩ_{m} ∩ O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R/(ζ^{m}). The latter equality follows from the commutative diagram

\xymatrix{0 \ar[r] & I_{m} \ar[r] \ar@{^{(}->}[d] & I_{m} ⊗_{R} R̃ \ar[r] \ar@{^{(}->}[d] & I_{m} ⊗_{R} R̃/R \ar[r] \ar@{^{(}->}[d] & 0 \\ 0 \ar[r] & O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R/(ζ^{m}) \ar[r] & O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R̃/(ζ^{m}) \ar[r] & O_{Fℓ{(m) \atop ℙ}} ⊗_{𝔽} R/(ζ^{m}) ⊗_{R} R̃/R \ar[r] & 0 ,}

in which the rows are exact and the vertical morphisms are injective because R̃/R and R̃ are finite free R-modules.

It follows that two closed ind-subschemes Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} and Ẑ′_{R′} ⊂ F̂ℓ_{ℙ,R′} are equivalent if and only if Ẑ_{R} ˆ×_{Spf R} Spf R̃ = Ẑ′_{R′} ˆ×_{Spf R′} Spf R̃ for every finite extension of discrete valuation rings 𝔽[[ζ]] ⊂ R̃ ⊂ 𝔽((ζ))^{alg} containing R and R′.

Another consequence is, that a morphism f: S → F̂ℓ_{ℙ,R} for S ∈ Nilp_{R} factors through Ẑ_{R} if and only if the morphism f × idR̃: S ˆ×_{R} Spf R̃ → F̂ℓ_{ℙ,R̃} factors through ẐR̃. Indeed, this can be checked by the vanishing of the ideals f^{∗}I_{m}, respectively (f × idR̃)^{∗}Ĩ_{m}, using the injectivity O_{S} ↪ O_{S} ⊗_{R} R̃.

Remark 4.7 (about Definition 4.5(b)). Let R ⊂ 𝔽((ζ))^{alg} be a finite extension of 𝔽[[ζ]] over which a representative Ẑ_{R} of Ẑ exists.

(a) γ(Ẑ) = Ẑ then means that γ(Ẑ_{R}) ⊂ F̂ℓ_{ℙ,γ(R)} is equivalent to Ẑ_{R}. In particular, if γ(R) = R then, by our previous remark, γ(Ẑ) = Ẑ means that γ(Ẑ_{R}) = Ẑ_{R}. For example if Frac(R) is a normal field extension of 𝔽((ζ)) then γ(R) = R.

(b) It follows that Aut_{R}(𝔽((ζ))^{alg}) ⊂ GẐ because all R-automorphisms fix Ẑ_{R}.

(c) We let R^{GẐ} := {x ∈ R: γ(x) = x for all γ ∈ GẐ }. It equals the intersection of R with the fixed field of GẐ in 𝔽((ζ))^{alg}.

(d) Let i(R) := [Frac(R) : 𝔽((ζ))]_{insep} be the inseparability degree. Then 𝔽[[ \sqrt[i(R)]{ζ}]] ⊂ R^{GẐ} and R^{GẐ} equals the ring of integers O_{K} in the fixed field K of GẐ inside the separable closure 𝔽(( \sqrt[i(R)]{ζ}))^{sep} of 𝔽(( \sqrt[i(R)]{ζ})). To prove this we use the fact from field theory, that 𝔽(( \sqrt[i(R)]{ζ})) is contained in Frac(R) and that this is a separable extension. In particular Frac(R) ⊂ 𝔽(( \sqrt[i(R)]{ζ}))^{sep} and R^{GẐ} ⊂ O_{K}. From (b) it follows that K ⊂ Frac(R), and hence O_{K} ⊂ R^{GẐ}. Finally, since Aut_{𝔽[[ζ]]}(𝔽((ζ))^{alg}) fixes all elements of 𝔽(( \sqrt[i(R)]{ζ})) we find 𝔽[[ \sqrt[i(R)]{ζ}]] ⊂ R^{GẐ}.

(e) If R′ is another finite extension of 𝔽[[ζ]] over which a representative Ẑ_{R′} of Ẑ exists, such that i(R) ≤ i(R′), then R^{GẐ} ⊂ (R′)^{GẐ} by (d). If i(R) = i(R′) then R^{GẐ} = (R′)^{GẐ}.

(f) We conclude from (e) that the ring of definition of Ẑ may be computed as follows. We choose a finite extension R ⊂ 𝔽((ζ))^{alg} of 𝔽[[ζ]] over which a representative Ẑ_{R} of Ẑ exists, and for which i(R) is minimal. Then RẐ = R^{GẐ}. Moreover, let R̃ be the ring of integers in the normal closure of Frac(R) over 𝔽((ζ)). Then i(R̃) = i(R) and therefore Frac(R̃) is Galois over Frac(RẐ) with Galois group Aut_{RẐ}(R̃) = {γ ∈ Aut_{𝔽[[ζ]]}(R̃) with γ(ẐR̃) = ẐR̃} ⊂ Aut_{𝔽[[ζ]]}(R̃). We conclude that

RẐ = {x ∈ R̃: γ(x) = x for all γ ∈ Aut_{𝔽[[ζ]]}(R̃) with γ(ẐR̃) = ẐR̃} .

(g) We do not know whether in general Ẑ has a representative Ẑ_{RẐ} over the ring of definition RẐ, although this is true in many cases; see our Examples 4.12 to 4.14.

(h) Note that our Definition 4.5(b) is a direct translation of the analogous situation over number fields, taking the inseparability problem into account. Namely in the number field case one considers cocharacters μ: 𝔾_{m,ℚ_{p}^{alg}} → G_{ℚ_{p}^{alg}} for a reductive group G over ℚ_{p}, and one considers a conjugacy

class C(μ) = {Int_{g} ∘μ: g ∈ G(ℚ_{p}^{alg})}; see [Del71, 3.7] or [RZ96, 1.31]. (Our Example 4.13 corresponds to this.) One defines the field of definition E_{μ} of C(μ) as the fixed field inside ℚ_{p}^{alg} of {γ ∈ Gal(ℚ_{p}^{alg}/ℚ_{p}): γ(C(μ)) := C(γ(μ)) = C(μ)}. The field of definition E_{μ} is automatically contained in every field over which a representative of C(μ) exists. Our above discussion applies mutatis mutandis. If the group G is quasi-split over E_{μ}, Kottwitz [Kot84, Lemma 1.1.3] proved that C(μ) has a representative over E_{μ}.

After these preparatory observations we finally come to the announced

Definition 4.8. (a) We define a bound to be an equivalence class Ẑ := [Ẑ_{R}] of closed ind-subschemes Ẑ_{R} ⊂ F̂ℓ_{ℙ,R}, such that all the ind-subschemes Ẑ_{R} are stable under the left L^{+}ℙ-action on Fℓ_{ℙ}, and the special fibers Z_{R} := Ẑ_{R} ˆ×_{Spf R} Spec κ_{R} are quasi-compact subschemes of Fℓ_{ℙ} ˆ×_{𝔽} Spec κ_{R}. The ring of definition RẐ of Ẑ is called the reflex ring of Ẑ. Since the Galois descent for closed ind-subschemes of Fℓ_{ℙ} is effective, the Z_{R} arise by base change from a unique closed subscheme Z ⊂ Fℓ_{ℙ} ˆ×_{𝔽} κ_{RẐ}. We call Z the special fiber of the bound Ẑ. It is a quasi-projective scheme over κ_{RẐ} by Remark 4.3 and [HV11, Lemma 5.4] which implies that every morphism from a quasi-compact scheme to an ind-quasi-projective ind-scheme factors through a quasi-projective subscheme. If ℙ is parahoric in the sense of Bruhat and Tits [BT72, Définition 5.2.6] and [HR08] then Z is projective.

(b) Let Ẑ be a bound with reflex ring RẐ. Let L_{+} and L′_{+} be L^{+}ℙ-torsors over a scheme S in Nilp_{RẐ} and let δ: L ^{∼} ⟶ L′ be an isomorphism of the associated LP -torsors. We consider an fppf -covering S′ → S over which trivializations α: L_{+} ^{∼} ⟶ (L^{+}ℙ)_{S′} and α′: L′_{+} ^{∼} ⟶ (L^{+}ℙ)_{S′} exist. Then the automorphism α′ ∘ δ ∘ α^{−1} of (LP)_{S′} corresponds to a morphism S′ → LP ˆ×_{𝔽} Spf RẐ. We say that δ is bounded by Ẑ if for every such trivialization and for every finite extension R of 𝔽[[ζ]] over which a representative Ẑ_{R} of Ẑ exists the induced morphism

S′ ˆ×_{RẐ} Spf R → LP ˆ×_{𝔽} Spf R → F̂ℓ_{ℙ,R}

factors through Ẑ_{R}. Furthermore we say that a local ℙ-shtuka (L_{+}, τ̂) is bounded by Ẑ if the isomorphism τ̂ is bounded by Ẑ.

Remark 4.9. The condition of Definition 4.8(b) is satisfied for all trivializations and for all such finite extensions R of 𝔽[[ζ]] if and only if it is satisfied for one trivialization and for one such finite extension. Indeed, by the L^{+}ℙ-invariance of Ẑ the definition is independent of the trivializations. That one finite extension suffices follows from Remark 4.6.

In Examples 4.12 to 4.14 below we discuss the motivation for this definition and the relation to other boundedness conditions like in [HV11]. Note that the definition of “bounds” given above suffices for our purposes in this article and in [AH13]. For other purposes one may need more restrictive hypotheses on bounds; see for example [HV15, Section 2].

Remark 4.10. Let the ind-scheme structure on Fℓ_{ℙ} be given as the limit Fℓ_{ℙ} = lim_{⟶} Fℓ_{ℙ}^{(m)} and the one

on F̂ℓ_{ℙ,R} as F̂ℓ_{ℙ,R} = lim_{⟶} Fℓ_{ℙ}^{(m)} ×_{𝔽} Spec R/(ζ^{m}). Let Ẑ_{R}^{(m)} := Ẑ_{R} ˆ×_{F̂ℓ_{ℙ,R}}(Fℓ_{ℙ}^{(m)} ×_{𝔽} Spec R/(ζ^{m})). Then

Ẑ_{R} = lim_{⟶} Ẑ_{R}^{(m)} and Ẑ_{R,red} := lim_{⟶}(Ẑ_{R}^{(m)})_{red} = lim_{⟶}(Ẑ_{R} ˆ×_{Spf R} Spec κ_{R} ×_{Fℓ_{ℙ}} Fℓ_{ℙ}^{(m)})_{red} = (Z_{R})_{red} is a scheme.

This means that Ẑ_{R} is a “reasonable formal scheme” over Spf R in the sense of [BD, 7.11.1 and 7.12.17], and hence a formal scheme in the sense of [EGA, I_{new}].

Proposition 4.11. Let Ẑ be a bound with reflex ring RẐ. Let L_{+} and L′_{+} be L^{+}ℙ-torsors over a scheme S ∈ Nilp_{RẐ} and let δ: L ^{∼} ⟶ L′ be an isomorphism of the associated LP -torsors. Then the condition that δ is bounded by Ẑ is represented by a closed subscheme of S.

Proof. We consider a representative Ẑ_{R} of the bound Ẑ over a finite extension RẐ ⊂ R ⊂ 𝔽((ζ))^{alg}. As in Definition 4.8 we consider trivializations of L_{+} and L′_{+} over an fppf -covering S′ → S and the induced morphism S′ ˆ×_{RẐ} Spf R → LP ˆ×_{𝔽} Spf R → F̂ℓ_{ℙ,R}. Due to the L^{+}ℙ-invariance of Ẑ_{R} the closed subscheme S′ ˆ×_{F̂ℓ_{ℙ,R}} Ẑ_{R} of S′ descends to a closed subscheme of S. By Remark 4.9 this closed subscheme represents

the boundedness by Ẑ.

Example 4.12. Assume that ℙ is parahoric in the sense of Bruhat and Tits [BT72, Définition 5.2.6] and [HR08]; see Remark 4.3. Consider the base change P_{L} of P to L = 𝔽^{alg}((z)). Let A be a maximal split torus in P_{L} and let T be its centralizer. Since 𝔽^{alg} is algebraically closed, P_{L} is quasi-split by [Ser97, § II.2.3, Théorème 1′ and Remarque 1, p. 140] and so T is a maximal torus in P_{L}. Let N = N(T) be the normalizer of T and let T^{0} be the identity component of the Néron model of T over O_{L} = 𝔽^{alg}[[z]].

The Iwahori–Weyl group associated with A is the quotient group W̃ = N(L)/T^{0}(O_{L}). It is an extension of the finite Weyl group W_{0} = N(L)/T (L) by the coinvariants X_{∗}(T)_{I} under I = Gal(L^{sep}/L):

0 → X_{∗}(T)_{I} → W̃ → W_{0} → 1.

By [HR08, Proposition 8] there is a bijection

L^{+}ℙ(𝔽^{alg})\LP (𝔽^{alg})/L^{+}ℙ(𝔽^{alg}) ^{∼} ⟶ W̃^{ℙ}\W̃/W̃^{ℙ}  (4.2)

where W̃^{ℙ} := (N(L) ∩ ℙ(O_{L}))/T^{0}(O_{L}).

Let ω ∈ W̃^{ℙ}\W̃/W̃^{ℙ} and let 𝔽_{ω} be the fixed field in 𝔽^{alg} of {γ ∈ Gal(𝔽^{alg}/𝔽): γ(ω) = ω}. We show that ω has a representative g_{ω} ∈ LP (𝔽_{ω}). Indeed, let g ∈ LP (𝔽^{alg}) be any representative of ω and let γ be the 𝔽_{ω}-Frobenius which generates Gal(𝔽^{alg}/𝔽_{ω}). Since γ(ω) = ω there are elements b_{1}, b_{2} ∈ L^{+}ℙ(𝔽^{alg}) with γ(g) = b_{1}^{−1}g b_{2}. By Lemma 2.8 we find elements c_{1}, c_{2} ∈ L^{+}ℙ(𝔽^{alg}) with b_{i} γ(c_{i}) = c_{i} for i = 1, 2. Then g_{ω} := c_{1}^{−1}g c_{2} = γ(c_{1}^{−1}g c_{2}) ∈ LP (𝔽_{ω}) is the desired representative of ω over 𝔽_{ω}. Clearly, by definition of 𝔽_{ω} there are no representatives of ω over proper subfields of 𝔽_{ω}.

We define the Schubert variety S(ω) associated with ω as the ind-scheme theoretic closure of the L^{+}ℙ-orbit of g_{ω} in Fℓ_{ℙ} ˆ×_{𝔽} 𝔽_{ω}. It is a reduced projective variety over 𝔽_{ω}. For further details see [PR08] and [Ric13a]. The equivalence class of Ẑ_{𝔽_{ω}[[ζ]]} := S(ω) ˆ×_{𝔽_{ω}} Spf 𝔽_{ω}[[ζ]] defines a bound with reflex ring 𝔽_{ω}[[ζ]]. Instead of “bounded by [Ẑ_{𝔽_{ω}[[ζ]]}]” we also say “bounded by ω” in this case.

Example 4.13. In [HV11], Viehmann and the second author considered the case where ℙ = G ×_{𝔽} 𝔻 for a split connected reductive group G over 𝔽. In this case W̃^{ℙ} = W_{0} and W̃^{ℙ}\W̃/W̃^{ℙ} = X_{∗}(T), and any element μ ∈ X_{∗}(T) has a representative over 𝔽_{μ} = 𝔽. If μ ∈ X_{∗}(T) one could consider the bound Ẑ := [S(μ) ˆ×_{𝔽} Spf 𝔽[[ζ]]] as in Example 4.12 above.

However, in [HV11] we proceeded differently and instead fixed a Borel subgroup B of G and its opposite Borel \overline{B}. We considered a finite generating system Λ of the monoid of dominant weights X^{∗}(T)_{dom}, and for all λ ∈ Λ the Weyl module V_{λ} := (IndG ; \overline{B}(−λ)_{dom})^{∨}. For the representation ρ_{λ}: G → GL(V_{λ}) we considered the sheaves of O_{S}[[z]]-modules ρ_{λ∗}L_{+} and ρ_{λ∗}L′_{+} associated in (3.1) with the L^{+}ℙ-torsors L_{+} and L′_{+} over S. (For the definition of O_{S}[[z]] see Chapter 3.) The isomorphism δ of the associated LP -torsors corresponds to an isomorphism ρ_{λ∗}δ: ρ_{λ∗}L_{+} ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ ρ_{λ∗}L′_{+} ⊗_{O_{S}[[z]]} O_{S}((z)). We said in [HV11, Definition 3.5] that “δ is bounded by (μ, z̃)”, where z̃ = z or z̃ = z − ζ, if for all λ ∈ Λ

ρ_{λ∗}δ (ρ_{λ∗}L_{+}) ⊂ z̃^{−⟨ (−λ)_{dom},μ⟩} (ρ_{λ∗}L′_{+}),  (4.3)

and if for all geometric points s̄ of S the image of the isomorphism δs̄ at s̄ under the isomorphism (4.2) has the same image in π_{1}(G) than μ. Note that the more important case z̃ = z − ζ is useful to define bounds on local ℙ-shtukas, while the case z̃ = z is only useful to define bounds on quasi-isogenies between local ℙ-shtukas. In that sense the bound [S(μ) ˆ×_{𝔽} Spf 𝔽[[ζ]]] from Example 4.12, which corresponds to z̃ = z, is not the right one to define bounds on local ℙ-shtukas. Further note that in case z̃ = z − ζ the term z̃^{−⟨ (−λ)_{dom},μ⟩} in (4.3) can be viewed as the image under (−λ)_{dom} : LG(S) → L𝔾_{m}(S) of the element μ(z −ζ)^{−1} ∈ LG(S), which itself is the image of μ(z −ζ)^{−1} ∈ G(𝔽[[ζ, z]][\frac{1}{z−ζ}]) in LG(S) = G(O_{S}[[z]][\frac{1}{z−ζ}]) using that ζ is locally nilpotent on S.

In terms of Definition 4.8 the boundedness condition (4.3) can be described as follows. Consider the universal matrix M ∈ L GL(V_{λ})(S_{λ}) over the ind-scheme S_{λ} := L GL(V_{λ}) ˆ×_{𝔽} Spf 𝔽[[ζ]]. Let \overline{S}_{λ} be the closed ind-subscheme of S_{λ} defined by the condition that the matrix z̃^{⟨ (−λ)_{dom},μ⟩}M has entries in O_{S_{λ}}[[z]], and let

Ẑ_{λ} := \overline{S}_{λ}/(L^{+} GL(V_{λ}) ˆ×_{𝔽} Spf 𝔽[[ζ]]) ⊂ F̂ℓ_{GL(V_{λ})} .

Let μ^{#} ∈ π_{1}(G) be the image of μ in the fundamental group π_{1}(G) and let (F̂ℓ_{ℙ})_{μ^{#}} be the connected component of F̂ℓ_{ℙ} corresponding to μ^{#} under the isomorphism π_{0}(F̂ℓ_{ℙ}) ≅ π_{1}(G); see [BD, Proposition 4.5.4] or [PR08, Theorem 0.1]. Write Λ = {λ_{1}, …, λ_{m}} and for each λ_{i} consider the morphism ρ_{λ_{i}∗}: (F̂ℓ_{ℙ})_{μ^{#}} → F̂ℓ_{GL(V_{λ_{i}})} induced from ρ_{λ_{i}}. Then the base change Ẑ of the closed ind-subscheme

Ẑ_{λ_{1}} ˆ×_{Spf 𝔽[[ζ]]} … ˆ×_{Spf 𝔽[[ζ]]} Ẑ_{λ_{m}} under the morphism ∏_{i} ρ_{λ_{i}∗}: (F̂ℓ_{ℙ})_{μ^{#}} → F̂ℓ_{GL(V_{λ_{1}})} ˆ×_{𝔽[[ζ]]} … ˆ×_{𝔽[[ζ]]} F̂ℓ_{GL(V_{λm})} is the bound representing the “boundedness by (μ, z̃)” from [HV11, Definition 3.5]. It has reflex ring 𝔽[[ζ]] and Ẑ is a representative of this bound over the reflex ring.

Instead of the Weyl modules V_{λ} one could of course also work with other representations of G. If for example one takes the induced modules IndG ; \overline{B}(−λ)_{dom}, or tilting modules, one obtains different indsubschemes Ẑ, but the underlying reduced ind-subschemes of these Ẑ still coincide. If z̃ = z, this reduced ind-subscheme equals the Schubert variety S(μ) ˆ×_{𝔽} Spec 𝔽[[ζ]]. This already indicates that it is reasonable to consider boundedness in terms of closed ind-subschemes of F̂ℓ_{ℙ}. Note that here also for z̃ = z − ζ the bound Ẑ only depends on μ and the class of modules considered (Weyl modules, etc.). This is no longer true for general ℙ as one sees from the next example.

Example 4.14. We discuss a special case of Example 4.12. Assume that char 𝔽_{q} ≠ 2 and set K := 𝔽_{q}((z)). Let E := K[y]/(y^{2}−z) be the ramified quadratic field extension with y^{2} = z. Let T be the one dimensional torus ker(N_{E/K}: Res_{E/K} 𝔾_{m} → 𝔾_{m}). Explicitly T = Spec K[a, b]/(a^{2} − b^{2}z − 1), with the multiplication (a, b) ∗ (c, d) = (ac + bdz, ad + bc). Sending a ↦ \frac{1}{2}(t + t^{−1}) and b ↦ \frac{1}{2y} (t^{−1} − t) defines an isomorphism 𝔾_{m,E} = Spec E[t, t^{−1}] ≅ T_{E} which we will use in the sequel to identify X_{∗}(T) := X_{∗}(T_{E}) with ℤ. Here the finite Weyl group W_{0} = (1) is trivial and the inertia group I = Gal(E/K) = {1, γ} acts on X_{∗}(T) = ℤ via γ(μ) = −μ. Therefore W̃ = X_{∗}(T)_{I} = ℤ/2ℤ.

Consider the Néron-model T = ker(N_{O_{E}/O_{K}}: Res_{O_{E}/O_{K}} 𝔾_{m} → 𝔾_{m}). As a scheme it is isomorphic to Spec 𝔽_{q}[[z]][a, b]/(a^{2} − b^{2}z − 1). Its special fiber has two connected components distinguished by a ≡ 1 or −1 mod z. Therefore the connected component of identity of T is

T^{0} := Spec 𝔽_{q}[[z]][a′, b]/(2a′ + z(a′)^{2} − b^{2}),

where a = 1 + za′. In particular (−1, 0) ∉ T^{0}(𝔽_{q}((z))). We take ℙ as the group scheme T^{0} over 𝔻 = Spec 𝔽_{q}[[z]]. Here 𝔽 = 𝔽_{q} and σ̂ = σ is the q-Frobenius. By [HR08, Lemma 5 and its proof] the group T^{0} is the unique parahoric group scheme with generic fiber T.

In this example the isomorphism (4.2) is given by the Kottwitz map κ_{T} : LT (𝔽_{q}^{alg}) → X_{∗}(T)_{I}, see [Kot85, 2.5]. Its inverse associates with each element of X_{∗}(T)_{I} a σ̂-conjugacy class in LT (𝔽_{q}^{alg}). For example if we take μ̄ = 1 ∈ X_{∗}(T)_{I} = ℤ/2ℤ one has to choose a lift μ ∈ X_{∗}(T_{E}). If we choose μ = 1 then with μ̄ = 1 it associates

N_{E/K}(μ(y)) = μ(y) · γ(μ(y))
= (\frac{1}{2}(y + y^{−1}), \frac{1}{2y} (y^{−1} − y)) · (\frac{1}{2}(−y − y^{−1}), \frac{1}{−2y} (−y^{−1} − (−y)))
= (−1, 0) ∈ T (𝔽_{q}^{alg}((z))).

The σ̂-conjugacy class given by (−1, 0) is independent of the choice of μ and of the uniformizer y (and of E) by [Kot85, 2.5]. The Schubert variety S(μ̄) for μ̄ = 1 from Example 4.12 therefore equals

L^{+}T^{0}(𝔽_{q}) · (−1, 0) · L^{+}T^{0}(𝔽_{q})/L^{+}T^{0}(𝔽_{q}).

However, as we have mentioned in the discussion after equation (4.3), the bound [S(μ̄) ˆ×_{𝔽_{q}} Spf 𝔽_{q}[[ζ]]] is only useful to bound quasi-isogenies between local ℙ-shtukas.

Instead we want to define a bound [Ẑ_{R}] which is useful to bound local ℙ-shtukas, and whose fiber Z_{R} := Ẑ_{R} ˆ×_{Spf R} Spec κ_{R} over κ_{R} equals the Schubert variety S(μ̄) ×_{𝔽_{q}} Spec κ_{R} ⊂ Fℓ ×_{𝔽_{q}} Spec κ_{R}. In Example 4.13 we were able to achieve this by lifting μ(z) ∈ LG(𝔽_{q}) to an element μ(z−ζ) ∈ G(𝔽_{q}[[z, ζ]][\frac{1}{z−ζ}]); see the discussion after (4.3). This is not possible here. We can only lift (−1, 0) over the ramified extension 𝔽_{q}[[ζ]][ξ]/(ξ^{2} − ζ) of 𝔽_{q}[[ζ]]. Namely, consider the isomorphism K = 𝔽_{q}((z)) → 𝔽_{q}((ζ)), z ↦ ζ. Fix an extension i: E ↪ 𝔽_{q}((ζ))^{alg} of this isomorphism, set ξ := i(y) and lift N_{E/K}(μ(y)) to g(E, μ, i) := N_{E/K}(μ(y − i(y))). For example if μ = 1 ∈ ℤ = X_{∗}(T_{E}) we compute

(α, β) := g(E, 1, i) = μ(y − ξ) · γ(μ(y − ξ)) =

(\frac{(y − ξ) + (y − ξ)^{−1}}{2} , \frac{(y − ξ)^{−1} − (y − ξ)}{2y}) · γ (\frac{(y − ξ) + (y − ξ)^{−1}}{2} , \frac{(y − ξ)^{−1} − (y − ξ)}{2y}) ,

with γ(y) = −y and γ|_{𝔽_{q}((ξ))} = id. Then

α = \frac{1}{4} ((y − ξ) + (y − ξ)^{−1}) ((−y − ξ) + (−y − ξ)^{−1})
−\frac{1}{4} ((y − ξ)^{−1} − (y − ξ)) ((−y − ξ)^{−1} − (−y − ξ))
= \frac{1}{2} (\frac{(y − ξ)^{2} + (−y − ξ)^{2}}{(−y − ξ)(y − ξ)} )

= \frac{ζ + z}{ζ − z} ,

and

β = \frac{−1}{4y} ((y − ξ) + (y − ξ)^{−1}) ((−y − ξ)^{−1} − (−y − ξ))
+\frac{1}{4y} ((y − ξ)^{−1} − (y − ξ)) ((−y − ξ) + (−y − ξ)^{−1})
= \frac{2ξ}{ζ − z} .

Thus, as a lift of (−1, 0) we get

g(E, 1, i) = N_{E/K}(μ(y − ξ)) = (\frac{ζ + z}{ζ − z} , \frac{2ξ}{ζ − z} ) .

This shows that we can define the desired bound by

Ẑ_{μ,i,𝔽_{q}[[ξ]]} := \overline{L^{+}T^{0} · g(E, μ, i) · L^{+}T^{0}/L^{+}T^{0}} ⊂ Fℓ_{T^{0},𝔽_{q}[[ξ]]} .

However, this bound depends on the choice of μ and of the embedding i. We first compute how g(E, μ, i)

depends on the chosen embedding i. For this purpose we compute \frac{N_{E/K}(μ(y − i(y)))}{N_{E/K}(μ(y − i ∘ γ(y))}. Changing i to

i ∘ γ replaces ξ by −ξ and we have

g(E, 1, i ∘ γ) = N_{E/K}(μ(y + ξ)) = (\frac{ζ + z}{ζ − z} , \frac{−2ξ}{ζ − z} ) .

Note that g(E, 1, i) = g(E, 1, i ∘ γ)^{−1}. Therefore

\frac{g(E, 1, i)}{g(E, 1, i ∘ γ)} = g(E, 1, i)^{2} = (\frac{(z + ζ)^{2} + 4ζz}{(ζ − z)^{2}} , \frac{4ξ(z + ζ)}{(ζ − z)^{2}}) .

This also shows what happens if we replace μ ∈ X_{∗}(T_{E}) = ℤ by another lift of μ̄ ∈ ℤ/2ℤ, i.e.

\frac{g(E, μ + 2, i)}{g(E, μ, i)} = g(E, 2, i) = g(E, 1, i)^{2}.

Observe that \frac{g(E, 1, i)}{g(E, 1, i ∘ γ)} ∈ T^{0}(𝔽_{q}[[z, \frac{ξ}{z}]]) ∖ T^{0}(𝔽_{q}[[ξ, z]]). So the element g(E, 1, γ ∘ i) does not lie in

the closure of the subscheme

L^{+}T^{0} · g(E, 1, i) · L^{+}T^{0} ⊂ LT^{0} ˆ×_{Spec 𝔽_{q}} Spf 𝔽_{q}[[ξ]].

In particular the bounds Ẑ_{μ,i} := [Ẑ_{μ,i,𝔽_{q}[[ξ]]}] depend on the chosen embedding i: E ↪ 𝔽_{q}((ζ))^{alg} and on the lift μ ∈ X_{∗}(T) of μ̄ ∈ X_{∗}(T)_{I}. For this reason we decided to treat bounds axiomatically in Definition 4.8. Our discussion also shows that the reflex ring of the bound Ẑ_{μ,i} is 𝔽_{q}[[ξ]], because γ̃(Ẑ_{μ,i}) = Ẑ_{μ,i∘γ} ≠ Ẑ_{μ,i} for the non-trivial element γ̃ ∈ Aut_{𝔽_{q}[[ζ]]}(𝔽_{q}[[ξ]]).

4.3 Representability of the Bounded Rapoport–Zink Functor

In this section we assume that ℙ is a smooth affine group scheme over 𝔻 with connected reductive generic fiber P. Let b ∈ LP (k) for some field k ∈ Nilp_{𝔽[[ζ]]}. With b Kottwitz associates a slope homomorphism

ν_{b}: D_{k((z))} → P_{k((z))},

called Newton polygon of b; see [Kot85, 4.2]. Here D is the diagonalizable pro-algebraic group over k((z)) with character group ℚ. The slope homomorphism is characterized by assigning the slope filtration of (V ⊗_{𝔽((z))} k((z)), ρ(b) · ( id ⊗σ̂)) to any 𝔽((z))-rational representation (V, ρ) of P; see [Kot85, Section 3]. We assume that b ∈ LP (k) satisfies a decency equation for a positive integer s, that is,

(bσ̂)^{s} = sν_{b}(z) σ̂^{s}  in LP (k) ⋊ ⟨σ̂⟩.  (4.4)

Remark 4.15. Assume that b ∈ LP (k) is decent with the integer s and let ℓ ⊂ k^{alg} be the finite field extension of 𝔽 of degree s. Then b ∈ LP (ℓ) because by (4.4) the element b has values in the fixed field of σ̂^{s} which is ℓ. Note that if k is algebraically closed and the generic fiber P of ℙ is connected reductive, any σ̂-conjugacy class in LP (k) contains an element satisfying a decency equation; see [Kot85, 4.3] and use [Ser97, § II.2.3, Théorème 1′ and Remarque 1, p. 140] instead of Steinberg’s theorem.

Remark 4.16. With the element b ∈ LP (k) one can associate a connected algebraic group J_{b} over 𝔽((z)) which is defined by its functor of points that assigns to an 𝔽((z))-algebra R the group

J_{b}(R) := {g ∈ P (R ⊗_{𝔽((z))} k((z))): g^{−1}bσ̂(g) = b}.

Let b satisfy a decency equation for the integer s and let F_{s} be the fixed field of σ̂^{s} in 𝔽^{alg}((z)). Then ν_{b} is defined over F_{s} and J_{b} ×_{𝔽((z))} F_{s} is the centralizer of the 1-parameter subgroup sν_{b} of P and hence a Levi subgroup of P_{F_{s}}; see [RZ96, Corollary 1.14]. In particular J_{b}(𝔽((z))) ⊂ P (F_{s}) ⊂ LP (ℓ) where ℓ is the finite field extension of 𝔽 of degree s.

In the remaining part of the chapter we consider the bounded Rapoport–Zink functor and prove that it is ind-representable by a formal scheme in the following important special situation. Let Ẑ be a bound with reflex ring RẐ = κ[[ξ]] and special fiber Z ⊂ Fℓ_{ℙ} ˆ×_{𝔽} Spec κ ; see Definition 4.8. Let \underline{𝕃}_{0} = (L^{+}ℙ, bσ̂^{∗}) be a trivialized local ℙ-shtuka over a field k in Nilp_{𝔽[[ζ]]}. Assume that b is decent with integer s and let ℓ ⊂ k^{alg} be the compositum of the residue field κ of RẐ and the finite field extension of 𝔽 of degree s. Then b ∈ LP (ℓ) by Remark 4.15. So \underline{𝕃}_{0} is defined over ℓ and we may replace k by ℓ. Note that ℓ[[ξ]] is the unramified extension of RẐ with residue field ℓ.

Definition 4.17. Keep the notation from above and set T̄ := Spec ℓ and \underline{L}_{0} := \underline{𝕃}_{0}.

(a) Consider the base change \underline{M}_{\underline{𝕃}_{0}} ˆ×_{ℓ[[ζ]]} Spf ℓ[[ξ]] of the functor \underline{M}_{\underline{𝕃}_{0}} from Definition 4.1 and its subfunctor

\underline{M}Ẑ ; \underline{𝕃}_{0} : (Nilp_{ℓ[[ξ]]})^{o} ⟶ Sets
S ⟼ \{ Isomorphism classes of (\underline{L}, δ̄) ∈ \underline{M}_{\underline{𝕃}_{0}}(S): \underline{L} is bounded by Ẑ\}

Note that the functor \underline{M}_{\underline{𝕃}_{0}} ˆ×_{ℓ[[ζ]]} Spf ℓ[[ξ]] is represented by the ind-scheme F̂ℓ_{ℙ,ℓ[[ξ]]} := Fℓ_{ℙ} ˆ×_{𝔽} Spf ℓ[[ξ]] by

Theorem 4.4, and \underline{M}Ẑ ; \underline{𝕃}_{0} is a closed ind-subscheme by Proposition 4.11.

(b) We define the associated affine Deligne–Lusztig variety over ℓ as the reduced closed ind-subscheme X_{Z}(b) ⊂ Fℓ_{ℙ} ˆ×_{𝔽} Spec ℓ whose K-valued points (for any field extension K of ℓ) are given by

X_{Z}(b)(K) := {g ∈ Fℓ_{ℙ}(K): g^{−1} b σ̂^{∗}(g) ∈ Z(K)}.

If ω ∈ W̃ and Z = S(ω) is the Schubert variety from Example 4.12, we set X_{⪯ω}(b) := X_{S(ω)}(b).

Theorem 4.18. If ℙ is a smooth affine group scheme over 𝔻 with connected reductive generic fiber, the functor \underline{M}Ẑ ; \underline{𝕃}_{0} : (Nilp_{ℓ[[ξ]]})^{o} → Sets is ind-representable by a formal scheme over Spf ℓ[[ξ]] which is locally formally of finite type and separated. Its underlying reduced subscheme equals X_{Z}(b). In particular X_{Z}(b) is a scheme locally of finite type and separated over ℓ. The formal scheme representing \underline{M}Ẑ ; \underline{𝕃}_{0} is called a bounded Rapoport–Zink space for local ℙ-shtukas.

Recall that a formal scheme over ℓ[[ξ]] in the sense of [EGA, I_{new}, 10] is called locally formally of finite type if it is locally noetherian and adic and its reduced subscheme is locally of finite type over ℓ. It is called formally of finite type if in addition it is quasi-compact.

Remark 4.19. By our assumptions QIsog_{ℓ}(\underline{𝕃}_{0}) equals the group J_{b}(𝔽((z))) from Remark 4.16. This group acts on the functor \underline{M}Ẑ ; \underline{𝕃}_{0} via g: (\underline{L}, δ̄) ↦ (\underline{L}, g ∘ δ̄) for g ∈ QIsog_{ℓ}(\underline{𝕃}_{0}).

Proof of Theorem 4.18. The proof will use a sequence of lemmas and will eventually be complete after Lemma 4.25. Consider the universal local ℙ-shtuka \underline{L}_{univ} over F̂ℓ_{ℙ,ℓ[[ξ]]} (see Theorem 4.4). Let MẐ ; \underline{𝕃}_{0}

be the closed ind-subscheme of F̂ℓ_{ℙ,ℓ[[ξ]]} over which \underline{L}_{univ} is bounded by Ẑ; see Proposition 4.11. By construction MẐ ; \underline{𝕃}_{0} ind-represents the functor \underline{M}Ẑ ; \underline{𝕃}_{0}. It is clear that the reduced ind-subscheme equals X_{Z}(b). We have to show that MẐ ; \underline{𝕃}_{0} is a formal scheme locally formally of finite type and separated. Note

that the separatedness over Spf ℓ[[ξ]] follows from the ind-separatedness of F̂ℓ_{ℙ,ℓ[[ξ]]}; see Theorem 4.4. By rigidity of quasi-isogenies the functor \underline{M}Ẑ ; \underline{𝕃}_{0} is equivalent to the following functor

(Nilp_{ℓ[[ξ]]})^{o} → Sets
S ⟼ \{ (\underline{L}, δ): \underline{L} is a local ℙ-shtuka over S bounded by Ẑ
and δ: \underline{L} → \underline{𝕃}_{0,S} is a quasi-isogeny \}/ ∼ .

We take a representation ι: ℙ → SL_{r,𝔻} =: H with quasi-affine quotient H/ℙ; see [PR08, Proposition 1.3] and [AH13, Proposition 2.1]. It induces a 1-morphism

ι_{∗} := ℋ ^{1}(ι): ℋ ^{1}(Spec 𝔽, L^{+}ℙ)(S) → ℋ ^{1}(Spec 𝔽, L^{+}H)(S).

For an L^{+}ℙ-torsor L_{+} over S we denote by V(ι_{∗}L_{+}) the sheaf of O_{S}[[z]]-modules associated with the image of L_{+} in ℋ ^{1}(Spec 𝔽, L^{+} GL_{r})(S) by Remark 3.2. In particular V(ι_{∗}(L^{+}ℙ)_{S}) = O_{S}[[z]]^{⊕r}. Let ρ^{∨} be the half-sum of all positive coroots of H with respect to the Borel subgroup of upper triangular matrices in H = SL_{r}.

For each non-negative integer n ∈ ℕ_{0} let M^{n} be the closed ind-subscheme of \underline{M}Ẑ ; \underline{𝕃}_{0} defined by the

following sub functor of \underline{M}Ẑ ; \underline{𝕃}_{0}

\underline{M}^{n}: (Nilp_{ℓ[[ξ]]})^{o} ⟶ Sets
S ⟼ \{ (\underline{L}, δ) ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(S): ℋ ^{1}(ι)(δ) is bounded by 2nρ^{∨} \},

where \underline{L} = (L_{+}, τ̂) and where we say that ℋ ^{1}(ι)(δ) is bounded by 2nρ^{∨} if for all j = 1, …, r

⋀_{O_{S}[[z]]}^{j} ℋ ^{1}(ι)(δ)(V(ι_{∗}L_{+})) ⊂ z^{n(j^{2}−jr)} · ⋀_{O_{S}[[z]]}^{j} V(ι_{∗}(L^{+}ℙ)_{S}) .  (4.5)

By [HV11, Lemma 4.3] the latter is equivalent to the boundedness condition considered in [HV11, Definition 3.5], see Example 4.13, because ρ^{∨} = (r − 1, …, 1 − r) and (4.5) is automatically an equality for j = r as ι factors through H.

Lemma 4.20. The ind-scheme M^{n} representing the functor \underline{M}^{n} is a ξ-adic noetherian formal scheme over ℓ[[ξ]], whose underlying topological space (M^{n})_{red} is a quasi-projective scheme over Spec ℓ and even projective if ℙ is parahoric in the sense of Bruhat and Tits [BT72, Définition 5.2.6] and [HR08].

Proof. Since H/ℙ is quasi-affine, the induced morphism ι_{∗}: F̂ℓ_{ℙ,ℓ[[ξ]]} → F̂ℓ_{H,ℓ[[ξ]]} is a locally closed embedding by Remark 4.3. The representation ι induces a functor ι_{∗} from local ℙ-shtukas to local H-shtukas. Consider the local H-shtuka \underline{ℍ}_{0} := ι_{∗}\underline{𝕃}_{0} = ((L^{+}H)_{ℓ}, ι(b)σ̂^{∗}) over ℓ and view F̂ℓ_{H,ℓ[[ξ]]} as a moduli space representing the functor \underline{M}_{\underline{ℍ}_{0}}, parametrizing local H-shtukas together with a quasi-isogeny δ_{H} to \underline{ℍ}_{0}; see

Theorem 4.4. Let F̂ℓ_{H,ℓ[[ξ]]}^{(n)} be the closed ind-subscheme of F̂ℓ_{H,ℓ[[ξ]]} defined by condition (4.5), that is, by bounding δ_{H} by 2nρ^{∨}. It is a ξ-adic noetherian formal scheme over Spf ℓ[[ξ]] by [HV11, Proposition 5.5] whose underlying topological space is a projective scheme over Spec ℓ. Thus for all i

M^{n}(i) := M^{n} ˆ×_{Spf ℓ[[ξ]]} Spec ℓ[[ξ]]/(ξ^{i}) = \underline{M}Ẑ ; \underline{𝕃}_{0} ˆ×_{F̂ℓ_{H,ℓ[[ξ]]}} F̂ℓ_{H,ℓ[[ξ]]}^{(n)} ˆ×_{Spf ℓ[[ξ]]} Spec ℓ[[ξ]]/(ξ^{i})

is a locally closed subscheme of F̂ℓ_{H,ℓ[[ξ]]}^{(n)} ˆ×_{Spf ℓ[[ξ]]} Spec ℓ[[ξ]]/(ξ^{i}), and hence a scheme of finite type over Spec ℓ[[ξ]]/(ξ^{i}) with underlying topological space M^{n}(1) independent of i. Moreover, M^{n}(1) is (quasi-) projective, because it is closed in the ind-quasi-projective ind-scheme Fℓ_{ℙ} ˆ×_{𝔽} Spec ℓ which is even projective if ℙ is parahoric. Now our claim follows from [EGA, I_{new}, Corollary 10.6.4].

For each non-negative integer n we define the following sub functor of \underline{M}Ẑ ; \underline{𝕃}_{0}

\underline{M}_{n}: (Nilp_{ℓ[[ξ]]})^{o} ⟶ Sets
S ⟼ \{ (\underline{L}, δ) ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(S): for any point s in S,
ℋ (ι)(δ_{s}) is bounded by 2nρ^{∨} }.

This functor is represented by an ind-scheme M_{n} which is the formal completion of \underline{M}Ẑ ; \underline{𝕃}_{0} along the quasi-compact closed subscheme (M^{n})_{red}.

Lemma 4.21. M_{n} is a formal scheme formally of finite type over Spf ℓ[[ξ]].

To prove the lemma we need to start with the following definition. Recall that RẐ = κ[[ξ]].

Definition 4.22. Let R = lim_{⟵} R_{m} ∈ Nilp_{κ[[ξ]]} where (R_{m}, u_{m,m′}) is a projective system of discrete rings indexed by ℕ_{0}. Suppose that all homomorphisms R → R_{m} are surjective, and that all kernels I_{m} := ker(u_{m,0} : R_{m} → R_{0}) ⊂ R_{m} are nilpotent. A local ℙ-shtuka over Spf R is a projective system (\underline{L}_{m})_{m∈ℕ_{0}} of local ℙ-shtukas \underline{L}_{m} over R_{m} with \underline{L}_{m−1} ≅ \underline{L}_{m} ⊗_{R_{m}} R_{m−1}.

Lemma 4.23. Let R in Nilp_{κ[[ξ]]} be as in the above definition. The pull back functor defines an equivalence between the category of local ℙ-shtukas over Spec R bounded by Ẑ and the category of local ℙ-shtukas over Spf R bounded by Ẑ.

Proof. Since R is in Nilp_{κ[[ξ]]} there is an integer e ∈ ℕ such that ξ^{e} = 0 on R. Let \underline{L} := (\underline{L}_{m})_{m∈ℕ_{0}} be a local ℙ-shtuka over Spf R. By Proposition 2.4 there is an étale covering R′_{0} → R_{0} which trivializes \underline{L}_{0}. By [SGA 1, Théorème I.8.3] there is a unique étale R-algebra R′ with R′ ⊗_{R} R_{0} ≅ R′_{0}. As in [HV11, Proposition 2.2(c)] this gives rise to compatible trivializations \underline{L}_{m} ⊗_{R_{m}} R′_{m} ≅ ((L^{+}ℙ)_{R′_{m}}, b_{m}σ̂^{∗}) over R′_{m} := R′ ⊗_{R} R_{m} for all m. Here b_{m} ∈ LP (R′_{m}) and b_{m} ⊗_{R′_{m}} R′_{m−1} = b_{m−1}.

By [PR08, Proposition 1.3] and [AH13, Proposition 2.1] we may take a faithful representation ι: ℙ ↪ SL_{r,𝔻} and consider the induced closed immersion LP ↪ L SL_{r,𝔻}. The ind-scheme structure on Fℓ_{SL_{r,𝔻}} is given by Fℓ_{SL_{r,𝔻}} = lim_{⟶} Fℓ_{SL_{r,𝔻}}^{(n)} where Fℓ_{SL_{r,𝔻}}^{(n)} is defined by condition (4.5). Let

L SL_{r,𝔻}^{(n)} = L SL_{r,𝔻} ×_{Fℓ_{_{SL}_{r,𝔻}}} Fℓ_{SL_{r,𝔻}}^{(n)}. Then

L SL_{r,𝔻}^{(n)}(S) = { g ∈ L SL_{r,𝔻}(S): all j × j-minors of g lie in z^{n(j^{2}−jr)}O_{S}(S)[[z]] ∀ j }.

This implies that L SL_{r,𝔻}^{(n)}, and hence also LP^{(n)} := LP ×_{L SL_{r,𝔻}} L SL_{r,𝔻}^{(n)} is an infinite dimensional affine scheme over 𝔽. By Remark 4.7(f) we may choose a representative ẐR̃ ⊂ F̂ℓ_{ℙ,R̃} of the bound Ẑ over a finite extension R̃ of RẐ = κ[[ξ]] with Frac(R̃) Galois over Frac(RẐ). By Remark 4.9 the boundedness by Ẑ can be checked using ẐR̃. Since Z_{R̃,e} := ẐR̃ ˆ×_{Spf R̃} Spec R̃/(ξ^{e}) has the same underlying topological space as

ZR̃ := ẐR̃ ˆ×R̃ Spec κR̃, it is quasi-compact. So there is an n ∈ ℕ such that Z_{R̃,e} ⊆ Fℓ_{SL_{r,𝔻}}^{(n)} ˆ×_{𝔽} Spec R̃/(ξ^{e}) by [HV11, Lemma 5.4]. As one sees from the following diagram

\xymatrix{Spec R′_{m} ⊗_{κ[[ξ]]} R̃ \ar[d] \ar@{-->}[r]^{b_{m}× id} & L SL_{r,𝔻}^{(n)} ˆ×_{𝔽} Spec R̃/(ξ^{e}) \ar[d] \ar[r] & L SL_{r,𝔻} ˆ×_{𝔽} Spec R̃/(ξ^{e}) \ar[d] \\ Z_{R̃,e} \ar[r] & Fℓ_{SL_{r,𝔻}}^{(n)} ˆ×_{𝔽} Spec R̃/(ξ^{e}) \ar[r] & Fℓ_{SL_{r,𝔻}} ˆ×_{𝔽} Spec R̃/(ξ^{e}) ,}

the morphism b_{m} × id: Spec R′_{m} ⊗_{κ[[ξ]]} R̃ → LP factors through LP^{(n)} = LP ×_{L SL_{r,𝔻}} L SL_{r,𝔻}^{(n)} for all m. Since LP^{(n)} is affine, the compatible collection of morphisms b_{m}: Spec R′_{m} ⊗_{κ[[ξ]]} R̃ → LP^{(n)} is given by a compatible collection of homomorphisms O(LP^{(n)}) → R′_{m} ⊗_{κ[[ξ]]} R̃. It corresponds to a homomorphism b̃_{∞}^{∗}: O(LP^{(n)}) → R′ ⊗_{κ[[ξ]]} R̃, because R′ = lim_{⟵} R′_{m} and R̃ is a finite free κ[[ξ]]-module. By Remark 4.7(f),

γ(ẐR̃) = ẐR̃ for all γ ∈ Aut_{κ[[ξ]]}(R̃) and thus by construction the homomorphism b̃_{∞}^{∗} is invariant under Aut_{κ[[ξ]]}(R̃). It follows that b̃_{∞}^{∗} factors through a homomorphism b_{∞}^{∗}: O(LP^{(n)}) → R′. The latter corresponds to a morphism b_{∞}: Spec R′ → LP^{(n)}. This gives the local ℙ-shtuka ((L^{+}ℙ)_{R′}, b_{∞}σ̂^{∗}) over Spec R′ bounded by Ẑ, which carries a descent datum from the \underline{L}_{m} and hence induces a local ℙ-shtuka over Spec R.

Let us come back to the

Proof of Lemma 4.21. For each m ≥ n let M_{n}^{m} be the formal completion of M^{m} along (M_{n})_{red}. It is a noetherian adic formal scheme over ℓ[[ξ]]. Let U be an affine open subscheme of (M_{n})_{red}. By [EGA, I_{new}, Proposition 2.3.5] this defines an affine open formal subscheme Spf R_{m} of M_{n}^{m} with underlying set U. Let R be the inverse limit of the projective system R_{m+1} → R_{m} and let 𝔞_{m} ⊂ R denote the ideal such that R_{m} = R/𝔞_{m}. Let J be the inverse image in R of the largest ideal of definition in R_{n}. We want to show that R is J-adic. We make the following

Claim. For any integer c > 0 there is an integer m_{0} such that for any m ≥ m_{0} the natural map R_{m}/J^{c}R_{m} → R_{m_{0}}/J^{c}R_{m_{0}} is an isomorphism.

To prove the claim let \underline{L}_{m} be the universal bounded local ℙ-shtuka over Spf R_{m}. Consider the local ℙ-shtuka (\underline{L}_{m})_{m} over Spf R and its pullback over Spf R/J^{c}. The latter comes from a local ℙ-shtuka \underline{L} over Spec R/J^{c} ∈ Nilp_{ℓ[[ξ]]} by Lemma 4.23. By rigidity of quasi-isogenies the quasi-isogeny δ_{n} over R/J = R_{n}/J lifts to a quasi-isogeny δ over R/J^{c}. Since Spec R/J^{c} is quasi-compact, the quasi-isogeny ℋ ^{1}(ι)(δ) satisfies condition (4.5) for some m_{0}, that is, it is bounded by 2m_{0}ρ^{∨}. By the universal property of M_{n}^{m_{0}} the tuple (\underline{L}, δ) induces a morphism R_{m_{0}} → R/J^{c} making the following diagram commutative, from which the claim follows

\xymatrix{R \ar@{->>}[d] \ar@{->>}[r] & R_{m} \ar@{->>}[d] \ar@{->>}[r] & R_{m_{0}} \ar@{-->}[dll] \ar@{->>}[d] \\ R/J^{c} \ar@{->>}[r] & R_{m}/J^{c}R_{m} \ar@{->>}[r] & R_{m_{0}}/J^{c}R_{m_{0}}.}

The claim implies that the chain 𝔞_{n} + J^{c} ⊇ 𝔞_{n+1} + J^{c} ⊇ … ⊇ 𝔞_{i} + J^{c} ⊇ … stabilizes. Now set J_{c} := ⋂_{i} 𝔞_{i} + J^{c} = 𝔞_{m} + J^{c} for m ≫ 0 and consider the descending chain J_{1} ⊇ J_{2} ⊇ . . .. Note that J_{1} = J and J_{c+1} + J_{1}^{c} = J_{c}. Since J_{1}/J_{2} ≅ JR_{m}/J^{2}R_{m} for m ≫ 0, it is a finitely generated R-module. Therefore M_{n} is a locally noetherian adic formal scheme over Spf ℓ[[ξ]] by [RZ96, Proposition 2.5]. It is formally of finite type because (M_{n})_{red} = (M^{n})_{red} is quasi-projective over ℓ by Lemma 4.20.

From now on we use that \underline{𝕃}_{0} is decent with the integer s. In the sequel we consider points x ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(K)

for varying field extensions K of ℓ. For two points x := (\underline{L}, δ) and x′ := (\underline{L}′, δ′) in \underline{M}Ẑ ; \underline{𝕃}_{0}(K) we define

d̃(x, x′) := min {n ∈ ℕ_{0} : ℋ ^{1}(ι)(δ^{−1}δ′) is bounded by 2nρ^{∨}}.  (4.6)

By the definition of “being bounded by 2nρ^{∨}” in (4.5) we conclude that if x′′ ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(K) is a third point then d̃(x, x′′) ≤ d̃(x, x′) + d̃(x′, x′′). Moreover, in the situation where δ = g and δ′ = g′ for g, g′ ∈ LP (K), as well as x = ((L^{+}ℙ)_{K}, g^{−1}bσ̂^{∗}(g), g) and x′ = ((L^{+}ℙ)_{K}, (g′)^{−1}bσ̂^{∗}(g′), g′), we also write d̃(g, g′) := d̃(x, x′). Note that a point x ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(K) belongs to M_{n} if and only if d̃((\underline{𝕃}_{0}, id), x) ≤ n.

Although we will not use this, note that d̃ is a metric on \underline{M}Ẑ ; \underline{𝕃}_{0}. This follows from the fact that (4.5) for n = 0 together with Cramer’s rule implies that ℋ ^{1}(ι)(δ_{1}^{−1}δ_{2}) is an isomorphism V(ι_{∗}\underline{L}_{2}) ^{∼} ⟶ V(ι_{∗}\underline{L}_{1}); compare the discussion around [AH13, Equations (3.2) and (3.3)].

Lemma 4.24. For every integer c > 0 there is an integer d_{0} > 0 with the following property. For every g ∈ LP (K) with d̃(g, b σ̂^{∗}(g)) < c there is a g′ ∈ LP (ℓ) with d̃(g, g′) ≤ d_{0}.

Proof. This is just a reformulation of [RZ99, Theorem 1.4 and Subsection 2.1] taking into account that by functorial properties of Bruhat–Tits buildings (see [Lan00]) the representation ι induces an injective isometric map of Bruhat–Tits buildings ℬ(P) → ℬ(H).

Lemma 4.25. There is an integer d_{0} ∈ ℕ_{0} such that sup \{d̃(x, \underline{M}Ẑ ; \underline{𝕃}_{0}(ℓ)): x ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}\} ≤ d_{0}.

Proof. Let x := (\underline{L}^{+}, δ) be a point in \underline{M}Ẑ ; \underline{𝕃}_{0}(K). After replacing K by a separable field extension, we take a trivialization (\underline{L}^{+}, δ) ≅ ((L^{+}ℙ)_{K}, hσ̂^{∗}, g) with δ = g ∈ LP (K) and h = g^{−1}bσ̂^{∗}(g). Since the local ℙ-shtuka ((L^{+}ℙ)_{K}, hσ̂^{∗}) is bounded by Ẑ and Z is quasi-compact by definition of the boundedness condition (see Definition 4.8), we have d̃(g, b σ̂^{∗}(g)) = min{n ∈ ℕ_{0} : ℋ ^{1}(ι)(h) is bounded by 2nρ^{∨}} < c for a natural number c which is independent of x. Let d_{0} = d_{0}(c) be the integer from Lemma 4.24. Then there is a g′ ∈ LP (ℓ) with d̃(g, g′) ≤ d_{0}. Now the associated point x′ := ((L^{+}ℙ)_{ℓ}, g′^{−1}bσ̂^{∗}g′, g′) of \underline{M}Ẑ ; \underline{𝕃}_{0}(ℓ) satisfies d̃(x, x′) ≤ d_{0} and the lemma follows.

For a point y ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(ℓ) set B(y, d_{0}) := \{x ∈ \underline{M}Ẑ ; \underline{𝕃}_{0} : d̃(x, y) ≤ d_{0}\}. Here x ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(K) for a field extension K of ℓ. We set B_{n}(y, d_{0}) = B(y, d_{0}) ∩ M_{n}. Note that these are closed subsets. For each integer r let

Z_{n}^{r} = ⋃_{y∈\underline{M}Ẑ ; \underline{𝕃}_{0}(ℓ), d̃((\underline{𝕃}_{0}, id),y)≥r} B_{n}(y, d_{0})

Then Z_{n}^{r} = Z_{n+1}^{r} ∩ M_{n}. If y ∉ M_{n+d_{0}}, that is d̃((\underline{𝕃}_{0}, id), y) > n + d_{0}, and if x ∈ M_{n}, that is d̃((\underline{𝕃}_{0}, id), x) ≤ n, then

d̃(x, y) ≥ d̃((\underline{𝕃}_{0}, id), x) − n + d̃(x, y) ≥ d̃((\underline{𝕃}_{0}, id), y) − n > d_{0}

and thus B_{n}(y, d_{0}) = ∅. We get

Z_{n}^{r} = ⋃_{y∈M_{n+d_{0}}(ℓ),d̃((\underline{𝕃}_{0}, id),y)≥r} B_{n}(y, d_{0}).

Since (M_{n+d_{0}})_{red} = (M^{n+d_{0}})_{red} is quasi-projective over ℓ by Lemma 4.20, this union is finite.

Let U_{n}^{r} be the open formal sub-scheme of M_{n} whose underlying reduced set is M_{n} ∖ Z_{n}^{r}. We claim that the chain U_{n}^{r} ↪ U_{n+1}^{r} ↪ … of open formal sub-schemes of M_{n} stabilizes. By the definition of M_{n} it is enough to verify this on the underlying set of points. Suppose that there is some element x ∈ U_{n+1}^{r}∖U_{n}^{r}. By Lemma 4.25 there exists a y ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(ℓ) such that d̃(x, y) ≤ d_{0}. Since x ∈ M_{n+1} ∖ Z_{n+1}^{r} we must have d̃((\underline{𝕃}_{0}, id), y) < r. Then

d̃((\underline{𝕃}_{0}, id), x) ≤ d̃((\underline{𝕃}_{0}, id), y) + d̃(x, y) < r + d_{0}.  (4.7)

Therefore, if n ≥ r +d_{0} then d̃((\underline{𝕃}_{0}, id), x) ≤ n and x ∈ M_{n} which is a contradiction. Consequently there is no such x.

Let U^{r} = ⋃_{n} U_{n}^{r} (which equals U_{n}^{r} for n ≥ r + d_{0}). Note that every point x of \underline{M}Ẑ ; \underline{𝕃}_{0} lies in the union of the U^{r}s. Indeed, if d̃((\underline{𝕃}_{0}, id), x) < r − d_{0} for some r, then x is contained in U^{r}, because otherwise there is a y ∈ \underline{M}Ẑ ; \underline{𝕃}_{0}(ℓ) with d̃(x, y) ≤ d_{0} and d̃((\underline{𝕃}_{0}, id), y) ≥ r, a contradiction. Now consider the chain

U^{r} ↪ U^{r+1} … ↪ \underline{M}Ẑ ; \underline{𝕃}_{0}

of open immersions of formal schemes formally of finite type, note that U^{r} is open in \underline{M}Ẑ ; \underline{𝕃}_{0}. Indeed the

underlying topological space of U^{r} is open in M_{n} for every n and the ind-scheme MẐ ; 𝕃_{0} carries the limit topology of the limit over the M_{n}. This shows that the formal scheme U^{r} equals the formal completion of the open ind-scheme \underline{M}Ẑ ; \underline{𝕃}_{0}|_{|U^{r}|} of \underline{M}Ẑ ; \underline{𝕃}_{0} supported on |U^{r}| along the whole set |U^{r}| and thus \underline{M}Ẑ ; \underline{𝕃}_{0}|_{|U^{r}|} = U^{r}.

Since U^{r} is locally formally of finite type this implies that \underline{M}Ẑ ; \underline{𝕃}_{0} = ⋃_{r} U^{r} is locally formally of finite type as well. This completes the proof of Theorem 4.18

Corollary 4.26. The irreducible components of the topological space \underline{M}Ẑ ; \underline{𝕃}_{0} are quasi-projective schemes over ℓ. In particular they are quasi-compact. They are projective if ℙ is parahoric in the sense of Bruhat and Tits [BT72, Définition 5.2.6] and [HR08].

Proof. Let T be an irreducible component and let x be its generic point. As in the proof of the theorem there is an r such that x ∈ U^{r} = U_{r+d_{0}}^{r} ⊂ M_{r+d_{0}}. Since the underlying topological spaces of M_{r+d_{0}} and M^{r+d_{0}} coincide, are closed in \underline{M}Ẑ ; \underline{𝕃}_{0}, and (quasi-)projective over ℓ by Lemma 4.20, we see that T ⊂ M^{r+d_{0}} is a closed subscheme and the corollary follows.

In the rest of this section we fix an integer n and consider complete discrete valuation rings 𝔽_{i}[[z_{i}]] for i = 1, …, n with finite residue fields 𝔽_{i}, and fraction fields Q_{i} = 𝔽_{i}((z_{i})). Let ℙ_{i} be a smooth affine group scheme over Spec 𝔽_{i}[[z_{i}]] with connected reductive generic fiber P_{i} := ℙ_{i} ×_{𝔽_{i}[[z_{i}]]} Spec 𝔽_{i}((z_{i})), and let Ẑ_{i} = [Ẑ_{i,R′_{i}}] with Ẑ_{i,R′_{i}} ⊂ F̂ℓ_{ℙ_{i},R′_{i}} := Fℓ_{ℙ_{i}} ˆ×_{𝔽_{i}} Spf R′_{i} be a bound in the sense of Definition 4.8 with reflex ring RẐ_{i} =: R_{i} = κ_{i}[[ξ_{i}]]. Let k be a field containing all κ_{i}. For all i let \underline{𝕃}_{i} be a local ℙ_{i}-shtuka over

k which is trivialized and decent. By Theorem 4.18 the Rapoport–Zink space MẐ_{i} ; \underline{𝕃}_{i} is a formal scheme

locally formally of finite type over Spf k[[ξ_{i}]]. Therefore the product ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} := MẐ_{1} ; \underline{𝕃}_{1} ˆ×_{k} … ˆ×_{k} MẐ_{n} ; \underline{𝕃}_{n} is a formal scheme locally formally of finite type over Spf k[[\underline{ξ}]] := Spf k[[ξ_{1}, …, ξ_{n}]]. Recall that the group

J_{\underline{𝕃}_{i}}(Q_{i}) = QIsog_{k}(\underline{𝕃}_{i}) of quasi-isogenies of \underline{𝕃}_{i} over k acts naturally on MẐ_{i} ; \underline{𝕃}_{i}; see Remark 4.19.

Let Γ ⊆ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{i}) be a subgroup which is discrete for the product of the z_{i}-adic topologies. We say that Γ is separated if it is separated in the profinite topology, that is, if for every 1 ≠ g ∈ Γ there is a normal subgroup of finite index that does not contain g.

Proposition 4.27. Let Γ ⊆ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{i}) be a separated discrete subgroup. Then the quotient Γ\ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} is a locally noetherian, adic formal algebraic Spf k[[\underline{ξ}]]-stack locally formally of finite type. Moreover, the

1-morphism ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} → Γ\ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} is adic and étale.

Here we say that a formal algebraic Spf k[[\underline{ξ}]]-stack X (see [Har05, Definition A.5]) is J -adic for a sheaf of ideals J ⊂ O_{X}, if for some (any) presentation X → X the formal scheme X is J O_{X}-adic, that is, J^{r}O_{X} is an ideal of definition of X for all r. We then call J an ideal of definition of X. We say that X is locally formally of finite type if X is locally noetherian, adic, and if the closed substack defined by the largest ideal of definition (see [Har05, A.7]) is an algebraic stack locally of finite type over Spec k. Before proving the above proposition let us state the following lemma. Recall that (also an infinite dimensional) scheme is quasi-compact if and only if it is a finite union of affine schemes. Further recall that every morphism from a quasi-compact scheme to an ind-quasi-projective ind-scheme factors through a quasi-projective subscheme by [HV11, Lemma 5.4].

Lemma 4.28. Let Γ ⊆ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{i}) be a separated discrete subgroup. Let U_{i} ⊂ Fℓ_{ℙ_{i}} ˆ×_{Spec 𝔽_{i}} Spec k be a quasi-compact subscheme and set U = U_{1} ×_{k} … ×_{k} U_{n}. Then the set {γ ∈ Γ: γU ∩ U ≠ ∅} is finite.

Proof. Note that J_{\underline{𝕃}_{i}}(Q_{i}) is contained in LP_{i}(k^{alg}). By Theorem 4.4 any point \underline{x} ∈ U(k^{alg}) can be represented by a tuple (\underline{L}_{i}, g_{i})_{i}, where \underline{L}_{i} is a trivialized local ℙ_{i}-shtuka over k^{alg} and the quasi-isogeny g_{i}: \underline{L}_{i} → \underline{𝕃}_{i} is given by an element g_{i} ∈ LP_{i}(k^{alg}). By [PR08, Theorem 1.4] the projection LP_{i} → Fℓ_{ℙ_{i}} admits sections locally for the étale topology, and hence étale locally on Fℓ_{ℙ_{i}} the loop group LP_{i} is isomorphic to the product Fℓ_{ℙ_{i}} ×_{𝔽_{i}} L^{+}ℙ_{i}. In particular, by [EGA, IV_{2}, Proposition 2.7.1] the projection ∏_{i} LP_{i} ˆ×_{∏_{i} Fℓ_{ℙ_{i}}} U → U is an affine morphism of schemes and therefore Ũ := ∏_{i} LP_{i} ˆ×_{∏_{i} Fℓ_{ℙ_{i}}} U ⊆ ∏_{i} LP_{i} is a quasi-compact scheme. Consider the morphism

Ũ ×_{𝔽} Ũ ⟶ _{∏i} Fℓ_{ℙ_{i}}  (4.8)
( g_{i} , g′_{i})_{i} ⟼ ( g′_{i} g_{i}^{−1}L^{+}ℙ_{i} / L^{+}ℙ_{i})_{i}.

Since the Fℓ_{ℙ_{i}} are ind-quasi-projective ind-schemes also ∏_{i} Fℓ_{ℙ_{i}} is. Therefore (4.8) factors through some quasi-projective subscheme V ⊂ ∏_{i} Fℓ_{ℙ_{i}} by [HV11, Lemma 5.4]. Since \underline{𝕃}_{i} is decent the group of quasiisogenies J_{\underline{𝕃}_{i}}(Q_{i}) ⊂ LP_{i}(k^{alg}) is contained in LP_{i}(ℓ) for some finite field ℓ; see Remark 4.16. Let γ ∈ Γ be such that \underline{x} = (\underline{L}_{i}, g_{i})_{i} ∈ U and γ\underline{x} = (\underline{L}_{i}, γ_{i}g_{i}) ∈ U where γ_{i} ∈ LP_{i}(ℓ) is the projection of γ onto the i-th factor. Then ( g_{i} , γ_{i}g_{i})_{i} ∈ Ũ × Ũ and the image of γ under the projection map π: ∏_{i} LP_{i} → ∏_{i} Fℓ_{ℙ_{i}} lies in the finite set V (ℓ). Thus γ lies in the compact set S = π^{−1}(V (ℓ)) ∩ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{i}). On the other hand Γ is discrete and thus has finite intersection with S.

Proof of Proposition 4.27. By Lemma 4.25 there is a finite field ℓ_{i} and a constant d_{i} such that any ball in MẐ_{i} ; \underline{𝕃}_{i} with radius d_{i} contains a rational point in MẐ_{i} ; \underline{𝕃}_{i}(ℓ_{i}). Let d be the maximum of the integers d_{i} and let

ℓ be the compositum of the fields ℓ_{i}. Let x := (x_{i})_{i} be a point of ∏_{i} MẐ_{i} ; \underline{𝕃}_{i}(ℓ). We use the notation of the

proof of Theorem 4.18 and consider in particular the metric d̃_{i} on MẐ_{i} ; \underline{𝕃}_{i} defined as in (4.6) after choosing

some representation ι_{i} of ℙ_{i}. For any positive integer c we define the closed subscheme of (MẐ_{i} ; \underline{𝕃}_{i})_{red} given by

B(x_{i}, c) := \{z ∈ MẐ_{i} ; \underline{𝕃}_{i}: d̃_{i}(z, x_{i}) ≤ c\}.

Then every z ∈ B(x_{i}, c) satisfies d̃_{i}((\underline{𝕃}_{i}, id), z) ≤ d̃_{i}((\underline{𝕃}_{i}, id), x_{i}) + c =: m and so B(x_{i}, c) ⊂ (M_{m})_{red} = (M^{m})_{red}. Therefore Lemma 4.20 implies that B(x_{i}, c) is a quasi-projective scheme over Spec ℓ. In particular the set of ℓ-valued points B(x_{i}, c)(ℓ) is finite. Thus for all n ≥ 2d the subscheme

U_{n}^{2d}(x_{i}) := B(x_{i}, n) ∖ ⋃_{y∈B(x_{i},n+d)(ℓ), d̃_{i}(y,x_{i})>2d} B(y, d) ⊂ (MẐ_{i} ; \underline{𝕃}_{i})_{red}

is open in B(x_{i}, n + d) and quasi-compact. Note that for n ≥ 3d all U_{n}^{2d}(x_{i}) coincide with U_{3d}^{2d}(x_{i}) by an argument similar to (4.7). Since (MẐ_{i} ; \underline{𝕃}_{i})_{red} = lim_{⟶} B(x_{i}, n + d) carries the limit topology, the subscheme

U^{2d}(x_{i}) := U_{3d}^{2d}(x_{i}) ⊂ (MẐ_{i} ; \underline{𝕃}_{i})_{red} is open and quasi-compact. By Lemma 4.25 the union of the U^{2d}(x_{i}) for

all x_{i} ∈ MẐ_{i} ; \underline{𝕃}_{i}(ℓ) equals (MẐ_{i} ; \underline{𝕃}_{i})_{red}.

For x = (x_{i})_{i} set U_{x} := ∏_{i} U^{2d}(x_{i}). Then γ.U_{x} = U_{γ.x} and the open subsets U_{x} cover ∏_{i} MẐ_{i} ; \underline{𝕃}_{i}, for

varying x ∈ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i}(ℓ). Let I ⊂ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i}(ℓ) be a set of representatives of the Γ-orbits in ∏_{i} MẐ_{i} ; \underline{𝕃}_{i}(ℓ). Fix an x ∈ I. Since Γ is separated, we may choose by Lemma 4.28 a normal subgroup Γ′_{x} ⊂ Γ of finite index in Γ such that U_{x} ∩ γ′U_{x} = ∅ for all γ′ ≠ 1 in Γ′_{x}. For all γ ∈ Γ the natural morphism γU_{x} → Γ′_{x}\ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} is an open immersion. Let V_{x} be the (finite) union of the images of these morphisms

for all γ ∈ Γ. Then (Γ′_{x}\Γ)\V_{x} is an open substack of Γ\ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i}. Moreover, (Γ′_{x}\Γ)\V_{x} is a finite étale quotient of V_{x} and the map ∐_{γ∈Γ} γU_{x} → (Γ′_{x}\Γ)\V_{x} is adic and étale. Therefore the morphism

∏_{i} MẐ_{i} ; \underline{𝕃}_{i} → Γ\ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} is adic and étale above (Γ′_{x}\Γ)\V_{x}. Since (V_{x})_{red} is a scheme of finite type over k, its quotient ((Γ′_{x}\Γ)\V_{x})_{red} = (Γ′_{x}\Γ)\(V_{x})_{red} is an algebraic stack of finite type over k. Since the open

subsets U_{x} cover ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} also the (Γ′_{x}\Γ)\V_{x} cover Γ\ ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} and the proposition follows.

5 THE RELATION BETWEEN GLOBAL 𝔊-SHTUKAS AND LOCAL ℙ-SHTUKAS 28

Remark 4.29. For us it does not make sense to strengthen Proposition 4.27 like Rapoport and Zink [RZ96, Proposition 2.37] do, who obtain that for Rapoport–Zink spaces for p-divisible groups the quotient is a formal algebraic space if Γ is torsion free. Namely in our case all unipotent subgroups of ∏_{i} J_{\underline{𝕃}_{i}}(Q_{i}) are

torsion. So they cannot act fixed point free on ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} and the corresponding quotients cannot be formal algebraic spaces.

5 The Relation Between Global 𝔊-Shtukas and Local ℙ-Shtukas

5.1 Preliminaries on 𝔊-Torsors

In Chapter 5 we assume that 𝔊 is a flat affine group scheme of finite type over C. Let ν ∈ C be a closed point of C and set C′ := C ∖ {ν}. We let ℋ_{e}^{1}(C′, 𝔊) denote the category fibered in groupoids over the category of 𝔽_{q}-schemes, such that ℋ_{e}^{1}(C′, 𝔊)(S) is the full subcategory of [C′_{S}/𝔊](C′_{S}) consisting of those 𝔊-torsors over C′_{S} that can be extended to a 𝔊-torsor over the whole relative curve C_{S}. We denote by ˙( ) the restriction morphism

˙( ): ℋ ^{1}(C, 𝔊) ⟶ ℋ_{e}^{1}(C′, 𝔊)

which assigns to a 𝔊-torsor G over C_{S} the 𝔊-torsor Ġ := G ×_{C_{S}} C′_{S} over C′_{S}. Let ℙ̃_{ν} := Res_{𝔽_{ν}/𝔽_{q}} ℙ_{ν} and P̃_{ν} := Res_{𝔽_{ν}/𝔽_{q}} P_{ν} be the Weil restrictions. Then ℙ̃_{ν} is a flat affine group scheme of finite type over

Spec 𝔽_{q}[[z]]. We apply Definition 2.2 for 𝔽 = 𝔽_{q} and let ˆ ℙ̃_{ν} := ℙ̃_{ν} ˆ×_{Spec 𝔽_{q}[[z]]} Spf 𝔽_{q}[[z]] = Res_{𝔽_{ν}/𝔽_{q}} ℙ̂_{ν} be the ν-adic completion. We write A_{ν} ≅ 𝔽_{ν}[[z]] for a uniformizer z ∈ 𝔽_{q}(C). Then for every 𝔽_{q}-algebra R we have

A_{ν} ˆ⊗_{𝔽_{q}}R ≅ (R ⊗_{𝔽_{q}} 𝔽_{ν})[[z]] = R[[z]] ⊗_{𝔽_{q}} 𝔽_{ν} and
Q_{ν} ˆ⊗_{𝔽_{q}}R ≅ (R ⊗_{𝔽_{q}} 𝔽_{ν})((z)) = R((z)) ⊗_{𝔽_{q}} 𝔽_{ν} .

This implies that

L^{+}ℙ̃_{ν}(R) = ℙ̃_{ν}(R[[z]]) = ℙ_{ν}(A_{ν} ˆ⊗_{𝔽_{q}}R) and LP̃_{ν}(R) = P̃_{ν}(R((z))) = P_{ν}(Q_{ν} ˆ⊗_{𝔽_{q}}R).

If G ∈ ℋ ^{1}(C, 𝔊)(S), its completion Ĝ_{ν} := G ˆ×_{C_{S}}(Spf A_{ν} ˆ×_{𝔽_{q}} S) is a formal ℙ̂_{ν}-torsor (Definition 2.2) over

Spf A_{ν} ˆ×_{𝔽_{q}} S. The Weil restriction Res_{𝔽_{ν}/𝔽_{q}} Ĝ_{ν} is a formal ˆ ℙ̃_{ν}-torsor over Spf 𝔽_{q}[[z]] ˆ×_{𝔽_{q}} S and corresponds by Proposition 2.4 to an L^{+}ℙ̃_{ν}-torsor over S which we denote L_{ν}^{+}(G). We obtain the morphism

L_{ν}^{+} : ℋ ^{1}(C, 𝔊) ⟶ ℋ ^{1}(Spec 𝔽_{q}, L^{+}ℙ̃_{ν}) , G ↦ L_{ν}^{+}(G) .

Finally there is a morphism

L_{ν}: ℋ_{e}^{1}(C′, 𝔊)(S) ⟶ ℋ ^{1}(Spec 𝔽_{q}, LP̃_{ν})(S) , Ġ ↦ L_{ν}(Ġ)

which sends the 𝔊-torsor Ġ over C′_{S}, having some extension G over C_{S}, to the LP̃_{ν}-torsor L(L_{ν}^{+}(G)) associated with L_{ν}^{+}G under (2.1). We claim that L_{ν}(G) is independent of the extension G, and that we therefore may write L_{ν}(Ġ) := L(L_{ν}^{+}(G)). Indeed, let G′ be a second extension of Ġ and let f: Ġ ^{∼} ⟶ Ġ′ be an isomorphism of 𝔊-torsors over C′_{S}. Without loss of generality S = Spec R is affine. We may choose an fppf -covering Spec R̃ → C_{S} over which trivializations α: G ×_{C_{S}} Spec R̃ ^{∼} ⟶ 𝔊 ×_{C} Spec R̃ and α′: G′ ×_{C_{S}} Spec R̃ ^{∼} ⟶ 𝔊 ×_{C} Spec R̃ exist. Then α′ ∘ f ∘ α^{−1} equals multiplication with an element g ∈ 𝔊(Spec R̃ ×_{C} C′). The ring homomorphism 𝔽_{ν} → A_{ν}/ν^{n} = 𝔽_{ν}[[z]]/(z^{n}) and the short exact sequence 0 → 𝔽_{ν} ⟶^{z^{n}·} A_{ν}/ν^{n+1} → A_{ν}/ν^{n} → 0 of A-modules yield by tensoring with the flat A-algebra R̃ the vertical maps and the bottom row in the following commutative diagram of A_{ν}-modules

[figure: 0; R̃/ν; z^{n}·; (R̃/ν)[[z]]/(z^{n+1}); (R̃/ν)[[z]]/(z^{n}); 0; 0; R̃/ν; z^{n}·; R̃/ν^{n+1}; R̃/ν^{n}; 0 ,]

which proves by induction that (R̃/ν)[[z]]/(z^{n}) ^{∼} ⟶ R̃/ν^{n} and (R̃/ν)[[z]] ^{∼} ⟶ lim_{⟵} R̃/ν^{n}. The ν-adic com-

pletion α̂ of α is an isomorphism between Ĝ_{ν} ˆ×_{Spf(R⊗_{𝔽q} 𝔽_{ν})[[z]]} Spf(R̃/ν)[[z]] = G ˆ×_{C_{S}} Spf(R̃/ν)[[z]] and the trivial formal ℙ̂_{ν}-torsor (𝔊 ×_{C} Spec R̃) ˆ×_{Spec R̃} Spf(R̃/ν)[[z]] = ℙ̂_{ν} ˆ×_{Spf A_{ν}} Spf(R̃/ν)[[z]] over Spf(R̃/ν)[[z]]. For the base change (Res_{𝔽_{ν}/𝔽_{q}} Ĝ_{ν}) ˆ×_{R[[z]]} Spf(R̃/ν)[[z]] = ∏_{Gal(𝔽_{ν}/𝔽_{q})} Ĝ_{ν} ˆ×_{Spf(R⊗_{𝔽q} 𝔽_{ν})[[z]]} Spf(R̃/ν)[[z]] we obtain an isomorphism

∏_{Gal(𝔽_{ν}/𝔽_{q})} α̂ : (Res_{𝔽_{ν}/𝔽_{q}} Ĝ_{ν}) ˆ×_{R[[z]]} Spf(R̃/ν)[[z]] ^{∼} ⟶ ∏_{Gal(𝔽_{ν}/𝔽_{q})} 𝔊 ×_{C} Spf(R̃/ν)[[z]] = ˆ ℙ̃_{ν} ˆ×_{𝔽_{q}[[z]]} Spf(R̃/ν)[[z]]

which under Proposition 2.4 corresponds to an isomorphism

∏_{Gal(𝔽_{ν}/𝔽_{q})} α̂ : L_{ν}^{+}(G) ×_{R} Spec(R̃/ν) ^{∼} ⟶ (L^{+}ℙ̃_{ν})_{Spec(R̃/ν)}.

We also have the analogous isomorphism for Ĝ′_{ν} := G′ ˆ×_{C_{S}}(Spf A_{ν} ˆ×_{𝔽_{q}} S) and L_{ν}^{+}(G′). Under the ν-adic completion morphism, g is mapped to an element ĝ ∈ 𝔊((R̃/ν)((z))) = P_{ν}((R̃/ν)((z))). It yields the element (ĝ, …, ĝ) ∈ ∏_{Gal(𝔽_{ν}/𝔽_{q})} P_{ν}((R̃/ν)((z))) = (Res_{𝔽_{ν}/𝔽_{q}} P_{ν})((R̃/ν)((z))) = LP̃_{ν}(R̃/ν). The composition (∏_{Gal(𝔽_{ν}/𝔽_{q})} α̂′)^{−1} ∘ (ĝ, …, ĝ) ∘ ∏_{Gal(𝔽_{ν}/𝔽_{q})} α̂ defines an isomorphism L(L_{ν}^{+}(G)) ×_{R} Spec(R̃/ν) ^{∼} ⟶ L(L_{ν}^{+}(G′)) ×_{R} Spec(R̃/ν) which inherits the descent datum from f and defines the desired isomorphism L(L_{ν}^{+}(G)) ^{∼} ⟶ L(L_{ν}^{+}(G′)). This proves our claim that L_{ν}(G) is independent of the extension G. We write L_{ν}(Ġ) := L(L_{ν}^{+}(G)).

Lemma 5.1. The above maps assign to each 𝔊-torsor G over C_{S} a triple (Ġ, L_{ν}^{+}(G), φ) where φ: L_{ν}(Ġ) ^{∼} ⟶ L(L_{ν}^{+}(G)) is the canonical isomorphism of LP̃_{ν}-torsors. ℋ ^{1}(C, 𝔊)(S) is equivalent to the category of such triples. In other words, the following diagram of groupoids is cartesian

ℋ ^{1}(C, 𝔊)  \underrightarrow{˙( )}  ℋ_{e}^{1}(C′, 𝔊)
L_{ν}^{+}↓  ↓L_{ν}
ℋ ^{1}(Spec 𝔽_{q}, L^{+}ℙ̃_{ν}) \underrightarrow{L} ℋ ^{1}(Spec 𝔽_{q}, LP̃_{ν}) .

Proof. This follows from the glueing result of Beauville and Laszlo [BL95]. Let us give more details. We construct the inverse of the morphism

ℋ ^{1}(C, 𝔊) ⟶ ℋ_{e}^{1}(C′, 𝔊) ×_{ℋ ^{1}(Spec 𝔽_{q},LP̃_{ν})} ℋ ^{1}(Spec 𝔽_{q}, L^{+}ℙ̃_{ν}).

Over an 𝔽_{q}-scheme S we consider S-valued points Ġ ∈ ℋ_{e}^{1}(C′, 𝔊)(S) and L_{ν}^{+} ∈ ℋ ^{1}(Spec 𝔽_{q}, L^{+}ℙ̃_{ν})(S), as well as an isomorphism φ: L_{ν}(Ġ) ^{∼} ⟶ L(L_{ν}^{+}) in ℋ ^{1}(Spec 𝔽_{q}, LP̃_{ν})(S). Let G be an extension of Ġ to C_{S}. There exists an fppf -covering S′ → S and trivializations α̂: (L_{ν}^{+})_{S′} ^{∼} ⟶ (L^{+}ℙ̃_{ν})_{S′} and β̂: L_{ν}^{+}(G)_{S′} ^{∼} ⟶ (L^{+}ℙ̃_{ν})_{S′} of the pullbacks to S′. We may assume that S′ is the disjoint union of affine schemes of the form Spec R′.

We glue Ġ_{S′} via the isomorphism α̂ ∘ φ: L_{ν}(Ġ)_{S′} ^{∼} ⟶ (LP̃_{ν})_{S′} to (L^{+}ℙ̃_{ν})_{S′} as follows. Consider the algebraic torsor G ×_{C_{S}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′) for the group scheme

𝔊 ×_{C} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′) = ℙ_{ν} ×_{Spec A_{ν}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′)

over Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′). By [EGA, Proposition 2.7.1] it is affine of the form Spec B. Its ν-adic completion (Ĝ_{ν})_{R′} = G ˆ×_{C_{S}} Spf(A_{ν} ˆ⊗_{𝔽_{q}}R′) = Spf B̂ is affine with B̂ = lim_{⟵} B/ν^{m}B. Recall that L_{ν}^{+}(G)_{R′} is the L^{+}ℙ̃_{ν}-

torsor associated with the Weil restriction Res_{𝔽_{ν}/𝔽_{q}}(Ĝ_{ν})_{R′} by Proposition 2.4. The trivialization β̂ induces

a trivialization β̂: Res_{𝔽_{ν}/𝔽_{q}}(Ĝ_{ν})_{R′} ^{∼} ⟶ ˆ ℙ̃_{ν} ˆ×_{Spf 𝔽_{q}[[z]]} Spf R′[[z]] which is determined by the section

β̂^{−1}(1) ∈ Hom_{Spf R′[[z]]}(Spf R′[[z]] , Res_{𝔽_{ν}/𝔽_{q}}(Ĝ_{ν})_{R′})
= Hom_{Spf(A_{ν} ˆ⊗_{𝔽q} R′)}(Spf(A_{ν} ˆ⊗_{𝔽_{q}}R′) , Spf B̂)
= Hom_{A_{ν} ˆ⊗_{𝔽q} R′}^{cont}(B̂ , A_{ν} ˆ⊗_{𝔽_{q}}R′)
= Hom_{A_{ν} ˆ⊗_{𝔽q} R′}(B , A_{ν} ˆ⊗_{𝔽_{q}}R′)
= Hom_{Spec(A_{ν} ˆ⊗_{𝔽q} R′)}(Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′) , Spec B).

The latter induces a trivialization β: G ×_{C_{S}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′) ^{∼} ⟶ ℙ_{ν} ×_{Spec A_{ν}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′). Similarly the automorphism ψ̂ := α̂φβ̂^{−1} of (LP̃_{ν})_{R′} is determined by the image

ψ̂(1) ∈ LP̃_{ν}(R′) = P_{ν}(Spec Q_{ν} ˆ⊗_{𝔽_{q}}R′)

and thus induces an automorphism ψ of P_{ν} ×_{Spec Q_{ν}} Spec(Q_{ν} ˆ⊗_{𝔽_{q}}R′). By [BL95] we may glue Ġ_{R′} with ℙ_{ν} ×_{Spec A_{ν}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′) via the isomorphism ψβ to obtain a uniquely determined 𝔊-torsor G′ on C_{R′}.

We descend G′ to C_{S} as follows. Let S′′ = S′ ×_{S} S′ and let p_{i}: S′′ → S′ be the projection onto the i-th factor. Consider the element h := (p_{1}^{∗}α̂ ∘ p_{2}^{∗}α̂^{−1})(1) ∈ L^{+}ℙ̃_{ν}(S′′) = ℙ_{ν}(A_{ν} ˆ⊗_{𝔽_{q}}Γ(S′′, O_{S′′})). The isomorphism

( idĠ_{S′′} , h·): p_{2}^{∗}(Ġ_{S′}, ℙ_{ν} ×_{Spec A_{ν}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′), ψβ) ^{∼} ⟶ p_{1}^{∗}(Ġ_{S′}, ℙ_{ν} ×_{Spec A_{ν}} Spec(A_{ν} ˆ⊗_{𝔽_{q}}R′), ψβ)

induces a descent datum on G′ which is effective by [BLR90, § 6.1, Theorem 7] because G′ is affine over C_{S′}. Thus G′ descends to a 𝔊-torsor G ∈ ℋ ^{1}(C, 𝔊)(S). This defines the inverse morphism and finishes the proof.

5.2 The Global-Local Functor

Analogously to the functor which assigns to an abelian variety A over a ℤ_{p}-scheme its p-divisible group A[p^{∞}] we introduce a global-local functor from global 𝔊-shtukas to local ℙ_{ν_{i}}-shtukas. But whereas abelian varieties only have one characteristic place, our global 𝔊-shtukas have n characteristic places \underline{ν} = (ν_{1}, …, ν_{n}). So the global-local functor will assign to each global 𝔊-shtuka of characteristic \underline{ν} an n-tuple of local ℙ_{ν_{i}}-shtukas. We begin with a

Remark 5.2. Let ν be a place on C and let 𝔻_{ν} := Spec A_{ν} and 𝔻̂_{ν} := Spf A_{ν}. Let deg ν := [𝔽_{ν} : 𝔽_{q}] and fix an inclusion 𝔽_{ν} ⊂ A_{ν}. Assume that we have a section s: S → C which factors through Spf A_{ν}, that is, the image in O_{S} of a uniformizer of A_{ν} is locally nilpotent. In this case we have

𝔻̂_{ν} ˆ×_{𝔽_{q}} S ≅ ∐_{ℓ∈ℤ/(deg ν)} V(𝔞_{ν,ℓ}) ≅ ∐_{ℓ∈ℤ/(deg ν)} 𝔻̂_{ν,S},  (5.1)

where we write 𝔻̂_{ν,S} := 𝔻̂_{ν} ˆ×_{𝔽_{ν}} S and where we denote by V(𝔞_{ν,ℓ}) the component identified by the ideal 𝔞_{ν,ℓ} = ⟨a ⊗ 1 − 1 ⊗ s^{∗}(a)^{q^{ℓ}}: a ∈ 𝔽_{ν}⟩. Note that σ cyclically permutes these components and thus the 𝔽_{ν}- Frobenius σ^{deg ν} =: σ̂ leaves each of the components V(𝔞_{ν,ℓ}) stable. Also note that there are canonical isomorphisms V(𝔞_{ν,ℓ}) ≅ 𝔻̂_{ν,S} for all ℓ.

Although we will not need it in the sequel, we note the following interpretation of the component V(𝔞_{ν,0}).

Lemma 5.3. The section s: S → C induces an isomorphism of the component V(𝔞_{ν,0}) with the formal completion Ĉ_{S}^{Γ_{s}} of C_{S} along the graph Γ_{s} of s. In particular Ĉ_{S}^{Γ_{s}} is canonically isomorphic to 𝔻̂_{ν,S}.

Proof. We first consider the formal completion Ĉ_{A_{ν}}^{Γs̃} of C_{A_{ν}} along the graph Γs̃ of the morphism s̃: Spec A_{ν} → C. Let Spec A ⊂ C be a neighborhood of ν such that a uniformizing parameter z of C at ν lies in A. Then I = (a ⊗ 1 − 1 ⊗ a: a ∈ A) ⊂ A ⊗_{𝔽_{q}} A_{ν} is the ideal defining Γs̃ and Ĉ_{A_{ν}}^{Γs̃} = Spf lim_{⟵}(A ⊗_{𝔽_{q}} A_{ν})/I^{n}. The module I/I^{2} is free of rank one over A_{ν} = (A ⊗_{𝔽_{q}} A_{ν})/I since C is a

smooth curve over 𝔽_{q}. We claim that I/I^{2} is generated by z ⊗ 1 − 1 ⊗ z. Let a ∈ A, let m ∈ 𝔽_{q}(z)[X] be the minimal polynomial of a over 𝔽_{q}(z), and multiply it with the least common denominator to obtain the polynomial F (X, z) ∈ 𝔽_{q}[X, z]. Note that the least common denominator lies in OC,ν^{×} because a is integral over 𝔽_{q}[z] near ν. In A ⊗_{𝔽_{q}} A_{ν}[X] we use the abbreviations ζ := 1 ⊗ z and α := 1 ⊗ a and we consider the two-variable Taylor expansion of F at (α, ζ)

F (X, z ⊗ 1) ≡ F (α, ζ) + \frac{∂F}{∂X} (α, ζ) · (X − α) + \frac{∂F}{∂z} (α, ζ) · (z ⊗ 1 − ζ) mod I^{2}.

Plugging in a ⊗ 1 for X yields F (a ⊗ 1, z ⊗ 1) = 0 in addition to F (α, ζ) = 0. Since A is unramified over 𝔽_{q}[z] at ν we have \frac{∂F}{∂X} (α, ζ) ∈ Aν^{×}. This shows that z ⊗ 1 − 1 ⊗ z generates the A_{ν}-module I/I^{2}.

By Nakayama’s Lemma [Eis95, Corollary 4.7] there is an element f ∈ 1 + I that annihilates the A ⊗_{𝔽_{q}} A_{ν}-module I/(z ⊗ 1 − 1 ⊗ z). We may replace Spec(A ⊗_{𝔽_{q}} A_{ν}) by the principal open subset Spec(A ⊗_{𝔽_{q}} A_{ν})[\frac{1}{f}] which contains the graph Γs̃ and on which I is generated by the non-zero divisor z ⊗ 1 − 1 ⊗ z. This implies that I^{n}/I^{n+1} is a free A_{ν}-module with generator (z ⊗ 1 − 1 ⊗ z)^{n}. Therefore the morphism A_{ν}[[t]] → lim_{⟵}(A ⊗_{𝔽_{q}} A_{ν})/I^{n}, t ↦ z ⊗ 1 − 1 ⊗ z induces an isomorphism on the associated graded rings, and hence is itself an isomorphism by [AM69, Lemma 10.23].

Now observe that V(𝔞_{ν,0}) is the formal scheme on the topological space S with structure sheaf O_{S}[[z]], and that s identifies the topological spaces S and Γ_{s}. Under base change to S this implies that the formal completion Ĉ_{S}^{Γ_{s}} has structure sheaf O_{S}[[t]] = O_{S}[[z − ζ]], where we write t = z − ζ. Since ζ is locally nilpotent in O_{S}, the latter is isomorphic to O_{S}[[z]] proving the lemma.

Definition 5.4. Fix a tuple \underline{ν} := (ν_{i})_{i=1...n} of places on C with ν_{i} ≠ ν_{j} for i ≠ j. Let A\underline{ν} be the completion of the local ring O_{C^{n},\underline{ν}} of C^{n} at the closed point \underline{ν}, and let 𝔽\underline{ν} be the residue field of the point \underline{ν}. Then 𝔽\underline{ν} is the compositum of the fields 𝔽_{ν_{i}} inside 𝔽_{q}^{alg}, and A\underline{ν} ≅ 𝔽\underline{ν}[[ζ_{1}, …, ζ_{n}]] where ζ_{i} is a uniformizing parameter of C at ν_{i}. Let the stack

∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} := ∇_{n}ℋ ^{1}(C, 𝔊) ˆ×_{C^{n}} Spf A\underline{ν}

be the formal completion of the stack ∇_{n}ℋ ^{1}(C, 𝔊) along \underline{ν} ∈ C^{n}. Although we will not need it in this article, the reader should note that ∇_{n}ℋ ^{1}(C, 𝔊) is an ind-algebraic stack over Spf A\underline{ν} which is ind-separated and locally of ind-finite type by [AH13, Theorem 3.15]. Set ℙ_{ν_{i}} := 𝔊 ×_{C} Spec A_{ν_{i}} and ℙ̂_{ν_{i}} := 𝔊 ×_{C} Spf A_{ν_{i}}.

Let (G, s_{1}, …, s_{n}, τ) ∈ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S), that is, s_{i}: S → C factors through Spf A_{ν_{i}}. By Remark 5.2 we may decompose

G ˆ×_{C_{S}}(Spf A_{ν_{i}} ˆ×_{𝔽_{q}} S) ≅ ∐_{ℓ∈ℤ/(deg ν_{i})} G ˆ×_{C_{S}} V(𝔞_{ν_{i},ℓ})

into a finite product with components G ˆ×_{C_{S}} V(𝔞_{ν_{i},ℓ}) ∈ ℋ ^{1}(𝔻̂_{ν_{i}}, ℙ̂_{ν_{i}}). Using Proposition 2.4 we view (G ˆ×_{C_{S}} V(𝔞_{ν_{i},0}), τ^{deg ν_{i}}) as a local ℙ_{ν_{i}}-shtuka over S, where τ^{deg ν_{i}}: (σ^{deg ν_{i}})^{∗}L_{i} ^{∼} ⟶ L_{i} is the 𝔽_{ν_{i}}-Frobenius on the loop group torsor L_{i} associated with G ˆ×_{C_{S}} V(𝔞_{ν_{i},0}). We define the global-local functor \underline{Γ̂} by

Γ̂_{ν_{i}}: ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) ⟶ Sht_{ℙ_{ν_{i}}}^{Spec A_{ν_{i}}} (S) , (G, τ) ⟼ (G ˆ×_{C_{S}} V(𝔞_{ν_{i},0}), τ^{deg ν_{i}}) and
\underline{Γ̂} := ∏_{i} Γ̂_{ν_{i}}: ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) ⟶ ∏_{i} Sht_{ℙ_{ν_{i}}}^{Spec A_{ν_{i}}} (S) .  (5.2)

Note that Γ̂_{ν_{i}} and \underline{Γ̂} also transform quasi-isogenies into quasi-isogenies, as can be seen by tracing through the proof of Lemma 5.1.

Remark 5.5. Consider the preimages in V(𝔞_{ν_{i},ℓ}) of the graphs Γ_{s_{j}} ⊂ C_{S} of s_{j}. Since ν_{i} ≠ ν_{j} for i ≠ j the preimage of Γ_{s_{j}} is empty for j ≠ i. Also the preimage of Γ_{s_{i}} equals V(𝔞_{ν_{i},0}) and does not meet V(𝔞_{ν_{i},ℓ}) for ℓ ≠ 0. Thus for ℓ ≠ 0 the restriction of τ to V(𝔞_{ν_{i},ℓ}) is an isomorphism

τ × id: σ^{∗}(G ˆ×_{C_{S}} V(𝔞_{ν_{i},ℓ−1})) = (σ^{∗}G) ˆ×_{C_{S}} V(𝔞_{ν_{i},ℓ}) ^{∼} ⟶ G ˆ×_{C_{S}} V(𝔞_{ν_{i},ℓ}) .  (5.3)

This allows to recover (G, τ) ˆ×_{C_{S}}(Spf A_{ν_{i}} ˆ×_{𝔽_{q}} S) from (G ˆ×_{C_{S}} V(𝔞_{ν_{i},0}), τ^{deg ν_{i}}) via the isomorphism

∏_{ℓ}(τ^{ℓ} ˆ× id): (∏_{ℓ} σ^{ℓ∗}(G ˆ×_{C_{S}} V(𝔞_{ν_{i},0})), ( 0 τ^{deg ν_{i}} 1 1 0 )) ^{∼} ⟶ (G, τ) ˆ×_{C_{S}}(Spf A_{ν_{i}} ˆ×_{𝔽_{q}} S).  (5.4)

Recall from Section 5.1 that the Weil restriction Res_{𝔽_{ν_{i}}/𝔽_{q}} Ĝ_{ν_{i}} of the torsor Ĝ_{ν_{i}} := G ˆ×_{C_{S}}(Spf A_{ν_{i}} ˆ×_{𝔽_{q}} S) corresponds by Proposition 2.4 to an L^{+}ℙ̃_{ν_{i}}-torsor L_{ν_{i}}^{+}(G). Then (L_{ν_{i}}^{+}(G), τ ˆ× id) is a local ℙ̃_{ν_{i}}-shtuka over S. We call it the local ℙ̃_{ν_{i}}-shtuka associated with \underline{G} at the place ν_{i}. By equation (5.4) there is an equivalence between the category of local ℙ_{ν_{i}}-shtukas over schemes S ∈ Nilp_{A_{ν_{i}}} and the category of local ℙ̃_{ν_{i}}-shtukas over S for which the Frobenius τ is an isomorphism outside V(𝔞_{ν_{i},0}). (Compare also [BH11, Proposition 8.8].)

Remark 5.6. Note that in a similar way one can associate a local ℙ̃_{ν}-shtuka L_{ν}^{+}(\underline{G}) with a global 𝔊-shtuka \underline{G} = (G, τ) at a place ν outside the characteristic places ν_{i}. Namely L_{ν}^{+}(\underline{G}) is the local ℙ̃_{ν}- shtuka associated with Res_{𝔽_{ν}/𝔽_{q}} (\underline{G} ˆ×_{C_{S}}(Spf A_{ν} ˆ×_{𝔽_{q}} S)) by Proposition 2.4. It is étale because τ is an isomorphism at ν. We call L_{ν}^{+}(\underline{G}) the étale local ℙ̃_{ν}-shtuka associated with \underline{G} at the place ν ∉ \underline{ν}. In [AH13, Chapter 6] it will become useful for considerations of Tate-modules (Definition 3.5).

For this purpose we write A_{ν} ≅ 𝔽_{ν}[[z]]. For every representation ρ: ℙ_{ν} → GL_{r,A_{ν}} in Rep_{A_{ν}} ℙ_{ν} we consider the representation ρ̃ ∈ Rep_{𝔽_{q}[[z]]} ℙ̃_{ν} which is the composition of Res_{𝔽_{ν}/𝔽_{q}}(ρ): ℙ̃_{ν} → Res_{𝔽_{ν}/𝔽_{q}} GL_{r,A_{ν}} followed by the natural inclusion Res_{𝔽_{ν}/𝔽_{q}} GL_{r,A_{ν}} ⊂ GL_{r·[𝔽_{ν}:𝔽_{q}] , 𝔽_{q}[[z]]}. We set \underline{L̃} = L_{ν}^{+}(\underline{G}) and define Ť\underline{L̃}(ρ) := Ť\underline{L̃}(ρ̃) := Ť_{ρ̃_{∗}\underline{L̃}}. Then there is a canonical isomorphism Ť\underline{L̃}(ρ) ≅ lim_{⟵_{n}} ρ_{∗}(\underline{G} ×_{C} Spec A_{ν}/(ν^{n}))^{τ} of

A_{ν}-modules. This will be used in [AH13, Chapter 6].

If 𝔽_{ν} ⊂ O_{S} there also exists the decomposition (5.1) and we can associate a local ℙ_{ν}-shtuka \underline{L} with L_{ν}^{+}(\underline{G}). The main difference to Definition 5.4 and Remark 5.5 is that there is no distinguished component of G ˆ×_{C_{S}}(Spf A_{ν} ˆ×_{𝔽_{q}} S), like the one given by the characteristic section at ν_{i}. But τ induces isomorphisms between all components as in (5.3). Therefore we may take any component and the associated local ℙ_{ν}- shtuka \underline{L}. Equation (5.4) shows that over 𝔽_{ν}-schemes S we obtain an equivalence between the category of étale local ℙ̃_{ν}-shtukas and the category of étale local ℙ_{ν}-shtukas. If ℙ_{ν} = 𝔊 ×_{C} 𝔻_{ν} is smooth with connected special fiber, the same is true for ℙ̃_{ν} by [CGP10, Prop. A.5.9], and then Corollary 2.9 applies also to étale local ℙ̃_{ν}-shtukas. There is also a canonical isomorphism of Tate functors Ť\underline{L} = Ť_{L_{ν}^{+}(\underline{G})}; compare [BH11, Proposition 8.5] for more details.

Like abelian varieties also global 𝔊-shtukas can be pulled back along quasi-isogenies of their associated local ℙ-shtukas as follows.

Proposition 5.7. Let \underline{G} ∈ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) be a global 𝔊-shtuka over S and let ν ∈ C be a place. Let L_{ν}^{+}(\underline{G}) be the local ℙ̃_{ν}-shtuka associated with \underline{G} at ν in the sense of Remark 5.5 (if ν ∈ \underline{ν}), respectively Remark 5.6 (if ν ∉ \underline{ν}). Let f̃: \underline{L̃}′_{ν} → L_{ν}^{+}(\underline{G}) be a quasi-isogeny of local ℙ̃_{ν}-shtukas over S. If ν ∈ \underline{ν} we assume that the Frobenius of \underline{L̃}′_{ν} is an isomorphism outside V(𝔞_{ν,0}). If ν ∉ \underline{ν} we assume that \underline{L̃}′_{ν} is étale. Then there exists a unique global 𝔊-shtuka \underline{G}′ ∈ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) and a unique quasi-isogeny g: \underline{G}′ → \underline{G} which is an isomorphism outside ν, such that the local ℙ̃_{ν}-shtuka associated with \underline{G}′ is \underline{L̃}′_{ν} and the quasi-isogeny of local ℙ̃_{ν}-shtukas induced by g is f̃. We denote \underline{G}′ by f̃^{∗}\underline{G}.

Remark 5.8. Note that if ν ∈ \underline{ν} then by Remark 5.5 there is an equivalence between the category of local ℙ̃_{ν}-shtukas over S for which the Frobenius τ is an isomorphism outside V(𝔞_{ν,0}) and the category of local ℙ_{ν}-shtukas over S. In particular, if Γ̂_{ν}(\underline{G}) is the local ℙ_{ν}-shtuka associated with \underline{G} in Definition 5.4, then every isogeny f: \underline{L}′_{ν} → Γ̂_{ν}(\underline{G}) corresponds under this equivalence to an isogeny f̃: \underline{L̃}′_{ν} → L_{ν}^{+}(\underline{G}) as in the proposition. We obtain a global 𝔊-shtuka f̃^{∗}\underline{G} which we also denote by f^{∗}\underline{G}. It satisfies Γ̂_{ν}(f^{∗}\underline{G}) = \underline{L}′_{ν} and Γ̂_{ν}(g: f^{∗}\underline{G} → \underline{G}) = f.

Proof of Proposition 5.7. Let us set \underline{G} := (G, τ). Let (Ġ, L_{ν}^{+}(G), φ) be the triple for the place ν associated with the 𝔊-torsor G by Lemma 5.1. Thus L_{ν}^{+}(\underline{G}) = (L_{ν}^{+}(G), τ). We also set \underline{L̃}′_{ν} = (L̃′_{ν}, τ′). Now the triple (Ġ_{,}L̃′_{ν}, f̃^{−1}φ) defines a 𝔊-bundle G′ over C_{S} which coincides with G over C′_{S} and inherits the Frobenius automorphism τ from \underline{G} over C′_{S} ∖ ⋃_{i} Γ_{s_{i}}. If ν ∉ \underline{ν} then this τ extends to an isomorphism over \{ν\} ×_{𝔽_{q}} S because \underline{L̃}′_{ν} is étale. If ν ∈ \underline{ν} then τ extends to an isomorphism over C_{S} ∖ ⋃_{i} Γ_{s_{i}} because τ′ is an isomorphism outside V(𝔞_{ν,0}). This defines the global 𝔊-shtuka \underline{G}′ ∈ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S). The quasi-isogeny g is obtained from the identification Ġ′ = Ġ. It has the desired properties.

Finally we want to prove rigidity for quasi-isogenies of global 𝔊-shtukas. This is the global counterpart of Proposition 2.11 and fits into the analogy between abelian varieties and global 𝔊-shtukas. It only holds over schemes S ∈ Nilp_{A\underline{ν}}, similarly to rigidity for abelian varieties which only holds over schemes S ∈ Nilp_{ℤ_{p}}.

Proposition 5.9. (Rigidity of quasi-isogenies for global 𝔊-shtukas) Let S be a scheme in Nilp_{A\underline{ν}} and let j: S̄ → S be a closed immersion defined by a sheaf of ideals I which is locally nilpotent. Let \underline{G} = (G, τ) and \underline{G}′ = (G′, τ′) be two global 𝔊-shtukas over S. Then

QIsog_{S}(\underline{G}, \underline{G}′) ⟶ QIsogS̄(j^{∗}\underline{G}, j^{∗}\underline{G}′), f ↦ j^{∗}f

is a bijection of sets. f is an isomorphism at a place ν ∉ \underline{ν} if and only if j^{∗}f is an isomorphism at ν.

Note that the last assertion need not be true for places ν ∈ \underline{ν}. This is similar to lifts of quasi-isogenies between abelian varieties over schemes S ∈ Nilp_{ℤ_{p}} which can acquire additional “poles” at p.

Proof of Proposition 5.9. It suffices to treat the case where I^{q} = (0). In this case the morphism σ_{S} factors through j: S̄ → S

σ_{S} = j ∘ σ′: S ⟶^{σ′} S̄ ⟶^{j} S.

Since C_{S} ∖ ∪_{i}Γ_{s_{i}} ⊃ C_{S} ∖ \underline{ν} ×_{𝔽_{q}} S, the morphism τ defines a quasi-isogeny τ: σ^{′∗}j^{∗}\underline{G} = σ_{S}^{∗}\underline{G} → \underline{G} which is an isomorphism outside \underline{ν} ×_{𝔽_{q}} S and similarly for \underline{G}′. We fix a finite closed subset D ⊂ C which contains all ν_{i} and consider quasi-isogenies which are isomorphisms outside D. Then the following diagram

G|_{C_{S}∖D_{S}}  ⟶_{≅}^{f} G′|_{C_{S}∖D_{S}}
τ↑≅  _{τ′}↑≅
σ_{S}^{∗}G|_{C_{S}∖D_{S}} ⟶_{≅}^{σ^{′∗}(j^{∗}f)} σ_{S}^{∗}G′|_{C_{S}∖D_{S}}

allows to recover f from j^{∗}f and this proves the bijectivity.

If ν ∉ \underline{ν} we take a subset D ⊂ C which does not contain ν. Chasing through the diagram again shows that f is an isomorphism at ν if and only if j^{∗}f is an isomorphism at ν.

5.3 The Analog of the Serre–Tate Theorem

The Serre–Tate Theorem relates the deformation theory of an abelian variety in characteristic p with the deformation theory of the associated p-divisible group. In this section we introduce the analogous situation over function fields and prove the analogous theorem relating the deformation theory of a global 𝔊-shtuka to the deformation theory of the associated n-tuple of local ℙ_{ν_{i}}-shtukas via the global-local functor.

Let S be in Nilp_{A\underline{ν}} and let j: S̄ → S be a closed subscheme defined by a locally nilpotent sheaf of ideals I. Let ¯ \underline{G} be a global 𝔊-shtuka in ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S̄). The category Defo_{S}(¯\underline{G}) of lifts of ¯ \underline{G} to S consists of all pairs (\underline{G}, α: j^{∗}\underline{G} ^{∼} ⟶ ¯ \underline{G}) where \underline{G} belongs to ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S), where α is an isomorphism of global 𝔊-shtukas over S̄, and where morphisms are isomorphisms between the \underline{G}’s that are compatible with the α’s.

Similarly for a local ℙ-shtuka \underline{L̄} in Sht_{ℙ}^{𝔻}(S̄) we define the category of lifts Defo_{S}(\underline{L̄}) of \underline{L̄} to S. Notice that according to the rigidity of quasi-isogenies (Propositions 5.9 and 2.11) all Hom-sets in these categories contain at most one element.

Theorem 5.10. Let ¯ \underline{G} := (Ḡ, τ̄) be a global 𝔊-shtuka in ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S̄). Let (\underline{L̄}_{i})_{i} = ˆ \underline{Γ}(¯\underline{G}). Then the functor

Defo_{S}(¯\underline{G}) ⟶ ∏_{i} Defo_{S}(\underline{L̄}_{i}) , (\underline{G}, α) ⟼ (\underline{Γ̂}(\underline{G}), \underline{Γ̂}(α))

induced by the global-local functor, is an equivalence of categories.

Proof. We proceed by constructing the inverse of the above functor. It suffices to treat the case where I^{q} = (0). In this case the morphism σ_{S} factors through j: S̄ → S

σ_{S} = j ∘ σ′: S ⟶^{σ′} S̄ ⟶^{j} S.

Let (\underline{L}_{i}, α̂_{i}: j^{∗}\underline{L}_{i} ^{∼} ⟶ \underline{L̄}_{i})_{i} be an object of ∏_{i} Defo_{S}(\underline{L̄}_{i}). Consider the global 𝔊-shtuka \underline{G̃}:= (G̃, τ̃) := σ^{′∗}\underline{Ḡ} over S. Since C_{S} ∖ ∪_{i}Γ_{s_{i}} ⊃ C_{S} ∖ \underline{ν} ×_{𝔽_{q}} S, the morphism τ̄ defines a quasi-isogeny τ̄: j^{∗}\underline{G̃} → ¯ \underline{G} which is an isomorphism outside the graphs of the characteristic sections as one sees from the following diagram

σ_{S}^{∗}j^{∗}G̃  (σ_{S}^{2})^{∗}Ḡ ⟶^{σ_{S}^{∗} τ̄} σ_{S}^{∗}Ḡ
j^{∗}τ̃↓  ^{σ_{S}^{∗} τ̄}↓  τ̄↓
j^{∗}G̃  σ_{S}^{∗}Ḡ ⟶^{τ̄} Ḡ .

We write \underline{Γ̂}(\underline{G̃}) = (\underline{L̃}_{i})_{i} and \underline{L̄}_{i} = (L̄_{i}, ˆ τ̄_{i}). We compose ˆ τ̄_{i}^{−1} with α̂_{i} to obtain the quasi-isogenies ˆγ̄_{i} := ˆ τ̄_{i}^{−1} ∘ α̂_{i}: j^{∗}\underline{L}_{i} → j^{∗}\underline{L̃}_{i}. By rigidity of quasi-isogenies (Proposition 2.11) they lift to quasi-isogenies γ̂_{i}: \underline{L}_{i} → \underline{L̃}_{i} with j^{∗}γ̂_{i} = ˆ γ̄_{i}. We put \underline{G} := γ̂_{n}^{∗} ∘ … ∘ γ̂_{1}^{∗}\underline{G̃} (see Remark 5.8) and recall that there is a quasi-isogeny γ: \underline{G} → \underline{G̃} of global 𝔊-shtukas with \underline{Γ̂}(γ) = (γ̂_{i})_{i} which is an isomorphism outside \underline{ν}, see Proposition 5.7. We may now define the functor

∏_{i} Defo_{S}(\underline{L̄}_{i}) → Defo(¯\underline{G})

by sending (\underline{L}_{i}, α̂_{i}: j^{∗}\underline{L}_{i} → \underline{L̄}_{i})_{i} to (\underline{G}, τ̄ ∘ j^{∗}γ). The quasi-isogeny α := τ̄ ∘ j^{∗}γ is an isomorphism outside the graphs of the s_{i} by construction, and also at these graphs because \underline{Γ̂}(α) = (α̂_{i})_{i}. It can easily be seen by the above construction that these functors are actually inverse to each other.

REFERENCES 35

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Esmail Arasteh Rad

Universität Münster

Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany Urs Hartl Universität Münster Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany www.math.uni-muenster.de/u/urs.hartl/