Langlands-Rapoport Conjecture Over Function Fields

Esmail Arasteh Rad and Urs Hartl

May 9, 2016

Abstract

In this article we prove the analogue of the Langlands-Rapoport conjecture for the moduli stacks of global G-shtukas. Here G is a parahoric Bruhat-Tits group scheme over a smooth projective curve C over a finite field 𝔽_{q}. Mathematics Subject Classification (2000): 11G09, (11G18, 14L05)

Contents

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

2 Moduli Stack of G-Shtukas as a moduli for (G−)motives 4 2.1 G-Shtukas and G-Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 ℙ-Shtukas and ℙ-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Crystalline Realization Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Etale Realization Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Poincare-Weil Theorem for C-Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Langlands-Rapoport Conjecture 13 3.1 The ∇ℋ -data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Langlands-Rapoport Conjecture for the Moduli Stack of Global G-Shtukas . . . . . . . . . . . . 18

References 21

1 Introduction

According to Langlands’s philosophy it is highly desirable to understand the cohomology of Shimura varieties, as they naturally carry both Hecke and Galois actions. Thus, in particular, it is interesting to ask about the possible descriptions of the zeta functions of these varieties. Concerning this, Langlands made a conjecture about the structure of mod p points of Shimura varieties, from which the expression of the zeta function of a Shimura variety as a product of automorphic L-functions follows. The conjecture was made more precise by Kottwitz [Kot90] and consequently refined by Langlands and Rapoport [LR87]. This conjecture provides a conceptual group theoretic description of mod p points of Shimura varieties. As an evidence for the conjecture, Langlands and Rapoport proved the conjecture for simple Shimura varieties of PEL types A and C, under the assumption of Grothendieck’s standard conjectures and the Tate conjecture for varieties over finite fields, and the Hodge conjecture for abelian varieties. Following work of many authors, including Morita, Milne, Pfau, Deligne, Reimann, Zink, Kottwitz [Kot92] proved the conjecture for PEL Shimura varieties of type A and C and according to technical issues arising in the general set up, Kisin, considerably later, proved the conjecture for Shimura varieties of abelian type with hyperspecial level structure [Kis13]. Recall that the Shimura varieties of abelian type are exactly those which fulfill the Deligne conception of Shimura varieties [Del79]. Indeed, concerning the issue of existence of canonical integral model for Shimura varieties, this assumption is essential even for stating the conjecture.

In this article we formulate the Langlands-Rapoport conjecture for the moduli stacks of global G-shtukas. We further implement the local theory of global G-shtukas and their uniformization theory, which we developed in [AH14a] and [AH14b], to establish a proof of the conjecture in the following general set up. Let C be a smooth projective, geometrically irreducible curve over a finite field 𝔽_{q} with q elements, and let Q = 𝔽_{q}(C) be its function field. Fix an n-tuple \underline{ν} = (ν_{1}, …, ν_{n}) of pairwise different closed points ν_{i} ∈ C. In the situation of Shimura varieties, ℤ and p are the analogue of our C and \underline{ν}. Consider a parahoric (Bruhat-Tits) group scheme G over C in the sense of [BT72, Définition 5.2.6] and [HR03]. According to the analogy between function fields and number fields, the moduli stacks of global G-shtukas appear as the function field counterparts of (the canonical integral model for) Shimura varieties. In analogy with the theory of Shimura varieties, we introduce ∇ℋ -data.

A ∇ℋ -data is a pair (G, Ẑ\underline{ν}) where G is a parahoric group scheme over the curve C and Ẑ\underline{ν} = (Ẑ_{ν_{i}})_{i} is an n-tuple of bounds Ẑ_{ν_{i}} which are closed subschemes of certain formal completion F̂ℓ_{ℙ_{ν_{i}}} of the twisted affine flag variety Fℓ_{ℙ_{ν_{i}}}, associated with the parahoric group ℙ_{ν_{i}}; see [PR08]. Here ℙ_{ν_{i}} denote the parahoric group obtained by base change of G to Spec Â_{ν_{i}}, the spectrum of the completion of the stalk O_{C,ν_{i}} at ν_{i}. To such a

pair (G, Ẑ\underline{ν}) one may associate a tower ∇_{n}^{∗,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} of formal algebraic stacks, which are moduli spaces for global G-shtukas bounded by Ẑ\underline{ν} with fixed characteristics \underline{ν} (also called paws by V. Lafforgue [Laf12]). This tower of formal algebraic stacks is defined over a reflex ring R := R(G, Ẑ\underline{ν}) with finite residue field κ := κ(G, Ẑ\underline{ν}) and naturally carries an action of the adelic group G(𝔸\underline{ν}), that operates through Hecke correspondences. These formal algebraic stacks may be viewed as the function field counterparts of integral models for Shimura varieties and the parahoric case may reflect the bad reduction at the characteristic place.

On the other hand, to such ∇ℋ -data, we associate a groupoid L^{∗}(G, Ẑ\underline{ν}) := ⨆_{M_{G}} 𝒮 (M_{G}), where the index M_{G} runs over a set of equivalence classes of G-motives and each groupoid 𝒮 (M_{G}) is naturally equipped with an action of the adelic group G(𝔸\underline{ν}) which again operates through Hecke correspondences, with the action of the Frobenius Φ in Gal(\overline{𝔽}/κ), and with the action of Z(Q) where Z ⊂ G is the center; see Definition 3.9 and Remark 3.10. In the following theorem we establish an equivalence of the functors L^{∗}(−) and ∇_{n}^{∗,−}ℋ ^{1}(C, −)\underline{ν}(\overline{𝔽}).

Theorem 3.11. (Langlands-Rapoport Conjecture For G-shtukas) There exist a canonical G(𝔸\underline{ν} ; Q) × Z(Q)- equivariant isomorphism of functors

L^{∗}(−) ^{∼} ⟶ ∇_{n}^{∗,−}ℋ ^{1}(C, −)\underline{ν}(𝔽).

on the category of ∇ℋ -data. Moreover via this isomorphism, the operation \underline{Φ} on the left hand side of the above isomorphism corresponds to the Frobenius endomorphism σ on the right hand side. Furthermore, when G (the generic fiber of G) is semi-simple, one may even replace L^{∗}(−) by L_{adm}^{∗}(−).

For the definition of the functor L_{adm}^{∗}(−) see Definition 3.9.

As an advantage of working over function fields one may observe that 𝒮 (M_{G}) are not only sets, but in fact, they are special fibers of formal algebraic stacks, which are certain quotients of Rapoport-Zink spaces for local ℙ-shtukas. As an application of our theorem, one may use this fact together with the theory of nearby cycles to derive a decomposition for the cohomology of the moduli stack of global G-shtukas.

The organization of the article is as follows. In section 2 we explain how one views the moduli stack of global G-shtukas as a moduli for motives. This is done in the following way. First, we introduce the realization categories in subsection 2.2 and crystalline (resp. étale) realization functors at characteristic places (resp. away from characteristic places) in subsection 2.3 (resp. 2.4). Then in subsection 2.5 we show that the kernel of the motivic groupoid 𝔓 associated with a refined version of the category of Anderson t-motives is pro-reductive, see Theorem 2.29, and further we observe that the category of G-motives, consisting of morphisms 𝔓 → 𝔊_{G} to the neutral groupoid 𝔊_{G} is equivalent with the category of G-shtukas (with morphisms enlarged to quasiisogenies); see 2.30. In section 3 we first discuss ∇ℋ -data which are the function field analogues of Shimura data. Then in subsection 3.2 we state and prove the Langlands-Rapoport conjecture for the moduli stacks of global G-shtukas. Note however that most of the way to the proof of the main theorem 3.11 has been paved according to our results in the previous articles [AH14a] and [AH14b].

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, 𝔽 a finite field containing 𝔽_{q}, \overline{𝔽} an algebraic closure of 𝔽, Â_{ν} the completion of the stalk O_{C,ν} at ν, Q̂_{ν} := Frac(Â_{ν}) its fraction field, Â:= 𝔽[[z]] the ring of formal power series in z with coefficients in 𝔽 , Q̂ := Frac(Â) its fraction field, 𝔸\underline{ν} the ring of integral adeles of C outside \underline{ν}, 𝔸\underline{ν} ; Q := 𝔸\underline{ν} ⊗_{O_{C}} Q the ring of adeles of C outside \underline{ν},

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,

Â\underline{ν}  the completion of the local ring O_{C^{n},\underline{ν}} of C^{n} at the closed point \underline{ν} = (ν_{i}),

NilpÂ\underline{ν} := Nilp_{Spf Â\underline{ν}} the category of schemes over C^{n} on which the ideal defining the closed point \underline{ν} ∈ C^{n} is locally nilpotent,

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}.

ℙ a smooth affine group scheme of finite type over 𝔻 = Spec 𝔽[[z]], P := ℙ ×_{𝔻} Spec Q̂ the generic fiber of ℙ over Spec Q̂.

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}.

2 MODULI STACK OF G-SHTUKAS AS A MODULI FOR (G−)MOTIVES 4

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.

Assume that P is a connected reductive linear algebraic group over Q̂. Let K be the completion of the maximal unramied extension of Q̂. Let B = B(P, Q̂) be the (enlarged) Bruhat-Tits building. Fix a maximal Q̂-split torus S. Let T be the centralizer (a maximal torus) of S, and let N be the normalizer of S. Denote by W_{0} = N(Q̂)/T (Q̂) the relative Weyl group of G with respect to S and denote by A = A(P, S, Q̂) the apartment of B corresponding to S. Let W̃ := W̃(P, S) = N(F)/T^{0}(O_{K}) denote the Iwahori-Weyl group of P with respect to S. Here T^{0} is the unique parahoric subgroup of T (Q̂).

G a parahoric (Bruhat-Tits) group scheme over C; see Definition 1.1 ,

G  the generic fiber of G over Q,  ,

ℙ_{ν} := G ×_{C} Spec Â_{ν} the base change of G to Spec Â_{ν},

P_{ν} := G ×_{C} Spec Q̂_{ν} the generic fiber of ℙ_{ν} over Spec Q̂_{ν},

K_{ν} the completion of the maximal unramied extension of Q̂_{ν} ,

S_{ν} a maximal K_{ν}-split torus that is defined over Q̂_{ν},

W̃_{ν} := W̃(P_{ν}, S_{ν}) the Iwahori-Weyl group of P_{ν} with respect to S_{ν}.

Definition 1.1. A smooth affine group scheme G over C is called a parahoric (Bruhat-Tits) group scheme if

(a) all geometric fibers of G are connected and the generic fiber of G is reductive over 𝔽_{q}(C),

(b) for any ramification point ν of G (i.e. those points ν of C, for which the fiber above ν is not reductive) the group scheme ℙ_{ν} := G ×_{C} Spec Â_{ν} is a parahoric group scheme over Â_{ν}, as defined by Bruhat and Tits [BT72, Définition 5.2.6]; see also [HR03].

We let ℋ ^{1}(C, G) denote the category fibered in groupoids over the category of 𝔽_{q}-schemes, such that the objects over S, ℋ ^{1}(C, G)(S), are G-torsors over C_{S} (also called G-bundles) and morphisms are isomorphisms of G-torsors. The resulting stack ℋ ^{1}(C, G) is a smooth Artin-stack locally of finite type over 𝔽_{q}; see [AH14b, Theorem 2.5].

2 Moduli Stack of G-Shtukas as a moduli for (G−)motives

Let us consider the category Mot(𝔽) of motives over 𝔽, in the sense of [Mil92]. According to the results of Janssen [Jan92] and Deligne [Del90, Théorème 7.1] it can be seen that this category is a semi-simple tannakian category. Furthermore, when one assumes Grothendieck’s standard conjectures in algebraic geometry, this category admits certain fiber functors ω_{ℓ}(−) at ℓ ≠ p, ∞ (resp. ℓ = p) which extend the étale cohomology (resp. crystalline cohomology) functors on the category of smooth schemes over 𝔽_{q}.

Recall that according to Deligne’s philosophy, Shimura varieties with rational weight, may appear as moduli varieties for motives (with additional structures) [Del79]. This conception of Shimura varieties fits the case of Shimura varieties of abelian type, and may fail in general case, namely for Shimura varieties attached to exceptional groups. Nevertheless, as we will see in this section, one may justify that the Deligne’s conception for function field counterparts of Shimura varieties, i.e. for the moduli stack of global G-shtukas. This can be done in the following way. We consider the category of C-motives Mot\underline{ν} ; C(S) over S; see Definition 2.1. This category is a slight modification of the category of Anderson t-motives. We prove that this category is semi-simple tannakian category. We define the category G-Mot\underline{ν} ; C of C-motives with G-structure, in analogy with the category of motives with G-structure in the sense of [Mil92]. We show that this category admits crystalline (resp. étale) fiber functors at characteristic places (away from characteristic places).

2.1 G-Shtukas and G-Motives

Let 𝔽_{q} be a finite field with q elements, let C be a smooth projective geometrically irreducible curve over 𝔽_{q}.

In the following definition we introduce the function field analog of Mot(𝔽). It is a refined version of the category of Anderson t-motives [And86].

Definition 2.1. Let S be a scheme over 𝔽_{q}. The category of C-motives Mot\underline{ν} ; C(S) with characteristic \underline{ν} over S is the category whose objects are tuples \underline{M} := (M, τ) where M is a locally free sheaf of finite rank on C_{S} and τ : σ^{∗}Ṁ ^{∼} ⟶ Ṁ is an isomorphism, where Ṁ denotes the restriction of M to Ċ_{S} := C_{S} ∖ ⋃_{i} ν_{i} ×_{𝔽_{q}} S. And whose morphisms are quasi-morphism. A quasi-morphism (resp. quasi-isogeny) from \underline{M} := (M, τ) to \underline{M}′ := \underline{M} := (M′, τ′) is a morphism (resp. an isomorphism) f: M|_{C_{S}∖D_{S}} → M′|_{C_{S}∖D_{S}} satisfying τ′∘σ^{∗}f = f∘τ, for some effective divisor D on C. Notice that when S = Spec L, for some field L over 𝔽_{q}, then the category Mot\underline{ν} ; C(S) is an abelian category.

The category Mot\underline{ν} ; C(𝔽) admits an obvious fiber functor ω over Q.\overline{𝔽} = \overline{𝔽}(C_{𝔽}). We let 𝔓 := Mot\underline{ν} ; C(𝔽)(ω) denote the corresponding motivic groupoid.

Definition 2.2. Let Rep G denote the category of representations of G in finite free O_{C}-modules V. By a G-motive (resp. G-motive) over S we mean a tensor functor \underline{M}_{G} : Rep G → Mot\underline{ν} ; C(S) (resp. \underline{M}_{G} : Rep_{Q} G → Mot\underline{ν} ; C(S)). We say that two G-motives (resp. G-motives) are isomorphic if they are isomorphic as tensor functors. We denote the resulting category of G-motives (resp. G-motives) over S by G-Mot\underline{ν} ; C(S) (resp.

G-Mot\underline{ν} ; C(S)).

Remark 2.3. Let (T, ⊗) be a Tannakian category over a field k_{0}. Any two fibre functors of T over S become isomorphic over some faithfully flat covering of S; see [Del90, §4].

Remark 2.4. Let 𝔊_{G} denote the neutral groupoid associated to G. According to tannakian theory, instead of working with the category G-Mot\underline{ν} ; C(\overline{𝔽}) one may equivalently work with the category G-Mot\underline{ν} ; C(\overline{𝔽})′ whose objects are morphism 𝔓 → 𝔊_{G} of groupoids and whose morphisms are the natural transformations.

Definition 2.5. 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-bundle 𝒢 over C_{S}, an n-tuple of (characteristic) sections (s_{1}, …, s_{n}) ∈ (C)^{n}(S) and an isomorphism τ: σ^{∗}𝒢 |_{C_{S}∖∪_{i}Γ_{s_{i}}} ^{∼} ⟶ 𝒢 |_{C_{S}∖∪_{i}Γ_{s_{i}}}. We write ∇_{n}ℋ ^{1}(C, G) for the stack of global G-shtukas. Sometimes we fix sections (s_{1}, …, s_{n}) ∈ C^{n}(S) and we simply call \underline{𝒢} = (𝒢 , τ) a global G-shtuka over S.

Definition 2.6. Fix a tuple \underline{ν} := (ν_{i})_{i=1...n} of places on C with ν_{i} ≠ ν_{j} for i ≠ j. Let Â\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 Â\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 Â\underline{ν}

be the formal completion of the ind-algebraic stack ∇_{n}ℋ ^{1}(C, G) along \underline{ν} ∈ C^{n}. It is an ind-algebraic stack over Spf A\underline{ν} which is ind-separated and locally of ind-finite type by [AH14b, Theorem 3.14].

Definition 2.7. Let \underline{ν} be as in the above definition and let \underline{𝒢} = (𝒢 , τ) and \underline{𝒢} ′ = (𝒢′, τ′) be two global G-shtukas in ∇_{n}ℋ ^{1}(C, G)(S)\underline{ν}. 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 from \underline{𝒢} to itself by QIsog_{S}(\underline{𝒢} ). We denote by ∇_{n}ℋ ^{1}(C, G)(S)\underline{ν} ; Q the category whose objects are the same as ∇_{n}ℋ ^{1}(C, G)(S)\underline{ν}, and whose morphisms are the quasi-isogenies between them.

Remark 2.8. The assignment

G ↦ ∇_{n}ℋ ^{1}(C, G)(S)

is functorial. Namely, to a given morphism of algebraic groups ρ : G → G′, we associate the following morphism

\xymatrix{∇_{n}ℋ ^{1}(C, G)(S) \ar[r] & ∇_{n}ℋ ^{1}(C, G′)(S) \\ \underline{𝒢} := (𝒢 , (s_{i})_{i}, τ) ⟼ & ρ_{∗}\underline{𝒢} := (ρ_{∗}𝒢 := 𝒢 ×^{G,ρ} G′, (s_{i})_{i}, τ′ := τ ×^{G,ρ} id_{G′})}

of stacks.

There is the following functor

\underline{M}_{−} : ∇_{n}ℋ ^{1}(C, G)(S)\underline{ν} ; Q → G − Mot\underline{ν} ; C(S)  (2.1)

that assigns to a G-shtuka \underline{𝒢} the functor which sends the representation ρ to ρ_{∗}\underline{𝒢} .

2.2 ℙ-Shtukas and ℙ-Crystals

Let 𝔽 be a finite field and Â := 𝔽[[z]] be the power series ring over 𝔽 in the variable z. We let ℙ be a smooth affine group scheme over 𝔻 := Spec Â with connected fibers, and we let P := ℙ ×_{𝔻} 𝔻̇ be the generic fiber of ℙ over 𝔻̇ := Spec Q̂. We are mainly interested in the situation where we have an isomorphism 𝔻 ≅ Spec Â_{ν} for a place ν of C and where ℙ = ℙ_{ν} := G ×_{C} Spec Â_{ν}.

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 free fpqc-locally on S is already free Zariski-locally on S by [HV11, Proposition 2.3]. We call those modules 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 M̂ of O_{S}[[z]]-modules on S we set σ̂^{∗}M̂ := M̂ ⊗_{O_{S}[[z]],σ̂^{∗}} O_{S}[[z]]. One may define the category of 𝔻-crystals in the following way. Note that this category also was called the category of local shtukas, see [HV11, Definition 4.1] and [Har11, Definition 2.1.1].

Definition 2.9. (a) A 𝔻-iso-crystal over S (resp. 𝔻-crystal over S) is a pair \underline{˙ˆM} := ( ˙ ˆM, τ̂) (resp. \underline{M̂} := (M̂, τ̂)) consisting of a locally free sheaf ˙ ˆM (resp. M̂) of O_{S}((z))-modules (resp. O_{S}[[z]]-modules) of finite rank on S and an isomorphism τ̂: σ̂^{∗} ˙ ˆM ^{∼} ⟶ ˙ ˆM (resp. τ̂: σ̂^{∗}M̂ ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M̂ ⊗_{O_{S}[[z]]} O_{S}((z))).

(b) A morphism (resp. quasi-morphism) between two 𝔻-iso-crystals ( ˙ ˆM, τ̂) → ( ˙ ˆM′, τ̂′) (resp. 𝔻-crystals (M̂, τ̂) → (M̂′, τ̂′)) is a morphism f: ˙ ˆM → ˙ ˆM′ (resp. f: M̂ ⊗_{O_{S}[[z]]} O_{S}((z)) → M̂′ ⊗_{O_{S}[[z]]} O_{S}((z))) of O_{S}((z))-modules with τ̂′ ∘ σ̂^{∗}f = f ∘ τ̂.

(c) A 𝔻-iso-crystal \underline{˙ˆM} := ( ˙ ˆM, τ̂) (resp. 𝔻-crystal \underline{M̂} := (M̂, τ̂)) is called étale if ˙ τ̂ (resp. τ̂) comes from an isomorphism of O_{S}[[z]]-modules σ̂^{∗}M̂ ^{∼} ⟶ M̂.

(d) We denote by ICrys_{𝔻}(S) (resp. Crys_{𝔻}(S)) the category of 𝔻-iso-crystals (resp. 𝔻-crystals) over S with morphisms (resp. quasi-morphism) as above and by Ét ICrys_{𝔻}(S) (resp. Ét Crys_{𝔻}(S)) the full sub-category of étale 𝔻-iso-crystals (resp. étale 𝔻-crystals) over S.

Remark 2.10. We say that ˆ \underline{M} = (M̂, τ̂) is topologically nilpotent if there is an integer n such that τ̂ ∘ σ̂^{∗}τ̂ ∘ ⋯ ∘ σ̂^{(n−1)∗}τ̂(σ̂^{n∗}M̂) ⊂ zM̂. If S is the spectrum of a field L, every 𝔻-crystal ˆ \underline{M} = (M̂, τ̂) is canonically an extension

0 → ˆ \underline{M}^{ét} → ˆ \underline{M} → ˆ \underline{M}^{nil} → 0,

where ˆ \underline{M}^{ét} (resp. ˆ \underline{M}^{nil}) is étale (resp. topologically nilpotent 𝔻-crystal). If L is perfect this extension splits canonically; see [HS15, Proposition 2.8] and also [Lau96, Lemma B.3.10].

Remark 2.11. Note that when S = Spec L, for some field extension 𝔽 ⊆ L, the category Crys_{𝔻}(S) of 𝔻crystals over S is an abelian category. We denote by Q End(\underline{M̂}) the 𝔽((z))-algebra of the self quasi-morphisms of \underline{M̂}.

One can assign a Galois representation to a given étale 𝔻-crystal as follows. Assume that S is connected. Let s̄ be a geometric point of S and let π_{1} := π_{1}(S, s̄) denote the algebraic fundamental group of S at s̄. We define the (dual) Tate functor from the category of étale crystals Ét Crys_{𝔻}(S) over S to the category 𝔉𝔐od_{Â[π_{1}]} of Â[π_{1}]-modules which are finite free over Â as follows

Ť_{−}: Ét Crys_{𝔻}(S) ⟶ 𝔉𝔐od_{Â[π_{1}]},
\underline{M̂} := (M̂, τ̂) ⟼ Ť\underline{M̂} := (M̂ ⊗_{O_{S}[[z]]} κ(s̄)[[z]])τ̂ .  (2.2)

Here the superscript τ̂ denotes τ̂-invariants. Inverting z we also consider the rational (dual) Tate functor

V̌_{−}: Ét Crys_{𝔻}(S) ⟶ 𝔉𝔐od_{Q̂[π_{1}]},
\underline{M̂} := (M̂, τ̂) ⟼ V̌\underline{M̂} := (M̂ ⊗_{O_{S}[[z]]} κ(s̄)[[z]])τ̂ ⊗Â Q̂.  (2.3)

where 𝔉𝔐od_{Q̂[π_{1}]} denotes the category of Q̂[π_{1}]-modules which are finite over Q̂. The functor V̌_{−} transforms

quasi-isogenies into isomorphisms. Sometimes we use H_{ét}^{1}(−, Â) (resp. H_{ét}^{1}(−, Q̂)) to denote the functor Ť_{−} (resp. V̌_{−}).

Proposition 2.12. Let S ∈ Nilp_{𝔽[[ζ]]} be connected. We have the following statements

(a) The functor V̌_{−} is an equivalence between the category of étale iso-crystals ICrys _{𝔻}(S) over S and the category 𝔉𝔐od_{Q̂[π_{1}]} with isomorphisms as the only morphisms.

(b) There is a canonical isomorphism of κ(s̄)[[z]]-modules Ť\underline{M̂} ⊗_{𝔽[[z]]} κ(s̄)[[z]] ^{∼} ⟶ \underline{M̂} ⊗_{O_{S}[[z]]} κ(s̄)[[z]] which is equivariant for the action of π_{1}(S, s̄) and τ̂, where π_{1}(S, s̄) acts trivially on \underline{M̂} and τ̂ acts trivially on Ť\underline{M̂}.

Proof. See [AH14a, Proposition 3.4].

Remark 2.13. Note that when we work over general S the category Crys_{𝔻}(S) may fail to be abelian. Nevertheless, the category of crystals Crys_{𝔻}(𝔽) over a finite field 𝔽 is indeed a tannakian category admitting a fiber functor over Q̂.\overline{𝔽}. We denote the corresponding motivic groupoid by 𝔓̂_{ν}.

In contrast with the global version in the sense of Definition 2.2, let us now discuss the local categories which are endowed with ℙ-structure.

Definition 2.14. By a ℙ-crystal (resp. étale ℙ-crystal, resp. P -iso-crystal) over S we mean a tensor functor ˆ\underline{M}_{ℙ} : RepÂ ℙ → Crys_{𝔻}(S) (resp. ˆ \underline{M}_{ℙ} : RepÂ ℙ → Ét Crys_{𝔻}(S), resp. ˆ \underline{M}_{P} : RepQ̂ P → Crys_{𝔻}(S)). We say that two ℙ-crystals (resp. étale ℙ-crystals, resp. P -crystals) are isomorphic if they are isomorphic as tensor functors. We denote the resulting category of ℙ-crystals (resp. étale ℙ-crystal, resp. P -crystals) over S by ℙ-Crys_{𝔻}(S) (resp. Ét ℙ-Crys_{𝔻}(S), resp P -Crys_{𝔻}(S)).

Remark 2.15. Over 𝔽 the category ℙ-Crys_{𝔻}(𝔽) (resp. P -Crys_{𝔻}(𝔽) ) is equivalent with the category { \atop Mot}_{ℙ}(S)′ (resp. { \atop Mot}_{P} (𝔽)′ ) whose objects are morphisms 𝔓̂_{ν} → 𝔊_{ℙ} (resp. 𝔓̂_{ν} → 𝔊_{P}) of groupoids and whose morphisms are the natural transformations.

To construct a moduli space parametrizing such local motivic objects, that are also equipped with ℙ-structure, one further requires a relevant geometric modification of the notion we introduced above. For this purpose, we first recall the following Definition 2.16. 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

Ł: ℋ ^{1}(Spec 𝔽, L^{+}ℙ) ⟶ ℋ ^{1}(Spec 𝔽, LP), L_{+} ↦ L .  (2.4)

Let us recall the definition of local ℙ-shtukas

Definition 2.17. (a) 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 (2.4).

(b) 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}. We denote by Sht_{ℙ}^{𝔻}(S) (resp. Sht_{ℙ}^{𝔻}(S)Q̂) the category of étale local ℙ-shtukas over S with isomorphisms (resp. quasi-isogenies) as its morphisms.

(c) 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) (resp. ÉtSht_{ℙ}^{𝔻}(S)Q̂) the category of étale local ℙ-shtukas over S with isomorphisms (resp. quasi-isogenies) as its morphisms.

Remark 2.18. 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 𝔻-crystals over S of rank r as its objects, with isomorphisms as the only morphisms; see [HV11, Lemma 4.2].

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 RepÂ ℙ be the category of representations ρ: ℙ → GL(V) of ℙ in finite free Â-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_{+} 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) .  (2.5)

The functor ρ_{∗}: ℋ ^{1}(Spec 𝔽, L^{+}ℙ) → Vect_{𝔻} induces a functor from the category of local ℙ-shtukas over S to the category of crystals over S which we likewise denote ρ_{∗}. Therefore we obtain the following functor

D : Sht_{ℙ}^{𝔻}(S) → ℙ-Crys_{𝔻}(S),  (2.6)

that sends \underline{L} to the functor which sends a representation ρ to ρ_{∗}\underline{L}. Let _{A[π_{1}]}{ \atop Mod}_{ℙ} denote the category of tensor functors F unct^{⊗}(RepÂ ℙ, 𝔉𝔐od_{Â[π_{1}]}). If we restrict the above functor 2.6 to the caregory of étale local ℙ-shtukas then this functor induces the following functor

Ét Sht_{ℙ}^{𝔻}(S) → Ét ℙ-Crys_{𝔻}(S) ^{∼} ⟶_{Â[π_{1}]} { \atop Mod}_{ℙ}.  (2.7)

The right hand side equivalence comes from 2.2; see Proposition 2.12. Note that when S ∈ Nilp_{𝔽[[ζ]]}, using tannakian theory, one may observe that the functor 2.6 is an equivalence of categories; see Remark 2.3. Also for the left hand side functor of 2.7 see Proposition 2.20 bellow. Regarding this discussion we make the following

Definition 2.19. Let S ∈ Nilp_{𝔽[[ζ]]} be a connected scheme, \overline{s} a geometric point of S and set π_{1} := π_{1}(S, \overline{s}).

We define _{Â[π_{1}]}{ \atop Mod}_{ℙ} := F unct^{⊗}(RepÂ ℙ, 𝔉𝔐od_{Â[π_{1}]}) (resp. _{Q̂[π_{1}]}{ \atop Mod}_{ℙ} := F unct^{⊗}(Rep_{A} ℙ, 𝔉𝔐od_{Q̂[π_{1}]}), as

the category whose objects are tensor functors from RepÂ ℙ to 𝔉𝔐od_{Â[π_{1}]} (resp. to 𝔉𝔐od_{Q̂[π_{1}]}), 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) ⟶ _{Â[π_{1}]}{ \atop Mod}_{ℙ},
\underline{L} ⟼ Ť\underline{L}: ρ ↦ Ť_{ρ_{∗}\underline{L}},
V̌_{−}: ÉtSht_{ℙ}^{𝔻}(S) ⟶ _{Q̂[π_{1}]}{ \atop Mod}_{P}
\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 2.12. If \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_{Â[π_{1}]} → 𝔉𝔐odÂ is isomorphic to the forgetful fiber functor ω^{∘}: RepÂ ℙ → 𝔉𝔐odÂ according to [AH14a, 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}(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}(S, s̄) → ℙ(Â). Now consider the full subcategory _{Â[π_{1}]}{ \atop Mod}_{ℙ,0} of _{Â[π_{1}]}{ \atop Mod}_{ℙ} consisting of those tensor functors F for which F ∘ F ≅ ω^{∘}. Then, as we proved in [AH14a, Proposition 3.6], Proposition 2.12 generalizes as follows.

Proposition 2.20. Let S ∈ Nilp_{𝔽[[ζ]]} be a connected scheme. Then the functor Ť_{−} (resp. V̌_{−}) between the category ÉtSht_{ℙ}^{𝔻}(S) (resp. ÉtSht_{ℙ}^{𝔻}(S)Q̂) and the category _{Â[π_{1}]}{ \atop Mod}_{ℙ,0} (resp. _{Q̂[π_{1}]}{ \atop Mod}_{ℙ}) is equivalence (resp. fully faithful).

Remark 2.21. 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 smooth affine group scheme over Spec 𝔽_{q}[[z]]. Let

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 (see [AH14a, Definition 2.2] for the definition of formal torsor) over Spf A_{ν} ˆ×_{𝔽_{q}} S. The Weil restriction Res_{𝔽_{ν}/𝔽_{q}} 𝒢̂_{ν} is a formal

ˆℙ̃_{ν}-torsor over Spf 𝔽_{q}[[z]] ˆ×_{𝔽_{q}} S and may be viewed as an L^{+}ℙ̃_{ν}-torsor over S which we denote Ł_{ν}^{+}(𝒢); see [AH14a, Proposition 2.4]. We obtain the functor

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

Finally there is a funcor

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

which sends the G-torsor 𝒢̇ over C′_{S}, having some extension 𝒢 over C_{S}, to the LP̃_{ν}-torsor Ł(Ł_{ν}^{+}(𝒢)) associated with Ł_{ν}^{+}𝒢 under (2.4). Notice that Ł_{ν}(𝒢) is independent of the extension 𝒢, and that we hence may write Ł_{ν}(𝒢̇) := Ł(Ł_{ν}^{+}(𝒢)). The above maps assign to each G-torsor 𝒢 over C_{S} a triple (𝒢̇, Ł_{ν}^{+}(𝒢), φ) where φ: Ł_{ν}(𝒢̇) ^{∼} ⟶ Ł(Ł_{ν}^{+}(𝒢)) is the canonical isomorphism of LP̃_{ν}-torsors. The groupoid ℋ ^{1}(C, G)(S) is equivalent to the category of such triples; see [AH14a, Lemma 5.1]. In other words, the following diagram of groupoids is cartesian

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

2.3 Crystalline Realization Functors

Let ν be a place on C and let 𝔻_{ν} := Spec Â_{ν} and 𝔻̂_{ν} := Spf Â_{ν}. Let deg ν := [𝔽_{ν} : 𝔽_{q}] and fix an inclusion 𝔽_{ν} ⊂ Â_{ν}. Assume that we have a section s: S → C which factors through Spf Â_{ν}, that is, the image in O_{S} of a uniformizer of Â_{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},  (2.8)

where 𝔻̂_{ν,S} := 𝔻̂_{ν} ˆ×_{𝔽_{ν}} S and where V(𝔞_{ν,ℓ}) denotes 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.

Remark 2.22. Let us recall the following interpretation of the component V(𝔞_{ν,0}). 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}. For a proof we refer the reader to [AH14a, Lemma 5.3].

Definition 2.23. We 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 Â_{ν_{i}}. By 2.8 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 [AH14a, Poposition 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 crystalline realization functor ω_{ν_{i}}(−) at ν_{i} by

ω_{ν_{i}}(−): ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ ℙ_{ν_{i}}-Crys_{𝔻}(S) ,
(𝒢 , τ) ⟼ D(𝒢 ˆ×_{C_{S}} V(𝔞_{ν_{i},0}), τ^{deg ν_{i}}) ,
ω\underline{ν} := ∏_{i} ω_{ν_{i}}: ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ ∏_{i} ℙ_{ν_{i}}-Crys_{𝔻}(S) ;  (2.9)

see 2.7.

Remark 2.24. 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}) .  (2.10)

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 ))  (2.11)

∼ ⟶ (𝒢 , τ) ˆ×_{C_{S}}(Spf A_{ν_{i}} ˆ×_{𝔽_{q}} S).

Recall from Remark 2.21 that the Weil restriction Res_{𝔽_{ν_{i}}/𝔽_{q}} 𝒢̂_{ν} of the torsor 𝒢̂_{ν} := 𝒢 ˆ×_{C_{S}}(Spf A_{ν_{i}} ˆ×_{𝔽_{q}} S) corresponds by to an L^{+}ℙ̃_{ν_{i}}-torsor Ł_{ν_{i}}^{+}(𝒢) according to [AH14a, Poposition 2.4]. Then (Ł_{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 (2.11) 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}).

2.4 Etale Realization Functors

Definition 2.25. Let \underline{𝒢} = (𝒢 , τ) be a global G-shtuka and let ν be a place on C outside the characteristic places ν_{i}. Let Ł_{ν}^{+}(\underline{𝒢} ) be 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 [AH14a, Poposition 2.4]. Hence we obtain a functor

Ł_{ν}^{+}(−): ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ ÉtShtSpec A_{ν} ; ℙ̃_{ν}(S) ,

(2.12)

note that Ł_{ν}^{+}(\underline{𝒢} ) is étale because τ is an isomorphism at ν. We call Ł_{ν}^{+}(\underline{𝒢} ) the étale local ℙ̃_{ν}-shtuka associated with \underline{𝒢} at the place ν ∉ \underline{ν}.

Set Â_{ν} ≅ 𝔽_{ν}[[z]]. For every representation ρ: ℙ_{ν} → GL_{r,Â_{ν}} in RepÂ_{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,Â_{ν}} followed by the natural inclusion Res_{𝔽_{ν}/𝔽_{q}} GL_{r,Â_{ν}} ⊂ GL_{r·[𝔽_{ν}:𝔽_{q}] , 𝔽_{q}[[z]]}. We set \underline{L̃} = Ł_{ν}^{+}(\underline{𝒢} ) and define ω^{ν}(\underline{𝒢} ) := Ť\underline{L̃}(ρ̃) := Ť_{ρ̃_{∗}\underline{L̃}} (resp. ω_{Q}^{ν}(\underline{𝒢} ) := V̌\underline{L̃}(ρ̃) := V̌_{ρ̃_{∗}\underline{L̃}}). According to this procedure we obtain the étale realization functor with integral (resp. rational) coefficients

ω^{ν}(−): ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ _{Â_{ν}[π_{1}]}{ \atop Mod}_{ℙ_{ν}} ,
(resp.  ω_{Q}^{ν}(−): ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ _{Q̂_{ν}[π_{1}]}{ \atop Mod}_{ℙ_{ν}} .  (2.13)

Note that there is a canonical isomorphism of A_{ν}-modules ω^{ν}(\underline{𝒢} )(ρ) ≅ lim_{⟵_{n}} ρ_{∗}(\underline{𝒢} ×_{C} Spec A_{ν}/(ν^{n}))^{τ}. Putting

all the étale realization functors together in the adelic way, we obtain

ω\underline{ν}(−): ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ _{𝔸\underline{ν}[π_{1}]}Mod_{G} ,
(resp.  ω\underline{ν} ; Q(−): ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) ⟶ _{𝔸\underline{ν} ; Q[π_{1}]}Mod_{G} .  (2.14)

Here_{𝔸\underline{ν}[π_{1}]}Mod_{G} (resp. _{𝔸\underline{ν} ; Q[π_{1}]}Mod_{G}) denote the category of tensor functors F unct^{⊗}(Rep_{𝔸\underline{ν}} G, 𝔉𝔐od_{𝔸\underline{ν}[π_{1}(S,s̄)]}) (resp. F unct^{⊗}(Rep_{𝔸\underline{ν}} G, 𝔉𝔐od_{𝔸\underline{ν} ; Q[π_{1}(S,s̄)]})). Note that there is a canonical isomorphism of 𝔸\underline{ν}-modules

ω\underline{ν}(\underline{𝒢} )(ρ) ≅ lim_{⟵_{D}} ρ_{∗}(\underline{𝒢} |_{D\overline{s}})^{τ}, where D runs over all divisors on C ∖ \{ν_{1}, …, ν_{n}\}.

Remark 2.26. If 𝔽_{ν} ⊂ O_{S} there also exists the decomposition (2.8) and we can associate a local ℙ_{ν}-shtuka \underline{L} with Ł_{ν}^{+}(\underline{𝒢} ). The main difference to Definition 2.23 and Remark 2.24 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 (2.10). Therefore we may take any component and the associated local ℙ_{ν}-shtuka \underline{L}. Equation (2.11) 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.

Let us recall the following theorem from [Tam94, § 2].

Theorem 2.27. (Tate Conjecture for Mot\underline{ν} ; C(L)) Let ν be a place on C different from the characteristic places ν_{i}. Let \underline{M} and \underline{M}′ be in Mot\underline{ν} ; C(L), where L is a finitely generated field over 𝔽_{q}. Then

Hom(\underline{M}, \underline{M}′) ⊗ Q̂_{ν} ≅ Hom_{Gal(L^{sep}/L)}(V̌_{\underline{M̂}_{ν}}, V̌_{\underline{M̂}}′_{ν}).

Theorem 2.28. (Tate Conjecture for G-C-Motives) Let \underline{M}_{G} and \underline{M}′_{G} be in G-Mot\underline{ν} ; C(L), where L is a finitely generated field. Then

Hom(\underline{M}_{G}, \underline{M}′_{G}) ⊗ Q̂_{ν} ≅ Hom(ω^{ν}(\underline{M}_{G}), ω^{ν}(\underline{M}′_{G})).

Proof. This follows from Theorem 2.27 and tannakian theory.

2.5 Poincare-Weil Theorem for C-Motives

Theorem 2.29. The category Mot\underline{ν} ; C(\overline{𝔽}) with the fiber functor ω, is a semi-simple tannakian category. In particular the kernel P := 𝔓^{Δ} of the corresponding motivic groupoid 𝔓 := Mot\underline{ν} ; C(ω) is a pro-reductive group.

Proof. According to [AH14b, Theorem 3.14], we may suppose that a given motive \underline{M} ∈ Mot\underline{ν}(\overline{𝔽}) comes from a motive over a finite extension L/𝔽_{q}, which we again denote by \underline{M}. It is enough to show that after a finite extension L ⊂ L′ ⊂ 𝔽, the image \underline{M}′ of \underline{M} under the obvious functor Mot\underline{ν}(L) → Mot\underline{ν}(L′) is semi-simple, or equivalently the endomorphism algebra E := QEnd(\underline{M}′) is a semisimple algebra over Q. Let ν be a place on C which is distinct from the characteristic places ν_{i}. Let ˆ \underline{M}′_{ν} denote the crystal associated to ω^{ν}(\underline{M}′) = (M̂′_{ν}, τ′_{ν}), see Remark 2.18 and Proposition 2.12. By [Bou58, Corollaire 7.6/4] it is enough to show that after such a field extension the endomorphism algebra E_{ν} := E ⊗ Q̂_{ν} = QEnd( ˆ\underline{M}′_{ν}) is semi-simple. Note that the last equality follows from Theorem 2.27.

By [Bou58, Corollaire the proposition 6.4/9] one can equivalently show that F = Q̂_{ν}(π′_{ν}) is semi-simple, where π′_{ν} := τ′ ∘ ⋯ ∘ (σ̂^{∗})^{[L′:𝔽_{q}]−1}τ′ is the Frobenius endomorphism of M̂′_{ν}. Take a representative matrix B_{π_{ν}} for π_{ν} ⊗ 1 : End(ˆ[\underline{M}]_{ν} ⊗_{A_{ν}} Q̂_{ν}^{alg}) and write B_{π_{ν}} in the Jordan normal form. We let L′/L to be a field extension such that [L′ : L] is a power of the characteristic of 𝔽_{q} and [L′ : L] ≥ rank M. Clearly B_{π_{ν}}^{[L′:L]} is diagonal. This represents the Frobenius endomorphism π′ ⊗ Q̂_{ν}^{alg} of M̂′_{ν} ⊗ Q̂_{ν}^{alg}. Since Q̂_{ν}^{alg}/Q is perfect we may argue by [Bou58, Proposition 9.2/4] that π′_{ν} is semi-simple, and as we mentioned above, this suffices to argue that the image of \underline{M}′ is semi-simple. Now the second statement follows from [DM82, Proposition 2.23].

Consider the following functor

\underline{M}_{−} : ∇_{n}ℋ ^{1}(C, G)(S)\underline{ν} → Mot\underline{ν} ; G(S)  (2.15)

that assigns to a G-shtuka \underline{𝒢} the functor which sends the representation ρ to ρ_{∗}\underline{𝒢} .

Proposition 2.30. Let 𝔽_{q} ⊂ L be a field extension. Then the functor 2.15 induces an equivalence between the following categories

(a) The category ∇_{n}ℋ ^{1}(C, G)(L)\underline{ν} ; Q of global G-shtukas over L,

(b) the category G-Mot\underline{ν} ; C(L).

Proof. (a)⇒(b) Fix a global G-shtuka \underline{𝒢} . Now for every representation in Rep_{Q} G we choose an extension to a representation ρ : G → GL(V) where V is a vector bundle on the whole curve C and we consider ρ_{∗}\underline{𝒢} ∈ Mot\underline{ν} ; C(𝔽)_{Q}. Note that if we choose a different representation ρ : G → GL(V), since ρ and ρ′ are generically equal, therefore there is a unique quasi-isogeny ρ_{∗}\underline{𝒢} → ρ′_{∗}\underline{𝒢} .

(b)⇒(a) We choose a lift Rep G → Mot\underline{ν}(𝔽), by tannakian formalism this yields a global G-shtuka. One can check that different choices for the lift differ by a unique quasi-isogeny.

Remark 2.31. According to the Proposition 2.30 above and Remark 2.4, the étale (resp. crystalline) realization functors ω^{ν}(−) (resp. ω_{ν_{i}}(−)) induce étale (resp. crystalline) realization ω^{ν}(−) (resp. ω_{ν_{i}}(−)) on the category G-Mot\underline{ν} ; C(L)′.

3 Langlands-Rapoport Conjecture

Let (G, X) be a Shimura data which roughly consists of a reductive group G over ℤ_{(p)} with center Z, and a G(ℝ)-conjugacy class X of homomorphisms 𝕊 → G_{ℝ} for the Deligne torus 𝕊, that satisfies certain conditions. Let E be the reflex field of (G, X), choose a p-adic completion E_{ν} of E, and let p^{m} be the cardinality of the residue field of E_{ν}. Let 𝔽 be an algebraic closure of 𝔽_{p}, let B(𝔽) be the fraction field of the ring W (𝔽) of Witt vectors with values in 𝔽, and let σ^{∗} be the p-Frobenius lift on B(𝔽) and W (𝔽). Langlands and Rapoport [LR87] construct a “pseudomotivic groupoid” 𝒫 over \overline{ℚ}/ℚ, together with homomorphisms ζ_{ℓ} : 𝔊_{ℓ} → 𝒫 for all primes ℓ including ∞. Here 𝒫 is a transitive \overline{ℚ}/ℚ-groupoid that can equivalently be described as a gerb. Moreover, 𝔊_{ℓ} for ℓ ≠ p, ∞ is the trivial \overline{ℚ}_{ℓ}/ℚ_{ℓ}-groupoid, and 𝔊_{p} is the \overline{ℚ}_{p}/ℚ_{p}-groupoid which is the fundamental groupoid of the category of isocrysttals over 𝔽. Finally 𝔊_{∞} is the ℂ/ℝ-groupoid whose group of complex points is the real Weil group; see [Mil92, 3.18-3.26]. However, the existence of ζ_{∞} relies on Grothendiecks standard conjectures and the remaining ζ_{ℓ} on the Tate conjecture (this issue has been avoided in [LR87, Kis13] by purposing a quasi-motivic Galois gerb 𝔔 together with morphisms ζ_{ℓ} : 𝔊_{ℓ} → 𝔔(ℓ) instead of working with the pseudomotivic groupoid (𝒫, ζ_{ℓ})). Furthermore, to a given homomorphism φ : 𝒫 → 𝔊_{G} (here 𝔊_{G} = G ×_{ℚ} (Spec\overline{ℚ} ×_{ℚ} Spec\overline{ℚ}) denotes the neutral groupoid over \overline{ℚ}/ℚ) they associate a set 𝒮 (φ), which carries a G(𝔸_{f}^{p}) × ℤ(ℚ_{p}) × ⟨Frob_{p^{m}}⟩-action. The Langlands-Rapoport conjecture predicts a canonical G(𝔸_{f}^{p}) × ℤ(ℚ_{p}) × ⟨Frob_{p^{m}}⟩-equivariant bijection

Sh_{p}(X, G)(𝔽) = ⨆_{φ} 𝒮 (φ)

which is functorial in (G, X). Here φ runs over the set of equivalence classes of “admissible” morphisms φ : 𝒫 → 𝔊_{G} and Sh_{p}(G, X) denotes the canonical integral model of the Shimura variety

lim_{⟵_{K}} G(ℚ)\X × G(𝔸_{f})/K

in the sense of Milne [Mil92]. Let us briefly recall the construction of the set 𝒮 (φ). By Tannakian theory φ corresponds to a unique tensor functor M : (Rep_{Q} G, ω^{∘}) → (Mot(𝔽), ω), where ω^{∘} is the forgetful fiber functor. Such M are called “G-motives. Then

X^{p}(φ) := X^{p}(M) := Isom^{⊗}(𝔸_{f}^{p} ⊗_{ℚ} ω^{∘}, ∏_{ℓ≠p,∞} ω_{ℓ} ∘ M)

is the set of tensor isomorphisms where 𝔸_{f}^{p} is the finite adeles of ℚ away from the place p. At the place p, the morphism φ defines an element b = b(φ) ∈ G(B(𝔽)) unique up to σ̂^{∗}-conjugation. Since G is unramied at p, the Hermitian symmetric domain X in the Shimura p-datum (G, X) gives rise to a double coset C_{p} ∈ G(W (𝔽))\G(B(𝔽))/G(W (𝔽)). Then X_{p}(φ) is defined as follows

X_{p}(φ) := X_{C_{p}}(b) := {g ∈ G(B(F))/G(W (F)) : g^{−1}bσ̂^{∗}g ∈ C_{p}};

see [Mil92, Chapter 4]. One lets Z^{p} be the closure of Z(ℤ_{(p)}) in Z(𝔸_{f}^{p}), set I_{φ}(ℚ) := Aut(φ)(ℚ) := {g ∈ G(\overline{ℚ}); ad_{g} ∘ φ = φ\} = Aut^{⊗}(M)(ℚ) and defines 𝒮 (φ) := I_{φ}(ℚ)\ (X^{p}(φ)/Z^{p} × X_{p}(φ)). This set is equipped with an action of G(𝔸_{f}^{p}) × Z(ℚ_{p}) × ⟨Frob_{p^{m}}⟩ where the first factor acts on X^{p}(φ) and the other two on X_{p}(φ).

We now return to the arithmetic function fields. Recall that (formal completion of) the moduli stack of global G-shtukas (at certain characteristic places) may be viewed as the function field counterpart of (the canonical integral model for) Shimura varieties. In this section we state and prove this conjecture for the moduli stack of global G-shtukas. To this purpose we first introduce the function field analogue of Shimura data.

3.1 The ∇ℋ -data

First let us introduce boundedness conditions for local ℙ-shtukas. 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 3.1. (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 let 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.

For further explanation about the above definition see [AH14a, Remark 4.6 and Remark 4.7]

Definition 3.2. (a) We define a bound to be an equivalence class Ẑ := [Ẑ_{R}] of closed ind-subschemes Ẑ_{R} ⊂ F̂ℓ_{ℙ,R}, such that for all R the ind-subscheme Ẑ_{R} is stable under the left L^{+}ℙ-action on Fℓ_{ℙ}, and the special fiber Z_{R} := Ẑ_{R} ˆ×_{Spf R} Spec κ_{R} is a quasi-compact subscheme 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 projective scheme over κ_{RẐ} by Remark [AH14a, Remark 4.3] and [HV11, Lemma 5.4], which implies that every morphism from a quasi-compact scheme to an ind-projective ind-scheme factors through a projective subscheme.

(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 étale 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 any such trivialization and for all finite extensions 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 3.3. Note that the condition of Definition 3.2(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 [AH14a, Remark 4.9].

Definition 3.4. (a) Fix an n-tuple \underline{ν} := (ν_{i}) of closed points of C. A ∇ℋ \underline{ν}-data is a tuple (G, (Ẑ_{ν_{i}})_{i}, H) consisting of

(i) a parahoric group scheme G over C,

(ii) an n-tuple (Ẑ_{ν_{i}})_{i} of bounds with reflex rings RẐ_{ν_{i}},

(iii) a compact open subgroup H ⊆ G(𝔸\underline{ν} ; Q) ≅ Aut^{⊗}(ω_{𝔸\underline{ν}}^{∘}).

We use the shorthand ∇ℋ -data when \underline{ν} is clear from the context. A morphism between two ∇ℋ -data (G, (Ẑ_{ν_{i}}), H) and (G′, (Ẑ′_{ν_{i}}), H′) is a morphism φ : G → G′ such that the induced morphism F̂ℓ_{ℙ_{ν_{i}}} → F̂ℓ_{ℙ′_{ν_{i}}} send Ẑ_{ν_{i}} to Ẑ′_{ν_{i}}, for every i, and φ(H) ⊆ H′.

(b) For a compact open subgroup H ⊆ G(𝔸\underline{ν} ; Q) we define a rational H-level structure γ̄ on a global G-shtuka \underline{𝒢} over S ∈ Nilp_{A\underline{ν}} as a π_{1}(S, s̄)-invariant H-orbit γ̄ = γH in Isom^{⊗}(ω^{∘}, ω\underline{ν}(\underline{𝒢} )).

(c) We denote by ∇_{n}^{H}ℋ ^{1}(C, G)\underline{ν} the category fibered in groupoids whose S-valued points ∇_{n}^{H}ℋ ^{1}(C, G)\underline{ν}(S) is the category whose objects are tuples (\underline{𝒢} , γ̄), consisting of a global G-shtuka \underline{𝒢} in ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) together with a rational H-level structure γ̄, and whose morphisms are quasi-isogenies of global G-shtukas (see Definition 2.7) that are isomorphisms at the characteristic places ν_{i} and are compatible with the H-level structures.

(d) Fix an n-tuple \underline{ν} = (ν_{i}) of places on the curve C with ν_{i} ≠ ν_{j} for i ≠ j and let ∇ℋ ^{1}(C, G)\underline{ν} denote the formal completion of the stack ∇ℋ ^{1}(C, G) at \underline{ν}. Let Ẑ\underline{ν} := (Ẑ_{ν_{i}})_{i} be a tuple of closed ind-subschemes Ẑ_{i} of F̂ℓ_{ℙ_{ν_{i}}} which are bounds in the sense of Definition 3.2. Let \underline{𝒢} be a global G-shtuka in ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S). We say that \underline{𝒢} is bounded by Ẑ\underline{ν} := (Ẑ_{ν_{i}})_{i} if for every i the associated local ℙ_{ν_{i}}−shtuka ω_{ν_{i}}(\underline{𝒢} ) is bounded by Ẑ_{ν_{i}}.

(e) To a ∇ℋ \underline{ν}-data (G, (Ẑ_{ν_{i}})_{i}, H) we associate a moduli stack ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} parametrizing G-shtukas bounded by Ẑ\underline{ν} at place ν which are additionally equipped with a level H-structure. Furthermore this correspondence is functorial, i.e. a morphism φ : (G, (Ẑ_{i}), H) → (G′, (Ẑ′_{i}), H′) between ∇ℋ -data induces the following morphism

φ_{∗} : ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} → ∇_{n}^{H′,Ẑ′ ; \underline{ν}}ℋ ^{1}(C, G′)\underline{ν}

(f) The group G(𝔸\underline{ν} ; Q) acts on ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} by means of Hecke correspondences which is explicitly given as follows. Let H, H′ ⊂ G(𝔸\underline{ν} ; Q) be compact open subgroups. Then the Hecke correspondences π(g)_{H,H′} are given by the diagrams

\xymatrix{ & ∇_{n}^{(H′∩g^{−1}Hg),Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} \ar[dl] \ar[dr] \\ ∇_{n}^{H′,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} &  & ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} \ar@{-->}[ll] \\  & (\underline{G}, (H′ ∩ g^{−1}Hg)γ) \ar@{|->}[dl] \ar[dr] \\ (\underline{G}, H′γ) &  & (\underline{G}, Hgγ).}  (3.16)

(g) As we may desire to think of Hecke operation as a true operation on the moduli stack of global G-shtukas, hence we must take care of all possible level structures. Accordingly, we sometimes don’t specify level structure H in the above notation for ∇ℋ -data and simply write (G, (Ẑ_{ν_{i}})_{i}). We define the following functorial assignment

(G, (Ẑ_{ν_{i}})_{i}) ↦ ∇_{n}^{∗,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} := lim_{⟵_{H}} ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν},

where H runs over all compact open subgroups of G(𝔸\underline{ν} ; Q) ≅ Aut^{⊗}(ω_{𝔸\underline{ν}}^{∘}).

Definition 3.5. A G-shtuka \underline{𝒢} is called special G-shtuka if it comes from a T -shtuka \underline{𝒯} , i.e. \underline{𝒢} = i_{∗}\underline{𝒯} , where T is a torus in G and i : T → G denotes the inclusion.

Definition 3.6. Let Ẑ be a bound with reflex ring RẐ = κ[[ξ]] and special fiber Z ⊂ Fℓ_{ℙ} ˆ×_{𝔽} Spec κ; see Definition 3.2. Let \underline{𝕃} = (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 3.13. 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. We define the associated affine Deligne-Lusztig variety X_{Z}(\underline{𝕃}) as the reduced closed ind-subscheme X_{Z}(b) ⊂ Fℓ_{ℙ} ˆ×_{𝔽} Spec k whose K-valued points (for any field extension K of k) are given by

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

We set X_{⪯ω}(b) := X_{S(ω)}(b) if Z is the affine Schubert variety S(ω) with ω ∈ W̃, see [PR08].

Remark 3.7. Notice that the following operator

Φ_{b}.g = N b.σ̂^{m}g

acts on X_{Z}(b). Here N b := b.σ̂b … σ̂^{m−1}b and m = [RẐ : 𝔽_{q}[[ζ]]].

Let B(P_{ν}) be the set of σ̂-conjugacy classes of elements in P (Q̂_{ν}). Let B(W̃_{ν}) be the set of σ̂-conjugacy classes in the Iwahori-Weyl group W̃_{ν} := W̃(P_{ν}, S_{ν}).

Definition 3.8. (a) 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}. There is a representative g_{ω} ∈ LP (𝔽_{ω}) of ω; see [AH14a, Example 4.12]. The Schubert variety S(ω) associated with ω is 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 [Ric10].

(b) Let Ψ_{ν} : B(W̃_{ν}) → B(P_{ν}) denote the natural morphism induced from the natural inclusion N(K_{ν}) → G(K_{ν}). We say that a global G-shtuka \underline{𝒢} ∈ ∇\underline{w} ; nℋ ^{1}(C, G)\underline{ν}(𝔽) is admissible if the σ̂-conjugacy class [b_{i}] := [ω_{ν_{i}}(τ\underline{𝒢} )] ∈ B(P_{ν_{i}}) lies in ⋃_{w′⪯w_{i}} Ψ(w′), for all 1 ≤ i ≤ n. Here ∇\underline{w} ; nℋ ^{1}(C, G)\underline{ν} denotes the stack

∇Ẑ\underline{ν} ; nℋ ^{1}(C, G)\underline{ν} with Ẑ\underline{ν} := (S(w_{i})ˆ×_{𝔽_{ω_{i}}} Spf 𝔽_{ω_{i}}[[ζ]])_{i}.

Definition 3.9. Keep the above notation. Define the functor L (resp. L_{spe}, resp. L_{adm}) which sends a ∇ℋ −data (G, (Ẑ_{ν_{i}})_{i}, H) (resp. (G, (Ẑ_{ν_{i}})_{i})) to the disjoint union ⨆\underline{𝒢} 𝒮 (\underline{𝒢} ) of the quotient stack

𝒮 (\underline{𝒢} ) := I(Q)(\underline{𝒢} )\ ∏_{ν_{i}} X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )) × X\underline{ν}/H,  (3.17)

where \underline{𝒢} runs over the quasi-isogeny classes of (resp. quasi-isogeny classes of special, resp. quasi-isogeny classes of admissible) G-shtukas and X\underline{ν} = Isom^{⊗}(ω^{∘}, ω\underline{ν}(\underline{𝒢} )). We define the functor L^{∗} (resp. L_{spe}^{∗}, resp. L_{adm}^{∗}) as the functor which sends (G, (Ẑ_{i})_{i}) to lim_{⟵_{H}} L(G, (Ẑ_{ν_{i}})_{i}, H) (resp. lim_{⟵_{H}} L_{spe}(G, (Ẑ_{ν_{i}})_{i}, H), resp. lim_{⟵_{H}} L_{adm}(G, (Ẑ_{ν_{i}})_{i}, H)).

Remark 3.10. Notice that the above stacks 𝒮 (\underline{𝒢} ) are equipped with the following actions

(a) \underline{Φ} = (Φ_{i}) operates on L(G, (Ẑ_{i})_{i}, H) via the action of Φ_{i} on X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )). Here Φ_{i} := Φ_{b_{i}}, where b_{i} := ω_{ν_{i}}(τ\underline{𝒢} ); see Remark 3.7.

(b) G(𝔸\underline{ν} ; Q) operates via the Hecke correspondence. More explicitly let H, H′ ⊂ G(𝔸\underline{ν} ; Q) be compact open subgroups. Then the Hecke correspondences π(g)_{H,H′} are given by the diagrams

\xymatrix{ & ∏_{i} X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )) × X\underline{ν}/(H′ ∩ g^{−1}Hg) \ar@{-->}[dl] \ar@{-->}[dr] \\ ∏_{i} X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )) × X\underline{ν}/H′ &  & ∏_{i} X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )) × X\underline{ν}/H \ar@{-->}[ll] \\  & (x_{i})_{i} × h(H′ ∩ g^{−1}Hg) \ar@{|->}[dl] \ar[dr] \\ (x_{i})_{i} × hH′ &  & (x_{i})_{i} × hg^{−1}H} (3.18)

This defines a true operation of G(𝔸\underline{ν} ; Q) on lim_{⟵_{H}} L(G, (Ẑ_{i})_{i}, H).

(c) Let Z denote the center of G. Then Z(Q) operates on ∏_{ν_{i}} X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )) × X\underline{ν}/H via the factor X_{Z_{ν_{i}}} (ω_{ν_{i}}(\underline{𝒢} )), according to the inclusion Z(Q) ⊆ I(\underline{𝒢} )_{ν_{i}} := QIsog(ω_{ν_{i}}(\underline{𝒢} )).

3.2 Langlands-Rapoport Conjecture for the Moduli Stack of Global G-Shtukas

Theorem 3.11. (Langlands-Rapoport Conjecture For G-shtukas) There exist a canonical G(𝔸\underline{ν} ; Q) × Z(Q)- equivariant isomorphism of functors

L^{∗}(−) ^{∼} ⟶ ∇_{n}^{∗,−}ℋ ^{1}(C, −)\underline{ν}(\overline{𝔽}).

Moreover via this isomorphism, the operation \underline{Φ} on the left hand side of the above isomorphism corresponds to the Frobenius endomorphism σ on the right hand side. Furthermore, when G is semi-simple, one may replace L^{∗}(−) by L_{adm}^{∗}(−).

Lemma 3.12. Let \underline{𝒢} ∈ ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) be a global 𝔊-shtuka over S. Let \underline{L}_{ν_{i}}(\underline{𝒢} ) be the local ℙ_{ν_{i}}-shtuka associated with \underline{𝒢} via the crystalline realization functor ω_{ν_{i}}(−) at the characteristic place ν_{i}. Let f̃: \underline{L̃}′_{ν_{i}} → \underline{L}_{ν_{i}}(\underline{𝒢} ) be a quasi-isogeny of local ℙ_{ν_{i}}-shtukas over S. Then there exists a unique global 𝔊-shtuka \underline{𝒢} ′ ∈ ∇_{n}ℋ ^{1}(C, G)\underline{ν}(S) and a unique quasi-isogeny g = gf̃: \underline{𝒢} ′ → \underline{𝒢} which is the identity outside ν_{i}, such that the local ℙ_{ν_{i}}-shtuka associated with \underline{𝒢} ′ is \underline{L̃}′_{ν} and the quasi-isogeny of local ℙ_{ν}-shtukas induced by g is f̃. We denote \underline{𝒢} ′ by f̃^{∗}\underline{G}.

Proof. It is a particular case of [AH14a, Proposition 5.7].

Recall that a formal scheme over k[[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 k. It is called formally of finite type if in addition it is quasi-compact.

Let P denote the generic fiber of ℙ and 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⟩.  (3.19)

Remark 3.13. 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 (3.19) the element b has values in the fixed field of σ̂^{s} which is ℓ. Note that if k is algebraically closed, any σ̂-conjugacy class in LP (k) contains an element satisfying a decency equation; see [Kot85, Section 4].

Remark 3.14. 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 k((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 1.9]. In particular J_{b}(𝔽((z))) ⊂ P (F_{s}) ⊂ LP (ℓ) where ℓ is the finite field extension of 𝔽 of degree s.

Let Ẑ be a bound with reflex ring RẐ = κ[[ξ]] and special fiber Z ⊂ Fℓ_{ℙ} ˆ×_{𝔽} Spec κ ; see Definition 3.2. 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 3.13. 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 3.15. Keep the notation from above and set \underline{L}_{0} := \underline{𝕃}_{0}.

(a) Define the functor

\underline{M}Ẑ ; \underline{𝕃}_{0} : (Nilp_{ℓ[[ξ]]})^{o} ⟶ Sets  (3.20)
S ⟼ \{ Isomorphism classes of (\underline{ℒ} , δ̄)where
\underline{ℒ} is a local ℙ-shtuka bounded by Ẑand
\overline{δ} : ℒ\overline{S} → 𝕃_{ν_{i},\overline{S}} is a quasi-isogeny\}.

Note that this functor is represented by a closed ind-subscheme of F̂ℓ_{ℙ,ℓ[[ξ]]} := Fℓ_{ℙ} ˆ×_{𝔽} Spf ℓ[[ξ]] by [AH14a, Proposition 4.11].

(b) We define the associated affine Deligne-Lusztig variety 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, we set X_{⪯ω}(b) := X_{S(ω)}(b).

The representablity of the above functor 3.20 was studied in [HV11] and [AH14a]. Let us recall the following theorem from [AH14a, Theorem 4.18].

Theorem 3.16. The functor \underline{M}Ẑ ; \underline{𝕃}_{0} : (Nilp_{ℓ[[ξ]]})^{o} → Sets is pro-representable by a formal scheme over Spf ℓ[[ξ]] which is locally formally of finite type. Its underlying reduced subscheme equals X_{Z}(b). In particular X_{Z}(b) is a scheme locally of finite type over ℓ. The formal scheme representing \underline{M}Ẑ ; \underline{𝕃}_{0} is called a bounded Rapoport-Zink space for local ℙ-shtukas.

Remark 3.17. By our assumptions QIsog_{ℓ}(\underline{𝕃}_{0}) equals the group J_{b}(𝔽((z))) from Remark 3.14. 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 3.11. Consider a compact open subgroup H ⊂ G(𝔸\underline{ν} ; Q). Let \underline{𝒢} _{0} be a global G-shtuka over an algebraically closed field k with characteristic \underline{ν}, bounded by Ẑ\underline{ν}. We let I(Q) denote the group QIsog_{k}(\underline{𝒢} _{0}) of quasi-isogenies of \underline{𝒢} _{0}; see Definition 2.7. Let (\underline{𝕃}_{i})_{i=1...n} := ω\underline{ν}(\underline{𝒢} _{0}) denote the associated tuple of local ℙ_{ν_{i}}-shtukas over k, where ω\underline{ν} is the crystalline realization functor from Definition 2.23. Let MẐ_{i} ; \underline{𝕃}_{i} denote the Rapoport-Zink space of \underline{𝕃}_{i}; see Definition 3.15 and Theorem 3.16. Since k is algebraically closed we may assume that all \underline{𝕃}_{i} are trivialized and decent by [AH14a, Remark 4.10]. By Theorem 3.16 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 Spec k ˆ×_{𝔽\underline{ν}} Spf A\underline{ν} = Spf k[[ζ_{1}, …, ζ_{n}]] =: Spf k[[\underline{ζ}]]. Recall that the group J_{\underline{𝕃}_{i}}(Q_{ν_{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 3.17. Especially we see that the group I(Q) acts on ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} via the natural morphism

I(Q) ⟶ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}), α ↦ (ω_{ν_{i}}(α))_{i} =: (α_{i})_{i}.  (3.21)

Consider the following morphism

∏_{i} MẐ_{ν_{i}} ; \underline{𝕃}_{i} × X\underline{ν}(\underline{𝒢} _{0})/H ⟶ ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} ˆ×_{𝔽\underline{ν}} Spec k,

which sends (\underline{L}_{ν_{i}}, δ_{i})_{i} × αH to (δ_{n}^{∗} … δ_{1}^{∗}\underline{G}_{0}, ω\underline{ν}(g_{δ_{n}})^{−1} ∘ ⋯ ∘ ω\underline{ν}(g_{δ_{1}})^{−1} ∘ α); see Lemma 3.12. This induces the following morphism

Θ = Θ(\underline{𝒢} _{0}): I(Q)\ ∏_{i} MẐ_{ν_{i}} ; \underline{𝕃}_{i} × X\underline{ν}(\underline{𝒢} _{0})/H ⟶ ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} ˆ×_{𝔽\underline{ν}} Spec k

of ind-algebraic stacks over Spf k[[\underline{ζ}]] which is ind-proper and formally étale. Now let \{T_{j}\} be a set of repre-

sentatives of I(Q)-orbits of the irreducible components of ∏_{i} MẐ_{i} ; \underline{𝕃}_{i} × G(𝔸\underline{ν} ; Q)/H. Then the image Θ(T_{j}) of T_{j} under Θ′ is closed and each Θ(T_{j}) intersects only finitely many others. Let Z denote the union of the Θ(T_{j})

and let ∇_{n}^{H,Ẑ\underline{ν}}, ℋ ^{1}(C, G)\underline{ν} ; /Z be the formal completion of ∇_{n}^{H,Ẑ\underline{ν}}, ℋ ^{1}(C, G)\underline{ν} ˆ×_{𝔽\underline{ν}} Spec k along Z. According to [AH14b, ], Θ induces an isomorphism of formal algebraic stacks over Spf k[[\underline{ζ}]]

Θ_{Z}: I(Q)\ ∏_{i} MẐ_{ν_{i}} ; \underline{𝕃}_{i} × X\underline{ν}(\underline{𝒢} )/H ^{∼} ⟶ ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} ; /Z.

This isomorphism is compatible with the action of G(𝔸\underline{ν} ; Q)×Z(Q)×\underline{Φ}. To see this notice that the first factor acts through Hecke-correspondences on source (3.18) and target (3.16). Moreover since Φ_{i}(g).b_{i}.Φ_{i}(g)^{−1} = σ^{m_{i}} (g^{−1}.b.σ̂(g)), where m_{i} := [RẐ_{i} : 𝔽_{q}[[ζ_{ν}]]], using Remark 2.21 we see that the operation \underline{Φ} on the left hand side of the above isomorphism corresponds to the Frobenius endomorphism σ on the right hand side. Now, after passing to the special fiber, the theorem follows from Theorem 3.16, Lemma 3.18 bellow and [He14, Theorem 2.1]

The following lemma shows that the map Θ covers the Newton stratum.

Lemma 3.18. Let \underline{𝒢} _{0}, \underline{𝒢} ′_{0} be G-shtukas in ∇_{n}ℋ ^{1}(C, G)\underline{ν}(𝔽). Then the following are equivalent

(a) im Θ_{\underline{𝒢} _{0}} ∩ im Θ_{\underline{𝒢}} ′_{0} ≠ ∅

(b) im Θ_{\underline{𝒢} _{0}} = im Θ_{\underline{𝒢}} ′_{0}

(c) There is a quasi-isogeny \underline{𝒢} _{0} → \underline{𝒢} ′_{0} over 𝔽.

Proof. (a) ⇒ (b) obvious, (b)⇒ (c) follows from the fact that the image of the morphism Θ(\underline{𝒢} _{0}) lies in the quasi-isogeny locus of \underline{𝒢} _{0}. It remains to show that (c) implies (a). Let f : \underline{𝒢} _{0} → \underline{𝒢} ′_{0} be a quasi-isogeny over 𝔽. The quasi-isogeny f induces an n-tuple of quasi-isogenies ω_{ν_{i}}(f) : \underline{L}_{ν_{i}} → \underline{L}′_{ν_{i}} of local ℙ_{ν_{i}}-shtukas and also a tensor isomorphism V̌_{f} : V̌_{\underline{𝒢} _{0}} ^{∼} ⟶ V̌_{\underline{𝒢}} ′_{0}. These data in turn define the vertical arrows which make the following

[figure: ∇_{n}^{H,Ẑ\underline{ν}}ℋ ^{1}(C, G)\underline{ν} ˆ×_{𝔽\underline{ν}} Spec k; ∏_{i} X_{ν_{i}}(\underline{L}_{ν_{i}}) × X\underline{ν}(\underline{𝒢} _{0})/H; ∏_{i} X_{ν_{i}}(\underline{L}′_{ν_{i}}) × X\underline{ν}(\underline{𝒢} ′_{0}.)/H; Θ_{\underline{𝒢} _{0}}; Θ_{\underline{𝒢} ′_{0}}]

commutative.

REFERENCES 21

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M. 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/