Uniformizing The Moduli Stacks of Global 𝔊-Shtukas

Esmail Arasteh Rad and Urs Hartl^{∗}

March 19, 2019

Abstract

This is the second in a sequence of articles, in which we explore moduli stacks of global 𝔊-shtukas, the function field analogs for Shimura varieties. Here 𝔊 is a flat affine group scheme of finite type over a smooth projective curve C over a finite field. Global 𝔊-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global 𝔊shtukas are algebraic Deligne-Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the first article we explained the relation between global 𝔊-shtukas and local ℙ-shtukas, which are the function field analogs of p-divisible groups. Here ℙ is the base change of 𝔊 to the complete local ring at a point of C. When ℙ is smooth with connected reductive generic fiber we proved the existence of Rapoport-Zink spaces for local ℙ-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global 𝔊-shtukas for smooth 𝔊 with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands-Rapoport conjecture for our moduli stacks. Mathematics Subject Classification (2000): 11G09, (11G18, 14L05)

Contents

1 Introduction 1

2 𝔊-Bundles 5

3 Moduli Stacks of Global 𝔊-Shtukas 10

4 Loop Groups and Local ℙ-Shtukas 20

5 Unbounded Uniformization 23

6 Tate Modules of Global 𝔊-Shtukas 25

7 The Uniformization Theorem in the Bounded Case 30

References 43

1 Introduction

Since Drinfeld’s celebrated work [Dri74, Dri77, Dri87, Dri89] it is clear that moduli spaces of shtukas are the function field analog of Shimura varieties and of similar importance in arithmetic algebraic geometry. To give their definition 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. Fix an integer n > 0. 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) called legs, and a Frobenius connection τ defined outside the graphs Γ_{s_{i}} of the legs 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})^{∗}. More generally, we also consider iterated global 𝔊-shtukas; see Definition 3.3.

∗Both authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) in form of the research grant HA3002/2-1, the SFB’s 478 and 878, and Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics– Geometry–Structure.

In Theorem 3.15 of this article we show that the moduli stack ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) of global 𝔊-shtukas, after imposing suitable boundedness conditions (by \underline{ω}) and D-level structures, is a Deligne-Mumford stack separated and locally of finite type over (C ∖ D)^{n}. (See the end of this introduction for an explanation of our notation for the moduli stack of global 𝔊-shtukas.) Dropping the boundedness condition by \underline{ω} we obtain ∇_{n}ℋ_{D}^{1}(C, 𝔊) = lim_{⟶} ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) as an ind-algebraic Deligne-Mumford stack; see Definition 3.14. Spelling out the Riemann-Hilbert correspondence for function fields, together with the Tannakian philosophy, one sees that ∇\underline{ω} ; nℋ_{D}^{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. On the other hand this analogy can be viewed as an attempt to build a bridge between the geometric Langlands program and the arithmetic Langlands program, where the role of global 𝔊-shtukas is played by abelian varieties (together with additional structures), respectively D-modules.

Note that our moduli stack ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) generalizes the space of F -sheaves F Sh_{D,r} which was considered by Drinfeld [Dri87] and Lafforgue [Laf02] in their proof of the Langlands correspondence for 𝔊 = GL_{2} (resp. 𝔊 = GL_{r}), and its generalization F Bun by Varshavsky [Var04]. It likewise generalizes the moduli stacks Eℓℓ_{C,𝒟,I} of Laumon, Rapoport and Stuhler [LRS93], their generalizations by L. Lafforgue [Laf97], Ngô [Ngô06] and Spieß [Spi10], the spaces Cht\underline{λ} of Ngô and Ngô Dac [NN08, Ngô13], and the spaces Ab-Sh_{H}^{r,d} of the second author [Har05]; see Remark 3.20. Varshavsky’s stacks F Bun were recently used by V. Lafforgue [Laf18] to prove Langlands parameterization for split groups over function fields. Lafforgue [Laf18, § 12.3.2] also indicates the generalization to non-split groups and a corresponding generalization of Varshavsky’s moduli stacks of shtukas. Our Theorem 3.15 gives a way to justify some of his assertions; see Remark 3.20. The proof of Theorem 3.15 is similar to Varshavsky’s algebraicity result of his stack F Bun. But beyond this, our focus in the present article is quite different from Varshavsky’s.

Namely, our approach to study the moduli stack of global 𝔊-shtukas is to relate it to certain moduli spaces for local objects, called 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 let ℙ = ℙ_{ν} := 𝔊 ×_{C} Spec A_{ν} and P_{ν} = 𝔊 ×_{C} Spec Q_{ν}. 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 Chapter 4), L denotes the LP_{ν}-torsor associated with L_{+}, and σ̂^{∗}L denotes the pullback of L under the absolute 𝔽_{ν}-Frobenius endomorphism Frob_{(#𝔽_{ν}),S}: S → S. Moreover, Nilp_{A_{ν}} denotes the category of A_{ν}-schemes on which the uniformizer ζ of A_{ν} is locally nilpotent. Local ℙ_{ν}-shtukas can be viewed as function field analogs of p-divisible groups; see also [HV11, HV12].

If ℙ_{ν} is smooth over A_{ν} we proved in [AH14, Theorem 4.4] for a fixed local ℙ_{ν}-shtuka \underline{𝕃} over a field k ∈ Nilp_{A_{ν}} that the Rapoport-Zink functor

M\underline{𝕃}: Nilp_{k[[ζ]]} ⟶ 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}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 also Theorem 4.4. To obtain a formal scheme locally formally of finite type, as in the analog for p-divisible groups, one has to assume that the generic fiber P_{ν} is connected reductive, and one has to impose a bound Ẑ on the Hodge polygon, that is on the relative position of σ̂^{∗}L_{+} and L_{+} under τ̂; see Definition 4.5. By [AH14, Theorem 4.18] the bounded Rapoport-Zink functor MẐ ; \underline{𝕃} is representable by a formal scheme locally formally of finite type over Spf RẐ, where RẐ is the reflex ring of the bound Ẑ; see Definition 4.5 and Theorem 4.8.

Consider a fixed n-tuple of pairwise different characteristic places \underline{ν} = (ν_{1}, …, ν_{n}) of C and 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 \underline{ν} ∈ C^{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 we associated in [AH14, Definition 5.4] 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). The relation between global 𝔊-shtukas and local ℙ-shtukas was thoroughly explained in [AH14, Chapter 5].

As was pointed out in [HV11] the true analogs of p-divisible groups are bounded local ℙ-shtukas. Nevertheless in Chapter 5 we prove that the product ∏_{i} M_{Γ̂_{ν_{i}}(\underline{G})} can be regarded as a uniformization space for ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} already in the unbounded situation. Namely, in Theorem 5.3 we construct the uniformization morphism ∏_{i} M_{Γ̂_{ν_{i}}(\underline{G})} → ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} and show that it is ind-proper and formally étale. However, it will

gain its full strength and provide a true (partial) uniformization only after we fix an n-tuple \underline{Ẑ} = (Ẑ_{i})_{i} of bounds and consider global 𝔊-shtukas \underline{G} whose associated local ℙ_{ν_{i}}-shtuka Γ̂_{ν_{i}}(\underline{G}) is bounded by Ẑ_{i} for all i = 1, …, n. We let RẐ_{i} = κ_{i}[[ξ_{i}]] be the reflex ring of the bound Ẑ_{i}, and we set R\underline{Ẑ} := κ[[ξ_{1}, …, ξ_{n}]], where κ is the compositum of all the κ_{i} in an algebraic closure of 𝔽_{q}. We also introduce the notion of a rational H-level structure for a compact open subgroup H ⊂ 𝔊(𝔸\underline{ν}) on a global 𝔊-shtuka \underline{G} using the Tannakian theory of Tate modules in Chapter 6. Here 𝔸\underline{ν} is the ring of adeles of C outside \underline{ν} and we assume that 𝔊 is smooth over C with connected fibers. We denote by ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} the stack of global 𝔊-shtukas with H-level structure fibered in groupoids over Nilp_{A\underline{ν}}, and by ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} the closed ind-substack of ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{A\underline{ν}} Spf R\underline{Ẑ} consisting of global 𝔊-shtukas bounded by \underline{Ẑ}. Both stacks are ind-algebraic Deligne-Mumford stacks, ind-separated and locally of ind-finite type; see Theorem 6.7 and Remark 7.3. To describe our main uniformization result let \underline{G}_{0} be a fixed global 𝔊-shtuka in ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(k) where the field k ∈ Nilp_{R\underline{Ẑ}} is an algebraic closure of 𝔽_{q}. Let (\underline{𝕃}_{i})_{i} := ˆ \underline{Γ}(G_{0}) be the associated n-tuple of local ℙ_{ν_{i}}-

shtukas, and let I_{\underline{G}_{0}}(Q) denote the group QIsog_{k}(\underline{G}_{0}) of quasi-isogenies of \underline{G}_{0}; see Definition 3.4. Let M̆Ẑ_{i} ; \underline{𝕃}_{i}

denote the base change of MẐ_{i} ; \underline{𝕃}_{i} to R̆Ẑ_{i} := k[[ξ]], and set R̆\underline{Ẑ} := k[[ξ_{1}, …, ξ_{n}]]. If 𝔊 is smooth over C with connected fibers and reductive generic fiber, we construct in Theorem 7.11 the uniformization morphism

Θ: I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}  (1.1)

and in addition we prove that it induces an isomorphism after passing to the formal completion along its image. Note that the reduced subscheme of the Rapoport-Zink space for local ℙ-shtukas is an affine Deligne-Lusztig variety; see Theorem 4.8. Thus as a consequence of the uniformization theorem one can relate the rational points in the quasi-isogeny class of \underline{G}_{0} inside the moduli stack of global 𝔊-shtukas to the rational points of certain affine Deligne-Lusztig varieties. This has applications to the analog of the Langlands-Rapoport conjecture for ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} which we discuss in a future article. The latter analog aims to give a description of the points of ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} with values in finite fields, which is functorial in all the data, including the curve C. The general functoriality properties in dependence of (C, 𝔊, H, \underline{Ẑ}) are studied by Breutmann [Bre19]. For this reason we like to carry all the data along in the notation. Moreover, for the classifying space of 𝔊-torsors we prefer Behrend’s [Beh91] notation ℋ ^{1}(C, 𝔊) instead of the symbol Bun_{𝔊}. Our notation ∇_{n}ℋ ^{1}(C, 𝔊) (instead of 𝔊-Sht) for the space of global 𝔊-shtukas indicates that 𝔊-shtukas consist of a 𝔊-torsor together with a Frobenius connection. It is reminiscent of Drinfeld’s original inspiration by the Riemann-Hilbert correspondence, and the relation to the geometric Langlands program.

Let us add that similar uniformization results were previously obtained by Drinfeld [Dri76] who uniformized the moduli spaces of Drinfeld modules at ∞, by the second author [Har05] who uniformized the moduli stacks Ab-Sh_{H}^{r,d} of abelian τ-sheaves at ∞, and by Blum and Stuhler [BS97] and Hausberger [Hau05] who uniformized the moduli spaces Eℓℓ_{C,𝒟,I} of 𝒟-elliptic sheaves from [LRS93] at ∞, respectively at finite places. Spieß [Spi10] related the latter two uniformizations to each other. The analogous situation for Shimura varieties was developed by Čerednik [Čer76], Drinfeld [Dri76], Rapoport and Zink [RZ96], Boutot and Zink [BZ95], Varshavsky [Var98], and Kudla and Rapoport [KR13].

Acknowledgements. We are grateful to Alain Genestier who suggested to the second author already many years ago to investigate the uniformization Theorem 7.11 for smooth group schemes. We also thank Vincent Lafforgue for important remarks and his explanations on his current application of moduli spaces of 𝔊-shtukas to Langlands parameterization. Moreover, we thank Jochen Heinloth, Linus Kramer, Tuân Ngô Dac, Timo Richarz and Eva Viehmann for helpful discussions, and the referee for his careful reading and many valuable remarks.

Notation and Conventions

Throughout this article we denote by 𝔽_{q} a finite field with q elements, 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, n ∈ ℕ_{>0} a positive integer, \underline{ν} := (ν_{i})_{i=1...n} an n-tuple of closed points of C, 𝕆\underline{ν} := ∏_{ν∉\underline{ν}} A_{ν} the ring of integral adeles of C outside \underline{ν}, 𝔸\underline{ν} := 𝕆\underline{ν} ⊗_{O_{C}} Q the ring of adeles of C outside \underline{ν}, 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,

When R = 𝔽 we drop the subscript R from the notation of 𝔻_{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 étale topology. We also denote by 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,

Nilp

𝔽[[ζ]]

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 smooth 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 over an fppf -covering of X. Here H is viewed as an H-torsor by right multiplication.

2 𝔊-BUNDLES 5

2 𝔊-Bundles

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. From Chapter 4 on we are mainly interested in the situation where 𝔊 is a smooth group scheme. Furthermore, we assume in Chapter 6 that all fibers of 𝔊 are connected (and smooth), and in Chapter 7 that its generic fiber is reductive. In the following we use the term vector bundle for a locally free sheaf of finite rank. To construct useful representations of 𝔊 on vector bundles over C we consider the following condition on a pair (H, G) of group schemes of finite type over C such that H ↪ G is a closed subgroup scheme and flat over C.

There is a scheme Y affine and of finite type over C with an action  (2.1)
G ×_{C} Y → Y of G and a G-equivariant open immersion G/H ↪ Y.

If (H, G) satisfies (2.1) then the quotient G/H is a scheme quasi-affine and of finite type over C. We do not know whether the converse is true. Note that for a quasi-affine scheme X of finite type over a field k it is in general false that Γ(X, O_{X}) is a finitely generated k-algebra; see the discussion on http://mathoverflow.net/questions/209443/. Therefore we do not know whether we can take Y = \underline{Spec}_{C} O_{G/H} in condition (2.1). For this reason we need the following

Proposition 2.1. Let H ↪ G be a closed immersion of group schemes of finite type over C and let H be flat over C.

(a) The quotient G/H is representable by a scheme separated and of finite type over C. It is flat (respectively smooth) over C if and only if G is.

(b) Let G′ ↪ G be a closed immersion of group schemes such that G′ is flat over C and H ↪ G factors through G′. If the schemes G/G′ and G′/H are (quasi-)affine over C then also G/H is (quasi-)affine over C. Moreover, if (H, G′) satisfies (2.1) and G/G′ is affine over C then also (H, G) satisfies (2.1).

(c) Let G be affine and smooth over C such that all its geometric fibers are reductive groups and G/G^{∘} → C is finite, where G^{∘} ⊂ G is the open normal subgroup scheme whose fiber (G^{∘})_{x} over any x ∈ C is the connected component of G_{x}; see [SGA 3, VI_{B}, Théorème 3.10]. Let H be smooth over C. Then G/H is affine over C if and only if all geometric fibers of H are reductive and H/H^{∘} → C is finite.

Proof. (a) was proved in [Ana73, Théorème 4.C] and [SGA 3, VI_{B}, Proposition 9.2(x,xi,xii)].

(b) The property of being (quasi-)affine is fpqc-local on the base by [EGA, IV_{2}, Proposition 2.7.1]. We apply the base change G → G/G′ which is faithfully flat by [SGA 3, VI_{B}, Proposition 9.2(xi)] to the morphism G/H → G/G′ to obtain (G/H)×_{G/G′} G → G. This morphism equals the projection (G′/H)×_{C} G → G under the isomorphism α: (G/H) ×_{G/G′} G ^{∼} ⟶ (G′/H) ×_{C} G, (g_{1}H, g_{2}) ↦ (g_{2}^{−1}g_{1}H, g_{2}) with inverse (g′H, g_{2}) ↦ (g_{2}g′H, g_{2}). Therefore G/H → G/G′ is (quasi-)affine and by [EGA, II, Proposition 5.1.10(ii)] this proves that G/H is (quasi-)affine over C.

Let moreover Y′ be a scheme affine and of finite type over C with an action G′ ×_{C} Y′ → Y′ of G′ and a G′-equivariant open immersion ι: G′/H ↪ Y′. Set U := G/G′ and let p: U′ := G → → G/G′ = U be the quotient morphism. Then G ×_{C} G′ ^{∼} ⟶ U′ ×_{U} U′ =: U′′ under the map (g_{1}, g′) ↦ (g_{1}, g_{1}g′) and U′′′ := U′ ×_{U} U′ ×_{U} U′ ^{∼} ⟶ G ×_{C} G′ ×_{C} G′. Let p_{i}: U′′ → U′, respectively p_{ij}: U′′′ → U′′ be the projections onto the i-th, respectively i-th and j-th component and set q := p_{1} ∘ p = p_{2} ∘ p. That is p_{1}(g_{1}, g′) = g_{1} and p_{2}(g_{1}, g′) = g_{1}g′. We consider the scheme Y′ ×_{C} G which is affine and of finite type over G = U′ with the induced open immersion

ι′ ∘ α: p^{∗}(G/H) := (G/H) ×_{G/G′} G ^{∼} ⟶ (G′/H) ×_{C} G ⟶ Y′ ×_{C} G ,

where ι′ := ι × id_{G}. Consider the U′′-isomorphism

φ: p_{2}^{∗}(Y′ ×_{C} G) = (Y′ ×_{C} G) ×_{G,p_{2}} U′′ ^{∼} ⟶ (Y′ ×_{C} G) ×_{G,p_{1}} U′′ = p_{1}^{∗}(Y′ ×_{C} G) ,
(y′, g_{2}, g_{1}, g_{2}) ⟼ ((g_{1}^{−1}g_{2}).y′, g_{1}, g_{1}, g_{2}) .

It satisfies the cocycle condition p_{13}^{∗}φ = p_{12}^{∗}φ ∘ p_{23}^{∗}φ over U′′′. Therefore Y′ ×_{C} G descends to a U-scheme Y → U = G/G′ with a U′-isomorphism γ: Y ×_{U} U′ ^{∼} ⟶ Y′ ×_{C} G satisfying p_{1}^{∗}γ = φ ∘ p_{2}^{∗}γ by [BLR90, § 6.1, Theorem 6]. Moreover, Y is affine and of finite type over G/G′ by [EGA, IV_{2}, Proposition 2.7.1]. In particular, Y → C is affine and of finite type because G/G′ → C is assumed to be affine. Since φ satisfies p_{1}^{∗}(ι′α) = φ ∘ p_{2}^{∗}(ι′α): q^{∗}(G/H) → Y′ ×_{C} G ×_{C} G′ the inclusion ι′α descends to a morphism G/H → Y by [BLR90, § 6.1, Theorem 6] which is an open immersion by [EGA, IV_{2}, Proposition 2.7.1].

To define a G-action on Y we consider G ×_{C} G/G′ as a scheme over U = G/G′ via the morphism μ̄: (g_{1}, g_{2}G′) ↦ g_{1}g_{2}G′. In this way the G ×_{C} G/G′-scheme G ×_{C} Y becomes a U-scheme and we want to construct a U-morphism G ×_{C} Y → Y. We construct it over U′ as the composition in the top row of the following diagram

[figure: (g_{1}, y, g_{2}); (g_{1}, y, g_{1}^{−1}g_{2}); (g_{1}, y′, g_{1}^{−1}g_{2}); (y′, g_{2}); (G ×_{C} Y) ×_{μ̄,U} U′; ∼; pr_{U′}; G ×_{C} (Y ×_{U} U′); ∼; id_{G} ×γ; μ∘(pr_{1},pr_{3}); G ×_{C} Y′ ×_{C} G; id_{Y′} ×μ; μ∘(pr_{1},pr_{3}); Y′ ×_{C} G; pr_{G}; ∼; γ^{−1}; Y ×_{U} U′; pr_{U′}; U′; G; G; G; U′.]

One checks that this morphism is compatible with the descent datum φ over U′′, because it is given by the action of G on Y′ ×_{C} G on the second factor, whereas φ is given by the action of G′ on the first factor. By [BLR90, § 6.1, Theorem 6] we obtain an action G ×_{C} Y → Y of G on Y and one also checks over U′ that the inclusion G/H → Y is G-equivariant. Therefore (H, G) satisfies condition (2.1).

(c) was proved by Alper [Alp14, Theorems 9.4.1 and 9.7.6].

Proposition 2.2. Let 𝔊 be a flat affine group scheme of finite type over C.

(a) There is a faithful representation 𝔊 ↪ GL(V) for a vector bundle V on C such that the quotient GL(V)/𝔊 is a quasi-affine scheme over C. Moreover, the pair (𝔊, GL(V)) satisfies (2.1).

(b) There is a faithful representation ρ: 𝔊 ↪ GL(V) for a vector bundle V on C together with an isomorphism α: ∧^{top} V ^{∼} ⟶ O_{C} such that ρ factors through SL(V) := ker(det: GL(V) → GL(∧^{top}V)) and the quotients SL(V)/𝔊 and GL(V)/𝔊 are quasi-affine schemes over C. Moreover, the pairs (𝔊, SL(V)) and (𝔊, GL(V)) satisfy (2.1).

(c) Let 𝔊 be smooth. Then the quotients in (a) and (b) are affine over C if and only if all geometric fibers of 𝔊 over C are reductive and 𝔊/𝔊^{∘} → C is finite.

Remark. The assumption that 𝔊 is smooth is necessary for the conclusion in (c) that the fibers of 𝔊 are reductive. This can be seen from the flat affine non-reductive group scheme lμ .. _{q} which is the kernel of the q-Frobenius endomorphism of the multiplicative group 𝔾_{m} and its faithful representation on 𝔾_{m} with 𝔾_{m}/lμ .. _{q} = 𝔾_{m}.

Proof. (a) can be proved as in [PR08, Proposition 1.3] and [Hei10, Example(1), page 504]. Namely, the argument of [PR08, Proposition 1.3] works over any covering C = U_{1} ∪ … ∪ U_{r} by affine open subsets U_{i} ⊂ C. For each i it first produces a faithful representation 𝔊 ×_{C} U_{i} ↪ GL(N_{i}) on a finitely generated O_{U_{i}}-submodule N_{i} ⊂ O_{𝔊}|_{U_{i}} of the “regular representation” 𝔊 → GL(O_{𝔊}). By Lemma 2.3(b) below we can extend this representation to a representation 𝔊 → GL(F_{i}) on a vector bundle F_{i} ⊂ O_{𝔊} over all of C with F_{i}|_{U_{i}} = N_{i}. The direct sum W := F_{1} ⊕ … ⊕ F_{r} is a faithful representation of 𝔊 over C.

Let G = GL(W). In the next step [PR08, Proposition 1.3] construct for each i a representation ρ_{i}: G×_{C} U_{i} → GL(V_{i}) on a vector bundle V_{i} ⊂ ∧^{ℓ_{i}}O_{G}|_{U_{i}} of some exterior power of O_{G} and a line bundle L_{i} ⊂ V_{i} such that for any U_{i}-scheme π: S → U_{i} one has

𝔊(S) = { g ∈ GL(W)(S): ∃ χ_{i}(g) ∈ O_{S}(S)^{×} with ρ_{i}(g)(v) = χ_{i}(g)v ∀ v ∈ Γ(S, π^{∗}L_{i}) } .

By Lemma 2.3(b) below we extend V_{i} and L_{i} to representations ρ_{i}: GL(W) → GL(Ṽ_{i}) and χ_{i}: 𝔊 → GL(L̃_{i}) = 𝔾_{m} on vector bundles Ṽ_{i} and L̃_{i} over all of C with L̃_{i} ⊂ Ṽ_{i} and Ṽ_{i}|_{U_{i}} = V_{i} and L̃_{i}|_{U_{i}} = L_{i}. Next [PR08] consider the representations ( id, χ_{i}): 𝔊 ↪ GL(W) ×_{C} 𝔾_{m} and ρ_{i} ⊗ (−1): GL(W) ×_{C} 𝔾_{m} →

GL(Ṽ_{i} ⊗_{O_{C}} L̃i^{∨}), (g, λ_{i}) ↦ ρ_{i}(g) ⊗ λ_{i}^{−1}. Consider the inclusion map h_{i}: L̃_{i} ↪ Ṽ_{i} ∈ Γ(C, Hom_{O_{C}}(L̃_{i}, Ṽ_{i})) =

Γ(C, Ṽ_{i} ⊗_{O_{C}} L̃i^{∨}) and let H_{i} ⊂ GL(W) ×_{C} 𝔾_{m} be the stabilizer of the section h_{i}, that is, H_{i}(S) = { (g, λ_{i}) ∈ GL(W)(S) × 𝔾_{m}(S): (ρ_{i}(g) ⊗ λ_{i}^{−1}) · h_{i} = h_{i} } for any C-scheme S. It is a closed subgroup scheme with 𝔊 ⊂ H_{i} and 𝔊 ×_{C} U_{i} = H_{i} ×_{C} U_{i}.

Now we consider the group morphisms ( id, χ_{1}, …, χ_{r}): 𝔊 ↪ GL(W) ×_{C} 𝔾_{m}^{r} and

⨁_{i} ρ_{i} ⊗ (−1): GL(W) ×_{C} 𝔾_{m}^{r} ⟶ GL(⨁_{i}Ṽ_{i} ⊗_{O_{C}} L̃i^{∨}) ,
(g, λ_{1}, …, λ_{r}) ⟼ ⨁_{i}(ρ_{i}(g) ⊗ λ_{i}^{−1}) ,

and the section ⨁_{i} h_{i} ∈ ⨁_{i}Ṽ_{i} ⊗_{O_{C}} L̃i^{∨}. By working on the covering C = U_{1} ∪ … ∪ U_{r} we compute that the stabilizer of ⨁_{i} h_{i} in GL(W) ×_{C} 𝔾_{m}^{r} equals 𝔊. Let Y be the reduced closure of the orbit of ⨁_{i} h_{i} in the geometric vector bundle Ṽ associated to the locally free sheaf ⨁_{i} Ṽ_{i} ⊗_{O_{C}} L̃i^{∨}. Then Y is an affine scheme of finite type over C, and the quotient (GL(W) ×_{C} 𝔾_{m}^{r})/𝔊, which is reduced by Proposition 2.1(a), is open in Y by the usual argument that orbits are open in their closure.

We show that Y carries an action of GL(W)×_{C}𝔾_{m}^{r} compatible with the action on the quotient (GL(W)×_{C} 𝔾_{m}^{r})/𝔊. The preimage of Y under the morphism d: GL(W) ×_{C} 𝔾_{m}^{r} ×_{C} Ṽ → Ṽ is closed and contains GL(W) ×_{C} 𝔾_{m}^{r} ×_{C} (GL(W) ×_{C} 𝔾_{m}^{r})/𝔊. To show that this preimage contains GL(W) ×_{C} 𝔾_{m}^{r} ×_{C} Y it suffices to show that the latter is irreducible and reduced. Reducedness follows from the smoothness of GL(W) ×_{C} 𝔾_{m}^{r} over C by [BLR90, § 2.3, Proposition 9]. Since Y is irreducible and reduced it is flat over C by [Har77, Proposition III.9.7]. Conversely the flatness of GL(W) ×_{C} 𝔾_{m}^{r} ×_{C} Y over C shows that it equals the closure of its fiber over the generic point η ∈ C. This fiber is irreducible, because Y_{η} is irreducible and (GL(W) ×_{C} 𝔾_{m}^{r})_{η} is geometrically irreducible. This proves that GL(W) ×_{C} 𝔾_{m}^{r} ×_{C} Y is irreducible and maps to Y under the action d. In particular, the pair (𝔊, GL(W) ×_{C} 𝔾_{m}^{r}) satisfies condition (2.1).

So for V := W ⊕ O_{C}^{⊕r} the pair (𝔊, GL(V)) also satisfies condition (2.1) by Proposition 2.1(c) and (b) applied to the closed immersions 𝔊 ↪ GL(W) ×_{C} 𝔾_{m}^{r} ↪ GL(W ⊕ O_{C}^{⊕r}).

(b) To pass to SL, we take a representation 𝔊 ↪ GL(V_{0}) as in (a) and we consider V := V_{0} ⊕∧^{top}V0^{∨} and the closed immersion id ⊕ det^{∨}: GL(V_{0}) ↪ GL(V) which factors through SL(V). Applying Proposition 2.1(c) and (b) to the closed immersions of group schemes 𝔊 ↪ GL(V_{0}) ↪ SL(V) ↪ GL(V) shows that the pairs (𝔊, SL(V)) and (𝔊, GL(V)) satisfy condition (2.1).

(c) This follows from Proposition 2.1(c).

Lemma 2.3. Let E be a quasi-coherent sheaf of O_{C}-modules on C. Let U ⊂ C be an open subscheme and let N ⊂ E|_{U} be a finitely generated O_{U}-submodule.

(a) There is a finitely generated O_{C}-submodule H ⊂ E with H|_{U} = N.

(b) Assume that E is a flat O_{C}-module. Let 𝔊 → GL(E) be a representation (possibly of infinite dimension) and assume that N is 𝔊 ×_{C} U-invariant. Then there is a finitely generated flat 𝔊-invariant O_{C}- submodule F ⊂ E with N = F|_{U}.

Proof. If U = C there is nothing to do. Otherwise we choose U′ = Spec A ⊂ C open such that C = U ∪ U′. By Riemann-Roch Ũ := U ∩ U′ is affine with O_{C}(Ũ) = A[\frac{1}{a}] for an element a ∈ A.

(a) Let (m_{i}) be a finite set of generators of the A[\frac{1}{a}]-module N |Ũ. After multiplying the m_{i} by suitable powers of a they lie in E|_{U′} and generate a finite O_{U′}-submodule N′ ⊂ E|_{U′} which by construction satisfies N′|Ũ = N |Ũ. Glueing N with N′ over Ũ produces the desired finitely generated submodule H.

(b) By (a) we may choose a finitely generated O_{C}-submodule H ⊂ E with H|_{U} = N. Let d_{E}: E → O_{𝔊} ⊗_{O_{C}} E be the co-multiplication of the representation 𝔊 → GL(E). For each open affine subset V ⊂ C let F(V) := { x ∈ Γ(V, E): d_{E}(x) ∈ Γ(V, O_{𝔊} ⊗_{O_{C}} H) }. Then F(V) is 𝔊 ×_{C} V -invariant and contained in H(V) by [Ser68, § 1.5, Proposition 1]. Since F(V) equals the kernel of the homomorphism Γ(V, E) ⟶^{d_{E}} Γ(V, O_{𝔊} ⊗_{O_{C}} E) → → Γ(V, O_{𝔊} ⊗_{O_{C}} E/H) the F(V) form a sheaf F ⊂ H of O_{C}-submodules. It follows that F is finitely generated and 𝔊-invariant. Since E is flat, whence O_{C}-torsion free, the same holds for F because C is a smooth curve. Moreover, N ⊂ F|_{U} ⊂ H|_{U} = N by the 𝔊 ×_{C} U-invariance of N.

Definition 2.4. 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 (also called 𝔊-bundles) over C_{S} and morphisms are isomorphisms of 𝔊-torsors.

Let D be a proper closed subscheme of C. A D-level structure on a 𝔊-bundle G on C_{S} is a trivialization ψ: G ×_{C_{S}} D_{S} ^{∼} ⟶ 𝔊×_{C} D_{S} along D_{S} := D ×_{𝔽_{q}} S. Let ℋ_{D}^{1}(C, 𝔊) denote the stack classifying 𝔊-bundles with D-level structure, that is, ℋ_{D}^{1}(C, 𝔊) is the category fibered in groupoids over the category of 𝔽_{q}-schemes, which assigns to an 𝔽_{q}-scheme S the category whose objects are

Ob(ℋ_{D}^{1}(C, 𝔊)(S)) := {(G, ψ): G ∈ ℋ ^{1}(C, 𝔊)(S), ψ: G ×_{C_{S}} D_{S} ^{∼} ⟶ 𝔊 ×_{C} D_{S}} ,

and whose morphisms are those isomorphisms of 𝔊-bundles that preserve the D-level structure.

While we adopt the notation of Behrend [Beh91], the stack ℋ ^{1}(C, 𝔊) is also often denoted Bun_{𝔊}. The following theorem is well known when 𝔊 is a parahoric group scheme over C; see Definition 3.11 below. A proof is given for example by Heinloth [Hei10, Proposition 1].

Theorem 2.5. Let 𝔊 be a flat affine group scheme of finite type over the curve C. Then the stack ℋ ^{1}(C, 𝔊) is an Artin-stack locally of finite type over 𝔽_{q}. It admits a covering by connected open substacks of finite type over 𝔽_{q}. If 𝔊 is smooth over C then ℋ ^{1}(C, 𝔊) is smooth over 𝔽_{q}.

The theorem can be proved following the argument of Behrend [Beh91]. Since we will need the technique in Theorem 3.15 below, we indicate Behrend’s argument and thereby generalize his Propositions 4.4.4 and 4.4.1 and 4.4.5.

Theorem 2.6. Let V be a vector bundle over C and let ρ: 𝔊 ↪ GL(V) be a closed subgroup scheme flat over C, such that (𝔊, GL(V)) satisfies condition (2.1). Then the natural morphism of stacks over 𝔽_{q}

ρ_{∗}: ℋ ^{1}(C, 𝔊) → ℋ ^{1}(C, GL(V))

is representable by a morphism of schemes which is quasi-affine and of finite presentation.

Proof. Let p_{S}: C_{S} → S be the projection map and view it as a morphism of big étale sites Ét(C_{S}) → Ét(S). For any scheme Y over C_{S} let p_{S∗}(Y) denote the sheaf which sends an S-scheme T to Hom_{C_{S}}(C_{T} , Y). Let G be a GL(V)-bundle in ℋ ^{1}(C, GL(V))(S). By [Beh91, Proposition 4.2.3] we have the following 2-cartesian diagram of stacks

\begin{matrix}p_{S∗} (G/𝔊_{S}) & ⟶ & S \\ ↓ & & ^{G}↓ \\ ℋ ^{1}(C, 𝔊) & ⟶ & ℋ ^{1}(C, GL(V)). \end{matrix}  (2.2)

(See the proof of Proposition 3.9 for more details on this diagram.) We must show that p_{S∗} (G/𝔊_{S}) is a quasi-affine S-scheme of finite presentation. We claim that there exists a scheme Y affine and of finite presentation over C_{S} and a quasi-compact open immersion of C_{S}-schemes G/𝔊_{S} ↪ Y. We will prove this over a Zariski covering U′→ C_{S} of C_{S} and then glue. In this way we can assume that there is a trivialization α: G ×_{C_{S}} U′ ^{∼} ⟶ GL(V) ×_{C} U′. Here a Zariski covering suffices, because every GL(V)-torsor is trivial Zariski locally by Hilbert 90, see [Mil80, Proposition III.4.9]. Since (𝔊, GL(V)) satisfies condition (2.1) there is a scheme Y_{0} affine and of finite presentation over C with an action GL(V) ×_{C} Y_{0} → Y_{0} of GL(V) and a GL(V)-equivariant open immersion ι: GL(V)/𝔊 ↪ Y_{0} which automatically is quasi-compact. We consider the quasi-compact open immersion ια: (G/𝔊_{S}) ×_{C_{S}} U′ ^{∼} ⟶ (GL(V)/𝔊) ×_{C} U′ ↪ Y_{0} ×_{C} U′. Let p_{i}: U′′ := U′ ×_{C_{S}} U′ → U′ be the projection onto the i-th factor. Over U′′ the automorphism p_{2}^{∗}α ∘ p_{1}^{∗}α^{−1} of the trivial GL(V)-torsor is given by multiplication with an element g ∈ GL(V)(U′′) satisfying the cocycle condition on U′′′ := U′ ×_{C_{S}} U′ ×_{C_{S}} U′. The isomorphism g: p_{1}^{∗}(Y_{0} ×_{C} U′) ^{∼} ⟶ p_{2}^{∗}(Y_{0} ×_{C} U′) allows to glue Y_{0} ×_{C} U′ to a C_{S}-scheme Y which is affine and of finite presentation over C_{S}. By construction, the open immersion ια glues to a quasi-compact open immersion G/𝔊_{S} ↪ Y of C_{S}-schemes. Thus the theorem follows from the next lemma.

Lemma 2.7. Let X be a projective scheme over a field k. Let p denote the structure morphism p: X → Spec k. Let S be a k-scheme, let Y be a scheme affine and of finite presentation over X_{S} := X ×_{k} S, and let Ỹ ↪ Y a quasi-compact open immersion. Then p_{S}_{∗}Ỹ is a quasi-affine S-scheme of finite presentation.

Proof. Since Y is of finite presentation over X_{S} and Ỹ ↪ Y is quasi-compact we may reduce to the case that S is of finite type over k and in particular noetherian; see [EGA, IV_{3}, Proposition 8.9.1]. By [Beh91, Proposition 4.4.1] the sheaf A := p_{S∗}Y is representable by an affine S-scheme of finite presentation. Consider the universal X_{S}-morphism X_{A} → Y corresponding to id_{A} ∈ A(A). Let Z ↪ X_{A} be the complement of X_{A} ×_{Y} Ỹ in X_{A}. By properness of X, Z maps to a closed subscheme of A. Let U denote the complement of p_{A}(Z) in A. It is quasi-compact because we assumed that S and hence A are noetherian. In particular, U is a quasi-affine S-scheme of finite presentation.

We claim that U represents p_{S∗}Ỹ. To see this, first observe that the open immersion X_{U} ↪ X_{A} ×_{Y} Ỹ gives a morphism X_{U} → Ỹ which induces a U-point in (p_{S∗}Ỹ)(U). Hence it is enough to check that for any scheme T we have the inclusion (p_{S∗}Ỹ)(T) ⊆ U(T). Any point of (p_{S∗}Ỹ)(T) is a morphism X_{T} → Ỹ. Composing with i: Ỹ → Y induces a T valued point α of A. We have to show that α: T → A factors through the open subscheme U. One can easily check this on the level of topological spaces. Namely, if T ×_{A}p_{A}(Z) is non-empty then so is X_{T} ×_{X_{A}} Z, which is a contradiction, since Z is defined as the complement of X_{A} ×_{Y} Ỹ in X_{A}. This shows that U represents p_{S∗}Ỹ.

For the proof of Theorem 2.5 we use the following

Remark 2.8. Let V be a vector bundle on C of rank r. The stack ℋ ^{1}(C, GL(V)) is isomorphic to the stack Vect_{C}^{r} whose S-valued points parameterize vector bundles of rank r on C_{S}. For a GL(V)-torsor G over C_{S} we let V(G) denote the associated vector bundle over C_{S}. By [LM00, Théorème 4.6.2.1] the stack Vect_{C}^{r} is an Artin-stack locally of finite type over 𝔽_{q} and by [Wan11, Theorem 1.0.1] it admits a covering by connected open substacks of finite type over 𝔽_{q}. (Note that even though [LM00, Théorème 4.6.2.1] states that Vect_{C}^{r} is of finite type, all that is true and proved is that it is locally of finite type.)

Proof of Theorem 2.5. By Proposition 2.2(a) there is a closed embedding 𝔊 ↪ GL(V) such that (𝔊, GL(V)) satisfies (2.1). By Theorem 2.6 and Remark 2.8 the stack ℋ ^{1}(C, 𝔊) is an Artin-stack locally of finite type over 𝔽_{q} and admits a covering by connected open substacks of finite type over 𝔽_{q}. Finally the smoothness of ℋ ^{1}(C, 𝔊) follows from the vanishing of the second cohomology of coherent sheaves on a curve; compare [Beh91, Corollary 4.5.2] or [Wan11, § 6].

Remark 2.9. A covering of ℋ ^{1}(C, 𝔊) by connected open substacks of finite type over 𝔽_{q} can be constructed as follows. Let ρ: 𝔊 ↪ GL(V) be a representation with quasi-affine quotient which factors through SL(V) as in Proposition 2.2(b) such that (𝔊, GL(V)) satisfies (2.1), and let μ = (μ_{1} ≥ … ≥ μ_{r}) with μ_{1} +…+μ_{r} = 0 be a dominant cocharacter of SL_{r}, where r = rk V. We use the notation of Remark 2.8. Let ℋ ^{1}(C, 𝔊)^{≤μ} be the sub-stack of ℋ ^{1}(C, 𝔊) consisting of those 𝔊-bundles G on C_{S} for 𝔽_{q}-schemes S such that for any geometric point s ∈ S and for any filtration (0) = F_{0} ⊂ … ⊂ F_{r} = V(ρ_{∗}G)_{s} of the vector bundle F := V(ρ_{∗}G)_{s} on C_{s} by sub-vector bundles F_{i} with line bundles as subquotients (such a filtration is called a B-structure) the degree of the line bundle F_{i}/F_{i−1} is at most μ_{i}. Since ρ factors through SL(V) there is a canonical isomorphism α_{G}: ∧^{r} V(ρ_{∗}G) ^{∼} ⟶ O_{C_{S}} and therefore the condition that deg(F_{i}/F_{i−1}) ≤ μ_{i} for all i is equivalent to the condition

deg ⨂_{i=1}^{r}(F_{i}/F_{i−1})^{⊗λ_{i}} ≤ ∑_{i=1}^{r} μ_{i}λ_{i} for all integers λ_{1} ≥ … ≥ λ_{r} with λ_{1} + … + λ_{r} = 0 .  (2.3)

3 MODULI STACKS OF GLOBAL 𝔊-SHTUKAS 10

It follows from [Var04, Lemma A.3], that the stack ℋ ^{1}(C, GL_{r})^{≤μ} defined by condition (2.3) (which for arbitrary vector bundles F is not equivalent to deg(F_{i}/F_{i−1}) ≤ μ_{i} for all i) is an open substack of ℋ ^{1}(C, GL_{r}) ≅ V ect_{C}^{r}. Its intersections with the connected components of ℋ ^{1}(C, GL_{r}) are Artin stacks of finite type over 𝔽_{q} by [Var04, Lemma 3.1(a)]. Thus also ℋ ^{1}(C, 𝔊)^{≤μ} is an open substack of ℋ ^{1}(C, 𝔊) and an Artin stack of finite type over 𝔽_{q}, being the preimage of ℋ ^{1}(C, GL_{r})^{≤μ} under the morphism of stacks ρ_{∗}: ℋ ^{1}(X, 𝔊) → ℋ ^{1}(X, GL(V)) from Theorem 2.6, which factors through the connected component on which the vector bundles have degree zero; compare [BD, § 2.1.1] or [Var04, Lemma 2.2].

Moreover, if the closed subscheme D ⊂ C is large enough compared to μ then the preimage ℋ_{D}^{1}(C, GL_{r})^{≤μ} of ℋ ^{1}(C, GL_{r})^{≤μ} in ℋ_{D}^{1}(C, GL_{r}) is a scheme over 𝔽_{q} with quasi-projective connected components by [Var04, Lemma 3.1(a)]. Therefore also ℋ_{D}^{1}(C, 𝔊)^{≤μ} is a quasi-projective scheme over 𝔽_{q} by Theorem 2.6.

3 Moduli Stacks of Global 𝔊-Shtukas

Definition 3.1. Let I be a finite set and let I = I_{1} ∪ … ∪ I_{k} with I_{j} ∩ I_{j′} = ∅ for j ≠ j′ be a partition of I. We write \underline{I} := (I_{1}, …, I_{k}). We let Hecke_{𝔊,D, \underline{I}} be the stack fibered in groupoids over the category of 𝔽_{q}-schemes, whose S valued points are tuples ((s_{i})_{i∈I}, (G_{j}, ψ_{j})_{j=0,...,k}, (τ_{j})_{j=1,...,k} ) where

s_{i} ∈ (C ∖ D)(S) for all i ∈ I are sections, called legs (or characteristic sections),

(G_{j}, ψ_{j}) for all j = 0, …, k are objects in ℋ_{D}^{1}(C, 𝔊)(S), and

τ_{j}: G_{j−1}|_{C_{S}∖⋃_{i∈I_{j}} Γ_{s_{i}}} ^{∼} ⟶ G_{j}|_{C_{S}∖⋃_{i∈I_{j}} Γ_{s_{i}}} for all j = 1, … k are isomorphisms preserving the D-level structures, that is, ψ_{j} ∘ τ_{j}|_{D_{S}} = ψ_{j−1}. Here Γ_{s_{i}} ⊂ C_{S} denotes the graph of the section s_{i}.

If D = ∅ we will drop it from the notation. For each j = 0, …, k we denote by pr_{j}: Hecke_{𝔊,D, \underline{I}} → ℋ_{D}^{1}(C, 𝔊) the morphism sending ((s_{i})_{i∈I}, (G_{j}, ψ_{j})_{j=0,...,k}, (τ_{j})_{j=1,...,k} ) to (G_{j}, ψ_{j}). If k = 1 and I = I_{1} = {1, …, n} we also write Hecke_{𝔊,D,n} := Hecke_{𝔊,D,(I)} := Hecke_{𝔊,D, \underline{I}}. In general there is a canonical isomorphism

Hecke_{𝔊,D, \underline{I}} ^{∼} ⟶ Hecke_{𝔊,D,(I_{1})} ×_{pr_{1},ℋ_{D}^{1}(C,𝔊),pr_{0}} … ×_{pr_{1},ℋ_{D}^{1}(C,𝔊),pr_{0}} Hecke_{𝔊,D,(I_{k})}  (3.1)

sending the tuple ((s_{i})_{i∈I}, (G_{j}, ψ_{j})_{j=0,...,k}, (τ_{j})_{j=1,...,k} ) ∈ Hecke_{𝔊,D, \underline{I}}(S) to the tuple in the j-th factor ((s_{i})_{i∈I_{j}}, (G_{j−1}, ψ_{j−1}), (G_{j}, ψ_{j}), τ_{j} ) ∈ Hecke_{𝔊,D,(I_{j})}(S).

Definition 3.2. 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 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.

Definition 3.3. Keep the notation of Definition 3.1. We define the moduli stack ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) of (iterated) global 𝔊-shtukas with D-level structure to be the preimage on Hecke_{𝔊,D, \underline{I}} of the graph of the Frobenius morphism on ℋ ^{1}(C, 𝔊). In other words

∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) := equi(σ_{ℋ_{D}^{1}(C,𝔊)} ∘ pr_{k}, pr_{0} : Hecke_{𝔊,D, \underline{I}} ⇉ ℋ_{D}^{1}(C, 𝔊)).

If k = 1 and I = I_{1} = \{1, …, n\} we also write ∇_{n}ℋ_{D}^{1}(C, 𝔊) := ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊). Each object \underline{G} in Ob(∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊)(S)) is called an (iterated) global 𝔊-shtuka with D-level structure over S and the corresponding sections (s_{i})_{i∈I} are called the legs (or characteristic sections) of \underline{G}, or of S.

More explicitly, a global 𝔊-shtuka \underline{G} ∈ ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊)(S) with D-level structure over an 𝔽_{q}-scheme S is an object \underline{G} = ((s_{i})_{i∈I}, (G_{j}, ψ_{j})_{j=0,...,k}, (τ_{j})_{j=1,...,k} ) of Hecke_{𝔊,D, \underline{I}}(S) together with an isomorphism τ_{0} : σ^{∗}(G_{k}, ψ_{k}) ^{∼} ⟶ (G_{0}, ψ_{0}) in ℋ_{D}^{1}(C, 𝔊), that is an isomorphism τ_{0} : σ^{∗}G_{k} ^{∼} ⟶ G_{0} of 𝔊-bundles satisfying ψ_{0} ∘ τ_{0}|_{D_{S}} = σ^{∗}ψ_{k}.

If k = 1 and I = I_{1} = \{1, …, n\} a global 𝔊-shtuka \underline{G} ∈ ∇_{n}ℋ_{D}^{1}(C, 𝔊)(S) with D-level structure over an 𝔽_{q}-scheme S is a tuple \underline{G} = (s_{1}, …, s_{n}, G, ψ, τ) consisting of an n-tuple of legs (s_{1}, …, s_{n}) ∈ (C ∖ D)^{n}(S), a 𝔊-bundle G := G_{1} over C_{S}, a trivialization ψ := ψ_{1} : G ×_{C_{S}} D_{S} ^{∼} ⟶ 𝔊 ×_{C} D_{S}, and an isomorphism τ := τ_{1} ∘ τ_{0} : σ^{∗}G|_{C_{S}∖∪_{i}Γ_{s_{i}}} ^{∼} ⟶ G|_{C_{S}∖∪_{i}Γ_{s_{i}}} with ψ ∘ τ|_{D_{S}} = σ^{∗}(ψ).

If D = ∅ we drop ψ from \underline{G} and write ∇_{\underline{I}}ℋ ^{1}(C, 𝔊), resp. ∇_{n}ℋ ^{1}(C, 𝔊) for the stack of global 𝔊-shtukas. If k = 1 and I = I_{1} = {1, …, n} we will sometimes fix the legs (s_{1}, …, s_{n}) ∈ C^{n}(S) and simply call \underline{G} = (G, τ) a global 𝔊-shtuka over S.

Global 𝔊-shtukas can be viewed as function field analogs of abelian varieties with extra structure. This inspires the following notions of quasi-isogenies.

Definition 3.4. Let k = 1 and I = I_{1} = {1, …, n}. Consider a scheme S together with legs 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 legs s_{i}. A quasi-isogeny from \underline{G} to \underline{G}′ is an isomorphism f: G|_{C_{S}∖E_{S}} ^{∼} ⟶ G′|_{C_{S}∖E_{S}} satisfying τ′σ^{∗}(f) = fτ, where E is some effective divisor on C. We denote the group of quasi-isogenies from \underline{G} to itself by QIsog_{S}(\underline{G}).

In order to obtain algebraic substacks locally of finite type of Hecke_{𝔊,D, \underline{I}} and ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) one has to bound the relative position of G_{j−1} and G_{j} under the isomorphism τ_{j}. For the rest of this section we fix a faithful representation ρ: 𝔊 → SL(V_{0}) ⊂ GL(V_{0}) for some vector bundle V_{0} of rank r, with quasi-affine quotient SL(V_{0})/𝔊 as in Proposition 2.2(b) such that the pairs (𝔊, SL(V_{0})) and (𝔊, GL(V_{0})) satisfy (2.1). In Remark 3.19 we explain to what extent our results depend on the choice of ρ.

Remark 3.5. We consider the induced morphisms of stacks

\xymatrix{ℋ ^{1}(C, 𝔊) & ρ_{∗} \ar[r] & ℋ ^{1}(C, SL(V_{0})) \ar[r] & ℋ ^{1}(C, GL(V_{0})) \ar[r]^{∼} & Vect_{C}^{r} \\ G \ar@{|->}[rr] &  & ρ_{∗}G \ar@{|->}[rr] &  & V(ρ_{∗}G) ,}

see Remark 2.8. Since ρ factors through SL(V_{0}) there is a canonical isomorphism α_{G}: ∧^{r} V(ρ_{∗}G) ^{∼} ⟶ O_{C_{S}}. Conversely, every pair (V, α) where V is a rank r vector bundle on C_{S} and α: ∧^{r}V ^{∼} ⟶ O_{C_{S}} is an isomorphism is induced from an SL(V_{0})-torsor over C_{S}.

As an auxiliary tool to prove the representability of Hecke_{𝔊,D, \underline{I}} and ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) we introduce the following stack.

Definition 3.6. Keep the notations of Remark 3.5. We define the relative affine Grassmannian Gr_{𝔊,I,r} as the stack over C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) which parametrizes tuples ((s_{i})_{i∈I}, G, V′, α′, φ), where

((s_{i})_{i∈I}, G, V′) ∈ C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) ×_{𝔽_{q}} Vect_{C}^{r},

α′: ∧^{r}V′ ^{∼} ⟶ O_{C_{S}} is an isomorphism and φ: V(ρ_{∗}G)|_{C_{S}∖⋃_{i∈I} Γ_{s_{i}}} ^{∼} ⟶ V′|_{C_{S}∖⋃_{i∈I} Γ_{s_{i}}} is an isomorphism between the vector bundle V(ρ_{∗}G) associated with ρ_{∗}G and V′ outside the graphs ⋃_{i∈I} Γ_{s_{i}} such that α_{G} = α′ ∘∧^{r}φ on C_{S} ∖∪Γ_{s_{i}}. In particular ∧^{r}φ = (α′)^{−1}∘α_{G} extends to an isomorphism ∧^{r}φ: ∧^{r} V(ρ_{∗}G) ^{∼} ⟶ ∧^{r}V′ on all of C_{S}. If Sl′ is the SL(V_{0})-torsor associated with (V′, α′) then φ induces an isomorphism φ: ρ_{∗}G|_{C_{S}∖⋃_{i∈I} Γ_{s_{i}}} ^{∼} ⟶ Sl′|_{C_{S}∖⋃_{i∈I} Γ_{s_{i}}}.

Note that for every j ∈ \{1, …, k\} the morphism ρ_{∗} yields a morphism Hecke_{𝔊, \underline{I}} → Gr_{𝔊,I_{j},r}, sending the tuple ((s_{i})_{i∈I}, (G_{j})_{j}, (τ_{j})_{j}) to the tuple ((s_{i})_{i∈I_{j}}, G_{j−1}, V(ρ_{∗}G_{j}), α_{G_{j}}, V(ρ_{∗}τ_{j})) where α_{G_{j}}: ∧^{r} V(ρ_{∗}G_{j}) ^{∼} ⟶ O_{C_{S}} is the canonical isomorphism induced from the fact that ρ factors through SL(V_{0}). The morphism Hecke_{𝔊, \underline{I}} → Gr_{𝔊,I_{j},r} factors through the j-th component in (3.1).

Let \underline{ω} := (ω_{i})_{i∈I} be a tuple of coweights of SL_{r} given as ω_{i}: x ↦ diag(x^{ω_{i,1}}, …, x^{ω_{i,r}}) for integers ω_{i,1} ≥ … ≥ ω_{i,r} with ω_{i,1} + … + ω_{i,r} = 0 for all i. (The inequality means that all ω_{i} are dominant with respect to the Borel subgroup of upper triangular matrices.) Let Gr\underline{ω} ; 𝔊,I,r denote the substack of Gr_{𝔊,I,r} defined by the condition that the isomorphism φ is bounded by \underline{ω}, that is satisfies

⋀_{C_{S}}^{ℓ} φ(V(ρ_{∗}G)) ⊂ (⋀_{C_{S}}^{ℓ} V′)(∑_{i∈I}(−ω_{i,r−ℓ+1} − … − ω_{i,r})·Γ_{s_{i}})  (3.2)

for all 1 ≤ ℓ ≤ r with equality for ℓ = r

where the notation (∧_{C_{S}}^{ℓ}V′)(∑_{i∈I}(−ω_{i,r−ℓ+1}−…−ω_{i,r})·Γ_{s_{i}}) means that we allow poles of order −ω_{i,r−ℓ+1}− … − ω_{i,r} along the Cartier divisor Γ_{s_{i}} on C_{S}; compare [HV11, Lemma 4.3]. Note that the condition for ℓ = r is equivalent to the requirement that ∧^{r}φ is an isomorphism on all of C_{S}, which in turn is equivalent to the condition that α_{G} = α′ ∘ ∧^{r}φ for an isomorphism α′: ∧^{r} V′ ^{∼} ⟶ O_{C_{S}}. By Cramer’s rule (e.g. [Bou70, III.8.6, Formulas (21) and (22)]) condition (3.2) is equivalent to

⋀_{C_{S}}^{ℓ} φ^{−1}(V′) ⊂ (⋀_{C_{S}}^{ℓ} V(ρ_{∗}G))(∑_{i}(ω_{i,1} + … + ω_{i,ℓ})·Γ_{s_{i}})  (3.3)

for all 1 ≤ ℓ ≤ r with equality for ℓ = r

Again the condition for ℓ = r is equivalent to the condition that α_{G} = α′ ∘ ∧^{r}φ for an isomorphism α′: ∧^{r} V′ ^{∼} ⟶ O_{C_{S}}.

Proposition 3.7. The relative affine Grassmannian Gr\underline{ω} ; 𝔊,I,r is relatively representable over C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) by a projective morphism of schemes.

Proof. We look at the fiber of Gr\underline{ω} ; 𝔊,I,r → C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) over an S-valued point ((s_{i})_{i∈I}, G) in (C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊))(S). Then (3.2) and (3.3) imply that V(ρ_{∗}G)(∑_{i∈I} ω_{i,1}·Γ_{s_{i}})/φ^{−1}(V′) must be a quotient of the sheaf

F := V(ρ_{∗}G)(∑_{i∈I} ω_{i,1}·Γ_{s_{i}})/V(ρ_{∗}G)(∑_{i∈I} ω_{i,r}·Γ_{s_{i}})

supported on the effective relative Cartier divisor X := ∑_{i∈I}(ω_{i,1} − ω_{i,r}) · Γ_{s_{i}} inside C_{S}. Note that X is a finite flat S-scheme. From the case ℓ = r in (3.2) and (3.3) we also obtain the isomorphism

α′ := α_{G} ∘ (∧^{r}φ)^{−1} : ∧^{r} V′ ^{∼} ⟶ O_{C_{S}}. Therefore Gr\underline{ω} ; 𝔊,I,r ×_{(C^{I}×_{𝔽q} ℋ ^{1}(C,𝔊))} S is represented by a closed sub-

scheme of Grothendieck’s Quot-scheme Quot_{F/X/S}^{Φ}, see [FGA, n^{∘}221, Théorème 3.1] or [AK80, Theorem 2.6], for constant Hilbert polynomial Φ = r · ∑_{i∈I} ω_{i,1}; compare [HH13, p. 5].

Definition 3.8. The stack Hecke\underline{ω} ; 𝔊, \underline{I} is defined by the cartesian diagram

\xymatrix{Hecke\underline{ω} ; 𝔊, \underline{I} \ar[r] \ar[d] & Gr\underline{ω} ; 𝔊,I_{1},r ×_{𝔽_{q}} … ×_{𝔽_{q}} Gr\underline{ω} ; 𝔊,I_{k},r \ar[d] \\ Hecke_{𝔊, \underline{I}} \ar[r] & Gr_{𝔊,I_{1},r} ×_{𝔽_{q}} … ×_{𝔽_{q}} Gr_{𝔊,I_{k},r} .}

Proposition 3.9. Let ρ: 𝔊 → SL(V_{0}) be a faithful representation as above with quasi-affine (resp. affine) quotient SL(V_{0})/𝔊. Then the morphism Hecke_{𝔊,(I)} → Gr_{𝔊,I,r} is represented by a locally closed and quasicompact (resp. a closed) immersion. In particular, the morphism of stacks Hecke\underline{ω} ; 𝔊, \underline{I} → C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) sending ((s_{i})_{i∈I}, (G_{j})_{j=0,...,k}, (τ_{j})_{j=1,...,k} ) to ((s_{i})_{i∈I}, G_{0}) is relatively representable by a morphism of schemes which is quasi-compact and quasi-projective, and even projective if there is a representation ρ with affine quotient SL(V_{0})/𝔊.

Proof. Note that for any SL(V_{0})-torsor Sl over a C-scheme U there is an isomorphism of stacks

[C/𝔊] ×_{ρ_{∗},[C/ SL(V_{0})], Sl} U ^{∼} ⟶ Sl/𝔊

where [C/𝔊] denotes the stack classifying 𝔊-torsors over C-schemes. This isomorphism was already used in diagram (2.2) for GL(V) instead of SL(V_{0}). It is given as follows. Let G be a 𝔊-torsor over a U-scheme T and let φ̃: ρ_{∗}G ^{∼} ⟶ Sl_{T} be an isomorphism of SL(V_{0})-torsors over T. We choose an fppf -covering T′ → T and a trivialization α: 𝔊_{T′} ^{∼} ⟶ G_{T′}. We consider the induced trivialization ρ_{∗}α: SL(V_{0})_{T′} ^{∼} ⟶ ρ_{∗}G_{T′} and let 1_{T′} ∈ SL(V_{0})(T′) be the identity section. Then the section \overline{φ_{T′} ∘ ρ_{∗}α(1_{T′})} ∈ (Sl/𝔊)(T′) descends to a section in (Sl/𝔊)(T) which is independent of our choices.

We now consider a morphism S → Gr_{𝔊,I,r} given by a tuple ((s_{i})_{i∈I}, G, V′, α′, φ) over a scheme S. We let Sl′ be the SL(V_{0})-torsor over C_{S} associated with the pair (V′, α′) by Remark 3.5, and we let φ̃: ρ_{∗}G|_{C_{S}∖∪Γ_{s_{i}}} ^{∼} ⟶ Sl′|_{C_{S}∖∪Γ_{s_{i}}} be the isomorphism induced by φ on C_{S} ∖ ∪_{i∈I}Γ_{s_{i}}. By the above, the pair (G, φ̃) defines a section x ∈ (Sl′/𝔊)(C_{S} ∖ ∪_{i∈I}Γ_{s_{i}}). Since Sl′ is isomorphic to SL(V_{0}) fppf -locally on C_{S} and SL(V_{0})/𝔊 is (quasi-)affine over C_{S}, also Sl′/𝔊 is (quasi-)affine over C_{S} by [EGA, IV_{2}, Proposition 2.7.1]. So by Lemma 3.10 below there exists a (locally) closed (and quasi-compact) immersion S′ ↪ S such that x extends over S′ to a section x′ ∈ (Sl′/𝔊)(C_{S′}). Again the section x′ corresponds to a 𝔊-torsor G′ over C_{S′} and an isomorphism ψ: ρ_{∗}G′ ^{∼} ⟶ Sl′ of SL(V_{0})-torsors over C_{S′}. The equality of sections x|_{C_{S′}∖∪Γ_{s_{i}}} = x′|_{C_{S′}∖∪Γ_{s_{i}}} induces an isomorphism τ: G|_{C_{S′}∖∪Γ_{s_{i}}} ^{∼} ⟶ G′|_{C_{S′}∖∪Γ_{s_{i}}} with ρ_{∗}τ = ψ^{−1}φ̃. Therefore S′ represents the fiber product Hecke_{𝔊,(I)} ×_{Gr_{𝔊,I,r}} S. This implies that the morphism Hecke_{𝔊,(I)} → Gr_{𝔊,I,r} is represented by a (locally) closed (quasi-compact) immersion.

Finally, Proposition 3.7 implies that Hecke\underline{ω} ; 𝔊,(I_{j}) is (quasi-)projective over C^{I_{j}} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) for all j = 1, …, k. So the (quasi-)projectivity of Hecke\underline{ω} ; 𝔊,n → C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) follows from the fact that

Hecke\underline{ω} ; 𝔊, \underline{I} ^{∼} ⟶ Hecke\underline{ω} ; 𝔊,(I_{1}) ×_{pr_{1},ℋ ^{1}(C,𝔊),pr_{0}} … ×_{pr_{1},ℋ ^{1}(C,𝔊),pr_{0}} Hecke\underline{ω} ; 𝔊,(I_{k})  (3.4)

under the isomorphism (3.1).

Lemma 3.10. Let Y be a (quasi-)affine scheme over C_{S}. Let D be an effective relative Cartier divisor on C_{S} over S and set U := C_{S} ∖D. Let x: U → Y be a section. Then there is a (locally) closed (quasi-compact) immersion S′ ↪ S such that for any S-scheme T the section x_{T} : U ×_{S} T → Y ×_{S} T extends uniquely to a section x′_{T} : C_{T} → Y ×_{S} T if and only if the structure morphism T → S factors through S′.

Proof. We first assume that Y is affine over C_{S}. By the uniqueness the question is local on S. So we can assume that S and T are affine and that there is an affine open subset Spec A ⊂ C_{S} containing D. Then Y ×_{C_{S}} Spec A and Y ×_{C_{S}} Spec A×_{S} T are affine, say of the form Spec B and Spec B_{T}, respectively. Let I ⊂ A be the invertible ideal defining D, which is of finite presentation by [EGA, I_{new}, Corollaire 1.4.3]. Then the section x corresponds to an A-morphism x^{∗}: B → Γ(Spec A ∖ D, O_{C_{S}}) by [EGA, I, Proposition 2.2.4]. For each element b ∈ B there is a positive integer m_{b} such that J_{b} := x^{∗}(b) · I^{m_{b}} ⊂ A by [EGA, I, Théorème 1.4.1]. The section x_{T} extends uniquely to a section x′_{T} : C_{T} → Y ×_{S} T if and only if x^{∗}(B_{T}) ⊂ A ⊗_{O_{S}} O_{T} ⊂ Γ(Spec A ∖ D, O_{C_{S}}) ⊗_{O_{S}} O_{T}. Since I is an invertible ideal and B_{T} = B ⊗_{O_{S}} O_{T}, this is the case if and only if J_{b} ⊗_{O_{S}} O_{T} ⊂ I^{m_{b}} ⊗_{O_{S}} O_{T} for all b ∈ B, that is, if and only if the image of J_{b} ⊗_{O_{S}} O_{T} in (A/I^{m_{b}}) ⊗_{O_{S}} O_{T} is zero for all b ∈ B. Now A/I^{m_{b}} is the structure sheaf of the relative Cartier divisor m_{b} · D. Since the scheme m_{b} · D is finite flat and of finite presentation over S the O_{S}-module A/I^{m_{b}} is finite locally free by [EGA, I_{new}, Proposition 6.2.10]. Therefore by [EGA, I_{new}, Lemma 9.7.9.1] there exists a closed subscheme S′_{b} ↪ S such that the image of J_{b} ⊗_{O_{S}} O_{T} in (A/I^{m_{b}}) ⊗_{O_{S}} O_{T} is zero if and only if the structure morphism T → S factors through S′_{b}. Taking S′ as the scheme theoretic intersection of the S′_{b} for all b ∈ B proves the lemma when Y is affine over C_{S}.

If Y is quasi-affine let π: Y → C_{S} be the structure morphism and set \overline{Y} := \underline{Spec}_{C_{S}}π_{∗}O_{Y}. Then Y ↪ \overline{Y} is a quasi-compact open subscheme by [EGA, II, Proposition 5.1.2(b)] and hence the complement \overline{Y} ∖ Y is a finitely presented closed subscheme of \overline{Y}. By the above there is a closed subscheme S′ of S over which the section x ∈ Y (U) extends uniquely to a section x′ ∈ \overline{Y} (C_{S′}), because \overline{Y} is affine over C_{S}. The preimage Z := (\overline{Y} ∖ Y) ×_{\overline{Y}, x′} C_{S′} is a finitely presented closed subscheme of C_{S′} and its image in S′ is closed and of finite presentation. The latter can be seen as follows. Since C_{S′} and Z are of finite presentation over S′ we may compute the image of Z in S′ by reducing to the case that S′ is noetherian, see [EGA, IV_{3}, Proposition 8.9.1], and then every closed subscheme of S′ is of finite presentation. Let S′′ be the complement of this image. Then S′′ ↪ S′ is a quasi-compact open immersion and for any S′-scheme T the structure morphism T → S′ factors through S′′ if and only if the section x′_{T} : C_{T} → \overline{Y} ×_{S} T factors through Y ×_{S} T. In other words, S′′ represents the question whether x extends uniquely to a section x′′ ∈ Y (C_{S}).

The criterion for Hecke\underline{ω} ; 𝔊, \underline{I} to be projective over C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) given in Proposition 3.9 is only sufficient. To give a necessary and sufficient criterion we recall the following

Definition 3.11. 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 A_{ν} is a parahoric group scheme over A_{ν}, as defined by Bruhat and Tits [BT84, Définition 5.2.6]; see also [HR03].

Note that by [BT84, §§ 4.6 and 5.1.9] every connected reductive group over Q has (in general many) integral models over C which are parahoric group schemes.

Proposition 3.12. The group scheme 𝔊 is parahoric if and only if the representable morphism of stacks

Hecke\underline{ω} ; 𝔊, \underline{I} → C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) sending ((s_{i})_{i∈I}, (G_{j})_{j=0,...,k}, (τ_{j})_{j=1,...,k} ) to ((s_{i})_{i∈I}, G_{0}) is projective.

Proof. To prove that the morphism is projective it suffices by Proposition 3.9 to show that it is proper. For this we use the valuative criterion for properness [LM00, Théorème 7.10]. So let R be a discrete valuation ring with fraction field K. Let ((s_{i})_{i}, G_{0}) be an R-valued point of C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) and let ((s_{i})_{i}, (G_{j})_{j}, (τ_{j})_{j}) be a K-valued point of Hecke\underline{ω} ; 𝔊, \underline{I} mapping to ((s_{i})_{i}, G_{0}). We must extend this K-valued point to an R-valued point. By looking at the decomposition (3.4) it suffices to treat the case with k = 1 and I = I_{1} = {1, …, n}. If s_{i} = s_{i′} for some i, i′ ∈ I we can replace I by I ∖{i′} and ω_{i} by ω_{i} +ω_{i′}. In this way we can assume that all s_{i} ∈ C(R) are pairwise different. Since C is separated over 𝔽_{q} this implies that all the induced s_{i,K} ∈ C(K) are pairwise different. We now consider Hecke\underline{ω} ; 𝔊, \underline{I}′ with I′ = I = \{1, …, n\} and I′_{j} = \{j\} for j = 1, …, n and the morphism of schemes

Hecke\underline{ω} ; 𝔊, \underline{I}′ ×_{C^{I}×_{𝔽q} ℋ ^{1}(C,𝔊)} Spec R ⟶ Hecke\underline{ω} ; 𝔊, (I) ×_{C^{I}×_{𝔽q} ℋ ^{1}(C,𝔊)} Spec R
((s_{i})_{i}, (G_{j})_{j=0,...,n}, (τ_{j})_{j=1,...,n}) ⟼ ((s_{i})_{i}, (G_{0}, G_{n}), τ := τ_{n} ∘ … ∘ τ_{1}).

Since all s_{i,K} ∈ C(K) are pairwise different, our K-valued point (G_{0}, G_{n}, τ) of Hecke\underline{ω} ; 𝔊, (I) lifts to a unique K-valued point of Hecke\underline{ω} ; 𝔊, \underline{I}′ by letting G_{j} be the 𝔊-bundle that coincides with G_{0} on C_{K} ∖ ⋃_{i≤j} Γ_{s_{i,K}} and with G_{n} on C_{K} ∖ ⋃_{i>j} Γ_{s_{i,K}}. To prove the properness of Hecke\underline{ω} ; 𝔊, (I) it therefore suffices to prove the properness of Hecke\underline{ω} ; 𝔊, \underline{I}′, and by looking at the decomposition (3.4) again, it suffices to prove the properness of Hecke\underline{ω} ; 𝔊,(I′_{j}) which has only one leg. By [AH16, Proposition 2.0.9] (respectively [Var04, Lemma 4.1] for

constant split reductive 𝔊) the base change Hecke\underline{ω} ; 𝔊,(I′_{j}) ×_{C×_{𝔽q} ℋ ^{1}(C,𝔊)} Spec R is isomorphic to the fiber over s_{j} ∈ C(R) of the BD-Grassmannian Gr(𝔊, C) from [Ric13, Definition 1.4]. The latter is proper over C by [Ric13, Theorem 1.18] if 𝔊 is parahoric.

Conversely, if the morphism of stacks is projective we fix a point x ∈ C(L) for L = 𝔽_{q}^{alg}, and we consider the fiber of Hecke\underline{ω} ; 𝔊, \underline{I} over the L-valued point ((s_{i})_{i}, 𝔊_{C_{L}}) of C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) where all the s_{i} are equal to x and 𝔊_{C_{L}} is the trivial 𝔊-bundle. Then (G_{0} = … = G_{k−1} = 𝔊_{C_{L}}, τ_{1} = … = τ_{k−1} = id) is a closed L-valued point of the first k − 1 factors of the decomposition (3.4). It defines a closed subscheme isomorphic to the fiber of Hecke\underline{ω} ; 𝔊,(I_{k}) over ((s_{i})_{i}, 𝔊_{C_{L}}). Therefore the latter is proper over Spec L. The limit of this fiber for varying \underline{ω} is ind-proper and equals the fiber of the BD-Grassmannian Gr(𝔊, C) over x. It now follows from [Ric13, Theorem 1.18] that 𝔊 is parahoric.

Definition 3.13. We denote by ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) the pull back of Hecke\underline{ω} ; 𝔊, \underline{I} under the morphism of stacks ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) → Hecke_{𝔊, \underline{I}}. We also denote by ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) the pull back of ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) under the (forgetful) morphism ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) → ∇_{\underline{I}}ℋ ^{1}(C, 𝔊).

We want to prove that the stack ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) is an ind-algebraic stack. We do not intend to give a general treatment of ind-algebraic stacks here. We just make the following tentative

Definition 3.14. Let T be a scheme.

(a) By an inductive system of Deligne-Mumford stacks over T we mean an inductive system (C_{a}, i_{ab}) indexed by a countable directed set A, such that each C_{a} is a Deligne-Mumford stack over T and i_{ab}: C_{a} ↪ C_{b} is a closed immersion of stacks for all a ≤ b in A.

(b) A stack C over T is an ind-DM-stack over T (or ind-algebraic Deligne-Mumford stack over T) if there is an inductive system of Deligne-Mumford stacks (C_{a}, i_{ab}) over T together with morphisms j_{a} : C_{a} → C satisfying j_{b} ∘ i_{ab} = j_{a} for all a ≤ b, such that for all quasi-compact T -schemes S and all objects c ∈ C(S) there is an a ∈ A, an object c_{a} ∈ C_{a}(S) and an isomorphism j_{a}(c_{a}) ≅ c in C. In this case we say that C is the inductive limit of (C_{a}, i_{ab}) and we write C = lim_{⟶} C_{a}.

(c) If in (b) all C_{a} are locally of finite type (resp. separated) over T we say that C is locally of ind-finite type (resp. ind-separated) over T.

Theorem 3.15. Let 𝔊 be a flat affine group scheme of finite type over the curve C and let D be a proper closed subscheme of C. Then the stack ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is a Deligne-Mumford stack locally of finite type and separated over (C ∖ D)^{I}. It is relatively representable over (C ∖ D)^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) by a separated morphism of finite type of algebraic spaces. In particular ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) = lim_{⟶} ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is an ind-

DM-stack over (C ∖ D)^{I} which is ind-separated and locally of ind-finite type. The forgetful morphism F: ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) → ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) ×_{C^{I}} (C ∖ D)^{I} is relatively representable by an affine étale morphism of schemes and above its image it is a torsor under the finite group 𝔊(D). If 𝔊 ×_{C} D → D is smooth, then F is finite. If the fibers of 𝔊 over the points in D are smooth and connected, then F is surjective.

Remark. If the fibers of 𝔊 over the points in D are not smooth and connected, F may fail to be surjective; see Example 3.18.

Proof of Theorem 3.15. We first show that the morphism

F: ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ⟶ ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) ×_{C^{I}} (C ∖ D)^{I}

is relatively representable by an affine étale morphism of schemes, which is in addition finite if 𝔊 ×_{C} D → D is smooth. Let S be a scheme and consider a morphism S → ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) ×_{C^{I}} (C ∖ D)^{I} given by a global 𝔊-shtuka ((s_{i})_{i∈I}, (G_{j})_{j}, (τ_{j})_{j}). For an S-scheme T the T -valued points of the fiber product ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ×_{∇_{\underline{I}}ℋ ^{1}(C,𝔊)} S equals the set of isomorphisms of 𝔊-torsors (ψ_{j}: G_{j} ×_{C_{S}} D_{T} ^{∼} ⟶ 𝔊 ×_{C} D_{T})_{j} satisfying ψ_{j} ∘ τ_{j}|_{D_{T}} = ψ_{j−1} for j = 1, …, k and ψ_{0} ∘ τ_{0}|_{D_{T}} = σ^{∗}(ψ_{k}). Since the characteristic morphism S → C^{I} factors through (C ∖ D)^{I}, the isomorphism τ := τ_{k} ∘ … ∘ τ_{0} induces an isomorphism τ|_{D_{S}}: σ^{∗}(G_{k} ×_{C_{S}} D_{S}) ^{∼} ⟶ G_{k} ×_{C_{S}} D_{S}. In particular ψ_{k} ∘ τ|_{D_{T}} = σ^{∗}ψ_{k} and the other ψ_{j} are uniquely determined by ψ_{k}. Let π_{S}: D_{S} → S be the structure morphism. Then ψ_{k} is a section over T of the the sheaf Y := π_{S∗}\underline{Isom}(G_{k} ×_{C_{S}} D_{S} , 𝔊 ×_{C} D_{S}), which is representable by an affine scheme of finite presentation over S by [Beh91, Corollary 4.4.2]. The condition ψ_{k} ∘ τ|_{D_{T}} = σ^{∗}ψ_{k}, and thus the fiber product ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ×_{∇_{\underline{I}}ℋ ^{1}(C,𝔊)} S is representable by the scheme equi( id_{Y} ∘τ, σ_{Y} : Y ⇉ Y), see Definition 3.2, which is a closed subscheme of finite presentation of Y, because Y is affine and of finite presentation over S. This shows that F is relatively representable by an affine morphism of schemes of finite presentations.

To show that F is étale let T be an affine S-scheme and let j: T̄ ↪ T be a closed subscheme defined by an ideal J with J^{2} = (0). Then the morphism σ_{T} factors through j: T̄ → T as

σ_{T} = j ∘ σ′: T ⟶^{σ′} T̄ ⟶^{j} T ,  (3.5)

where σ′ is the identity on the underlying topological space |T̄| = |T | and on the structure sheaf this factorization is given by

O_{T} ⟶^{j^{∗}} OT̄  ⟶^{σ′^{∗}} O_{T}
b ↦ b mod J ↦ b^{q} .

Note that σT̄ = σ′ ∘ j: T̄ → T̄. Any ψ̄_{k} defined over T̄ with ψ̄_{k} ∘ τ|_{DT̄} = σ^{∗}ψ̄_{k} lifts uniquely to ψ_{k} :=

σ′^{∗}ψ̄_{k} ∘ τ|_{D_{T}}^{−1} with j^{∗}ψ_{k} = j^{∗}σ′^{∗}ψ̄_{k} ∘ j^{∗}τ|_{D_{T}}^{−1} = σ∗ ; T̄ψ̄_{k} ∘ τ|_{DT̄}^{−1} = ψ̄_{k} and ψ_{k} ∘ τ|_{D_{T}} = σ′^{∗}ψ̄_{k} = σ′^{∗}j^{∗}ψ_{k} = σ^{∗}ψ_{k}.

This shows that F is étale, and hence faithfully flat over its image.

If 𝔊_{D} := 𝔊 ×_{C} D → D is smooth we show that ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ×_{∇_{\underline{I}}ℋ ^{1}(C,𝔊)} S is finite over S. By [EGA, IV_{2}, Proposition 2.7.1] this may be checked over an étale covering S̃ → S. By fpqc-descent [EGA, IV_{4}, Corollaire 17.7.3] the scheme G_{k} ×_{C_{S}} D_{S} is smooth over D_{S} and the Weil restriction Res_{D_{S}/S}(G_{k} ×_{C_{S}} D_{S}) is a smooth S-scheme by [BLR90, § 7.6, Theorem 4 and Proposition 5]. Therefore, there exists an étale covering S̃ → S and a section in Res_{D_{S}/S}(G_{k} ×_{C_{S}} D_{S})(S̃) = (G_{k} ×_{C_{S}} D_{S})(DS̃). This section yields a trivialization α: 𝔊 ×_{C} DS̃ ^{∼} ⟶ G_{k} ×_{C_{S}} DS̃. The Weil restriction Res_{D/𝔽_{q}} 𝔊_{D} is a smooth affine group scheme over 𝔽_{q} by [BLR90, § 7.6, Theorem 4 and Proposition 5]. Therefore, the morphism − id + Frob_{q}: Res_{D/𝔽_{q}} 𝔊_{D} → Res_{D/𝔽_{q}} 𝔊_{D} given by g ↦ g^{−1} · σ(g) is finite étale. The composition α^{−1} ∘ τ|_{DS̃} ∘ σ^{∗}α equals multiplication with an element g̃ ∈ 𝔊(DS̃) = (Res_{D/𝔽_{q}} 𝔊_{D})(S̃). There is an isomorphism

[figure: ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ×_{∇_{\underline{I}}ℋ ^{1}(C,𝔊)} S̃; ∼; Res_{D/𝔽_{q}} 𝔊_{D} × − id + Frob_{q}, Res_{D/𝔽_{q}} 𝔊_{D} S̃ ,; ψ_{k} with ψ_{k} ∘ τ|_{DS̃} = σ^{∗}ψ_{k}; g := ψ_{k} ∘ α with g^{−1}σ^{∗}g = g̃]

This proves that F is finite.

If in addition the fibers of 𝔊 over the points of D are connected (and smooth), the Weil restriction Res_{D/𝔽_{q}} 𝔊_{D} is connected by [CGP10, Proposition A.5.9]. Then Lang’s Theorem [Lan56, Corollary on p. 557] for Res_{D/𝔽_{q}} 𝔊_{D} shows that − id + Frob_{q} is surjective, and hence F is surjective.

We show that F: ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) → ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) ×_{C^{I}} (C ∖ D)^{I} is a torsor over its image under the finite group 𝔊(D), which we consider as a finite étale group scheme \underline{𝔊(D)} over Spec 𝔽_{q}. Indeed, g ∈ 𝔊(D) acts on ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) by sending ψ_{j} to g ∘ ψ_{j}. Here we observe that g is invariant under σ, and hence σ^{∗}(g ∘ ψ_{k}) = g∘σ^{∗}ψ_{k}. The product ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊)×_{∇_{\underline{I}}ℋ ^{1}(C,𝔊)} ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) classifies data ((s_{i})_{i∈I}, (G_{j})_{j}, (τ_{j})_{j}, (ψ_{j})_{j}, (ψ′_{j})_{j}) over 𝔽_{q}-schemes S where (ψ_{j})_{j} and (ψ′_{j})_{j} are two D-level structures on the global 𝔊-shtuka ((s_{i})_{i∈I}, (G_{j})_{j}, (τ_{j})_{j}). This means that ψ_{j}, ψ′_{j}: G_{j} ×_{C_{S}} D_{S} ^{∼} ⟶ 𝔊 ×_{C} D_{S} are isomorphisms of torsors with ψ_{j} ∘ τ_{j}|_{D_{S}} = ψ_{j−1} for j = 1, …, k and ψ_{0} ∘ τ_{0}|_{D_{S}} = σ^{∗}(ψ_{k}) and similarly for ψ′_{j}. In particular, ψ′_{k}ψ_{k}^{−1} is an automorphism of 𝔊 ×_{C} D_{S} given by left translation with an element g ∈ 𝔊(D_{S}). Since ψ′_{k}ψ_{k}^{−1} = … = ψ′_{0}ψ_{0}^{−1} = σ^{∗}(ψ′_{k}ψ_{k}^{−1}) we have g = σ^{∗}(g) ∈ \underline{𝔊(D)}(S). This shows that ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ×_{∇_{\underline{I}}ℋ ^{1}(C,𝔊)} ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) ≅ \underline{𝔊(D)} ×_{𝔽_{q}} ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) and ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) → ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) ×_{C^{I}} (C ∖ D)^{I} is a 𝔊(D)-torsor over its image.

To prove that the stack ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is an Artin stack locally of finite type over C^{I} we observe that its defining morphism ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) → Hecke\underline{ω} ; 𝔊, \underline{I} is representable, separated and of finite type, because it arises by base change from the diagonal morphism Δ_{ℋ ^{1}(C,𝔊) /𝔽_{q}} : ℋ ^{1}(C, 𝔊) → ℋ ^{1}(C, 𝔊) ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊). The latter is representable, separated and of finite type by [LM00, Lemma 4.2], because ℋ ^{1}(C, 𝔊) is algebraic (Theorem 2.5). Note that in general Δ_{ℋ ^{1}(C,𝔊) /𝔽_{q}} , and hence also ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) → Hecke\underline{ω} ; 𝔊, \underline{I} is not a closed immersion, and not even proper. Since Hecke\underline{ω} ; 𝔊, \underline{I} is quasi-projective and quasi-compact over C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) by Proposition 3.9, it follows that ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is relatively representable over (C ∖ D)^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) by a separated morphism of finite type of algebraic spaces. In particular it is an Artin stack locally of finite type over (C ∖ D)^{I}.

We prove that ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is Deligne-Mumford and separated over C^{I}. This means that the diagonal morphism Δ: ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) ⟶ ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) ×_{C^{I}} ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is unramified and proper; see [LM00, Théorème 8.1 and Définition 7.6]. By the algebraicity of ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) over C^{I} the diagonal Δ is already relatively representable by algebraic spaces, separated and of finite type; see [LM00, Lemme 4.2]. Consider an S-valued point of ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) ×_{C^{I}} ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) given by two global 𝔊-shtukas \underline{G} and \underline{G}′ over a C^{I}- scheme S. The base change of Δ to S is represented by the algebraic space \underline{Isom}_{S}(\underline{G}, \underline{G}′). Being unramified means that whenever S is affine and j: S̄ ↪ S is a closed subscheme defined by an ideal J with J^{2} = (0), then every isomorphism f̄ between \underline{G}S̄ and \underline{G}′ ; S̄ over S̄ comes from at most one isomorphism f between \underline{G} and \underline{G}′ over S. Such an isomorphism is a tuple f = (f_{0}, …, f_{k}) of isomorphisms f_{j}: G_{j} ^{∼} ⟶ G′_{j} compatible with the maps τ_{j} of \underline{G} and τ′_{j} of \underline{G}′. So assume that there are two isomorphisms f, f′ over S with j^{∗}f = f̄ = j^{∗}f′. Since J^{2} = (0) the morphism σ_{S} factors through j: S̄ → S as σ_{S} = j ∘ σ′ like in (3.5). From the diagram

G_{k}|_{C_{S}∖∪Γ_{s_{i}}}  ⟶_{≅}^{f_{k}} G′_{k}|_{C_{S}∖∪Γ_{s_{i}}}
τ_{k}∘...∘τ_{0}↑≅  ^{τ′_{k}∘...∘τ′_{0}}↑≅
σ_{S}^{∗}G_{k}|_{C_{S}∖∪Γ_{s_{i}}} ⟶_{≅}^{σ^{′∗}(j^{∗}f_{k})} σ_{S}^{∗}G′_{k}|_{C_{S}∖∪Γ_{s_{i}}}

we obtain that f_{k} and f′_{k} coincide on C_{S} ∖ ⋃_{i∈I} Γ_{s_{i}} and likewise f_{j} = (τ′_{k} ∘ … ∘ τ′_{j+1})^{−1} ∘ f_{k} ∘ τ_{k} ∘ … ∘ τ_{j+1} and f′_{j} coincide on C_{S} ∖ ⋃_{i∈I} Γ_{s_{i}} for all j. Since G′_{j} is separated over C_{S} and ⋃_{i∈I} Γ_{s_{i}} is a Cartier divisor on C_{S} we have f_{j} = f′_{j} for all j on all of C_{S} by [Bre19, Lemma 3.11]. This shows that the diagonal Δ is unramified and ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is Deligne-Mumford.

To prove that ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is separated over C^{I} we choose a representation ρ : 𝔊 ↪ GL(V) as in Proposition 2.2(a) such that the pair (𝔊, GL(V)) satisfies (2.1), and consider the commutative diagram

\xymatrix{∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) \ar[r] \ar[d] & ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, GL(V)) \ar[d] \\ C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, 𝔊) \ar[r] & C^{I} ×_{𝔽_{q}} ℋ ^{1}(C, GL(V)) .}

The vertical morphisms and the lower horizontal morphism are separated by what we have proved above and by Theorem 2.6. By [LM00, Remarque 7.8.1(3)] also the upper horizontal morphism is separated, and it suffices to prove the statement for 𝔊 = GL(V). We use the equivalence ℋ ^{1}(C, GL(V)) ^{∼} ⟶ Vect_{C}^{r} from Remark 2.8 under which \underline{G} corresponds to (M_{j}, τ_{j})_{j=0,...,k} consisting of locally free sheaves M_{j} of rank r on C_{S} and isomorphisms τ_{j}: M_{j−1} ^{∼} ⟶ M_{j} on C_{S} ∖ ⋃_{i∈I_{j}} Γ_{s_{i}} for all j = 1, …, k and an isomorphism τ_{0} : σ^{∗}M_{k} ^{∼} ⟶ M_{0} on C_{S}. Similarly \underline{G}′ corresponds to (M′_{j}, τ′_{j})_{j}. We now use the valuative criterion for properness [LM00, Théorème 7.10] to show that \underline{Isom}_{S}((M_{j}, τ_{j})_{j}, (M′_{j}, τ′_{j})_{j}) is proper over S. We may assume that S is the spectrum of a discrete valuation ring R with fraction field K, uniformizer π, and residue field k. Let f_{j}: M_{j} ⊗_{R} K ^{∼} ⟶ M′_{j} ⊗_{R} K with τ′_{j} ∘ f_{j−1} = f_{j} ∘ τ_{j} for all j and τ′_{0} ∘ σ^{∗}f_{k} = f_{0} ∘ τ_{0} be isomorphisms over C_{K}. We view f_{j} and its inverse as sections over C_{K} of the locally free sheaves M′_{j} ⊗ Mj^{∨} and M_{j} ⊗ (M′_{j})^{∨}. We have to show that both extend uniquely to C_{R}. The local ring O_{C_{R},η} at the generic point η of the special fiber C_{k} is a discrete valuation ring with uniformizer π. It suffices to extend f_{j} to O_{C_{R},η}, because then f_{j} is defined on an open set whose complement has codimension 2, and therefore f_{j} extends to all of C_{R} by [Eis95, Discussion after Corollary 11.4]. To extend f_{k} to O_{C_{R},η} we choose bases of M_{k,η} and M′_{k,η} and write f_{k} = π^{m}f̃_{k} with an r × r matrix f̃_{k} ∈ (O_{C_{R},η})^{r×r} ∖ (πO_{C_{R},η})^{r×r}. Now the equation π^{m}f̃_{k}τ_{k} … τ_{0} = τ′_{k} … τ′_{0} σ^{∗}f_{k} = π^{qm}τ′_{k} … τ′_{0} σ^{∗}f̃_{k} shows that m must be zero, because all τ_{j} and τ′_{j} belong to GL_{r}(O_{C_{R},η}). Therefore f_{k} and with it all f_{j} extend uniquely to C_{R} and this proves that ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is separated over C^{I}.

To prove that ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) is an ind-DM-stack, we consider the countable set of tuples \underline{ω} = (ω_{i})_{i∈I} of coweights of SL_{r} which are dominant with respect to the Borel subgroup of upper triangular matrices. We make this set into a directed set by equipping it with the partial order \underline{ω} ⪯ \underline{ω}′ whenever ω_{i} ⪯ ω′_{i} for all i in the Bruhat order, that is ω′_{i} − ω_{i} is a positive linear combination of positive coroots of SL_{r}. When \underline{ω} runs through this directed set, the ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) form an inductive system of algebraic stacks which are separated

and locally of finite type over (C ∖ D)^{I}. Indeed, we have to show that ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) → ∇\underline{ω}′ ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is a

closed immersion for \underline{ω} ⪯ \underline{ω}′. This follows from the fact that condition (3.2) for \underline{ω} on Gr_{𝔊,I_{j},r}^{\underline{ω}′} is equivalent to the vanishing of the image of ⋀_{C_{S}}^{ℓ} φ(V(ρ_{∗}G)) inside the finite locally free sheaf

(⋀_{C_{S}}^{ℓ} V′)(∑_{i∈I_{j}}(−ω′_{i,r−ℓ+1} − … − ω′_{i,r})·Γ_{s_{i}})/(⋀_{C_{S}}^{ℓ} V′)(∑_{i∈I_{j}}(−ω_{i,r−ℓ+1} − … − ω_{i,r})·Γ_{s_{i}})

on Gr_{𝔊,I_{j},r}^{\underline{ω}′}. Therefore condition (3.2) defines a closed immersion Gr\underline{ω} ; 𝔊,I_{j},r ↪ Gr_{𝔊,I_{j},r}^{\underline{ω}′} by [EGA, I_{new},

Lemma 9.7.9.1] for all j. Therefore, its base change ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) → ∇\underline{ω}′ ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is a closed immersion.

We prove that ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) = lim_{⟶} ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊). For this purpose let ((s_{i})_{i}, (G_{j})_{j}, (τ_{j})_{j}) be a global

𝔊-shtuka in ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊)(S) for a quasi-compact scheme S and let ((s_{i})_{i∈I_{j}}, G_{j−1}, V′_{j}, α′_{j}, φ_{j}) ∈ Gr_{𝔊,I_{j},r}(S) be the induced S-valued point of Gr_{𝔊,I_{j},r} for all j. Since S is quasi-compact there is for each i ∈ I_{j} an integer N_{i} ≥ 0 with φ_{j}(V(ρ_{∗}G_{j−1})) ⊂ V′_{j}(∑_{i∈I_{j}} N_{i}·Γ_{s_{i}}) by [EGA, I, Théorème 1.4.1]. We set ω_{i,ℓ} := −N_{i} for all i ∈ I and ℓ = 2, …, r and ω_{i,1} := (r − 1)N_{i}. Then the tuples ((s_{i})_{i∈I_{j}}, G_{j−1}, V′_{j}, α′_{j}, φ_{j}) satisfy condition (3.2). It follows that ((s_{i})_{i}, (G_{j})_{j}, (τ_{j})_{j}) ∈ ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊)(S) and the theorem is proved.

In the course of the proof we have established the following corollary which we formulate for later reference.

Corollary 3.16. Let \underline{G} and \underline{G}′ be global 𝔊-shtukas of the same characteristic over an 𝔽_{q}-scheme S. Then the sheaf of sets on S_{fpqc} given by \underline{Isom}_{S}(\underline{G}, \underline{G}′): T ↦ Isom_{T} (\underline{G}_{T} , \underline{G}′_{T}) is representable by a scheme which is finite and unramified over S. In particular the group of automorphisms Aut_{S}(\underline{G}) of \underline{G} is finite.

Proof. We have seen in the proof of Theorem 3.15 that \underline{Isom}_{S}(\underline{G}, \underline{G}′) is an algebraic space which is unramified and proper over S. In particular it is finite and affine over S and hence a scheme; compare [LM00, Lemma 4.2].

Remark 3.17. Note that the corollary says that the diagonal morphism

Δ: ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) ⟶ ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) ×_{C^{I}} ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊)

is representable by a finite and unramified morphism of schemes. However, this does not imply that for large enough level D the stack ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is a scheme. In general it is not even an algebraic space. By [LM00, Proposition 4.4] the latter holds if and only if the diagonal Δ is a closed immersion on ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊). This is not the case because ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊) is not quasi-compact and on it the fibers of Δ may grow arbitrarily. In particular, the order of the finite group Aut_{S}(\underline{G}) from the corollary is not bounded on ∇\underline{ω} ; \underline{I}ℋ ^{1}(C, 𝔊).

For example let C = ℙ^{1}, let ∞ ∈ C(𝔽_{q}) with C∖{∞} = Spec 𝔽_{q}[t], and let 0 be the point where t = 0. Let 𝔊 = GL_{2} and consider the global 𝔊-shtuka over S = Spec 𝔽_{q} with I = I_{1} = {1, 2} and legs s_{1} : S → {0} ⊂ C and s_{2} : S → {∞} ⊂ C, which is given by the vector bundle M = M_{1} = O_{ℙ^{1}}(m) ⊕ O_{ℙ^{1}}(0) and the Frobenius τ = τ_{1} ∘ τ_{0} = t · id_{M}: σ^{∗}M = M ^{∼} ⟶ M over C_{S} ∖ {0, ∞}. For any fixed level D = Spec 𝔽_{q}[t]/(a) with a ∈ 𝔽_{q}[t]∖(t) we can choose m ≥ deg_{t}(a) and the automorphism f = ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) of (M, τ) which also is compatible with any D-level structure ψ on (M, τ) over a field extension of 𝔽_{q}. This shows that there is no level D that is able to kill the automorphisms of every global 𝔊-shtuka.

Nevertheless ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is locally the quotient of a scheme separated and of finite type over 𝔽_{q} by a finite group whose order, however, grows arbitrarily as one moves on ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊). Indeed, by Remark 2.9 the stack ℋ ^{1}(C, 𝔊) can be covered by connected open substacks ℋ ^{1}(C, 𝔊)^{≤μ} of finite type over 𝔽_{q}, and the preimage ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊)^{≤μ} of ℋ ^{1}(C, 𝔊)^{≤μ} in ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) is separated and of finite type over (C ∖ D)^{I} by Theorem 3.15. On this preimage the automorphism group is bounded and can be killed by suitably enhancing the level D to some D′. Then ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊)^{≤μ} is the quotient by the finite group ker(𝔊(D′) → 𝔊(D)) of ∇\underline{ω} ; \underline{I}ℋ_{D′}^{1}(C, 𝔊)^{≤μ}. The latter is an algebraic space separated and of finite type over 𝔽_{q} by [LM00, Corollaire 8.1.1]. It is even a quasi-projective scheme over (C ∖ D′)^{I}, because it is constructed as in the proof of Theorem 3.15 from the quasi-projective scheme ℋ_{D′}^{1}(C, 𝔊)^{≤μ}; see Remark 2.9.

Example 3.18. We show that the morphism F: ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) → ∇_{\underline{I}}ℋ ^{1}(C, 𝔊) ×_{C^{I}} (C ∖ D)^{I} does not have to be surjective for non-connected groups 𝔊. For example let G_{0} = \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix})\} ∐\{( \begin{matrix}0 & ∗ \\ ∗ & 0 \end{matrix} )\} ⊂ GL_{2} be the normalizer of the diagonal torus and let 𝔊 := G_{0} ×_{𝔽_{q}} C. Let C = ℙ^{1} and consider two points ∞ ∈ ℙ^{1} and 0 = V(t) ∈ ℙ^{1}, where ℙ^{1} ∖ {∞} = Spec 𝔽_{q}[t], and let D = V(t − 1). For an algebraically closed field k we consider the k-valued point x = (0, ∞, 𝔊_{C_{k}}, τ = ( \begin{matrix}0 & 1 \\ t & 0 \end{matrix} )) of ∇_{2}ℋ ^{1}(C, 𝔊) ×_{C^{2}} (C ∖ D)^{2}. Then there is no ψ_{2} : 𝔊 ×_{C} D_{k} ^{∼} ⟶ 𝔊 ×_{C} D_{k} with ψ_{2} ∘ τ|_{D_{k}} = σ^{∗}ψ_{2}. Namely ψ_{2} equals multiplication with an element g ∈ (𝔊×_{C} D)(k) = G_{0}(k) and then ψ_{2}^{−1} ∘σ^{∗}ψ_{2} equals multiplication with g^{−1}σ^{∗}g which lies in \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix})\} ⊂ G_{0}. On the other hand, τ|_{D_{k}} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). This proves the non-existence of ψ_{2}, and hence the non-existence of a point of ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) above x.

Remark 3.19. Let us explain to what extent the results of Theorem 3.15 depend on the choice of the representation ρ. The stacks Gr_{𝔊,I_{j},r} and all stacks involving a bound by \underline{ω} depend in their definition on ρ. The others are independent. However, the structure of ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) = lim_{⟶} ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) as an

ind-DM-stack over (C ∖ D)^{I} depends on ρ. Nevertheless, one can choose a covering of ℋ_{D}^{1}(C, 𝔊) by quasi-compact open substacks U; see Theorem 2.5. If we choose a different representation ρ′ and corresponding coweights \underline{ω}′, the restriction of ∇\underline{ω}′ ; \underline{I}ℋ_{D}^{1}(C, 𝔊) to U is quasi-compact by Theorem 3.15. So above U the morphism ∇\underline{ω}′ ; \underline{I}ℋ_{D}^{1}(C, 𝔊) ⟶ ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) = lim_{⟶} ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊) factors through some ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊)

by Definition 3.14(b). In that sense the structure of ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) as an ind-DM-stack over (C ∖ D)^{I} is independent of the representation ρ.

Another more intrinsic way to obtain the strcture of ind-DM-stack on ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) would be to use an analog of \underline{ω} for the group 𝔊. Since the fibers of 𝔊 over C are not reductive in general this analog cannot be given by coweights. Instead one should work in the affine flag variety of 𝔊. To be more precise, in the local situation in Chapter 4 we consider bounds for local ℙ_{ν}-shtukas, see Definition 4.5, and we apply these in Chapter 7 to global 𝔊-shtukas whose characteristic sections (s_{1}, …, s_{n}): S → C^{n} factor through the (formal spectrum of a) complete local ring of a closed point \underline{ν} ∈ C^{n}; see Definition 7.1. These bounds are given by closed formal subschemes of the local affine flag varieties; see after Definition 4.3. If one wants to avoid the restriction on the caracteristic sections, one can work with the global affine flag variety and use it to define bounds that then provide the structure of ind-DM-stack on ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) in a way intrinsic to the group 𝔊. For more details we refer to [AH16].

Remark 3.20. Our moduli spaces for global 𝔊-shtukas generalize Varshavsky’s [Var04] moduli stacks F Bun, which in turn are a generalization of the moduli spaces of F -sheaves F Sh_{D,r} considered by Drinfeld [Dri87] and Cht_{N}^{r} considered by Lafforgue [Laf02] in their proof of the Langlands correspondence for 𝔊 = GL_{2} (resp. 𝔊 = GL_{r}). Namely Varshavsky considers the situation where G is a split reductive group over 𝔽_{q} and 𝔊 = G×_{𝔽_{q}} C is constant. Let T be a maximal split torus in G and let Λ be a finite generating system of the monoid of dominant weights X^{∗}(T)_{dom}, dominant with respect to the choice of a Borel subgroup B ⊂ G containing T. Let \overline{B} be the opposite Borel. For all λ ∈ Λ let V_{λ} := (IndG ; \overline{B}(−λ)_{dom})^{∨} be the Weyl module of highest weight λ and set V := ⨁_{λ∈Λ} V_{λ}. The representation ρ: G → GL(V) ↪ SL(V ⊕ ∧^{top}V^{∨}) is faithful by [HV11, Proposition 3.14] and we take it as the representation fixed before Remark 3.5 using Proposition 2.1(c). Varshavsky considers an n-tuple of dominant coweights \underline{ω̃} = (ω̃_{1}, …, ω̃_{n}) of G and the induced coweights ρ\underline{ω̃} = (ρ∘ω̃_{1}, …, ρ∘ω̃_{n}) of SL(V ⊕∧^{top}V^{∨}). Then his stack F Bun_{D,n,\underline{ω̃}}, see [Var04, Proposition 2.16], coincides with our stack ∇_{n}^{ρ\underline{ω̃}}ℋ_{D}^{1}(C, 𝔊). The stacks F Sh_{D,r} and Cht_{N}^{r} of Drinfeld and Lafforgue are obtained by taking G = GL_{r}, n = 2, the dominant coweights ω̃_{1} = (1, 0, …, 0) and ω̃_{2} = (0, …, 0, −1) of GL_{r}, and using the correspondence between GL_{r}-torsors and locally free sheaves of rank r; see Remark 2.8.

Varshavsky’s moduli stacks were further generalized by Bao Châu Ngô and Ngô Dac Tuân [NN08, Ngô13] who dropped the assumption that 𝔊 is constant split and allowed 𝔊 to be a smooth group scheme over C all of whose fibers are reductive. Other variants considered before include the moduli spaces Eℓℓ_{C,𝒟,I} of 𝒟elliptic sheaves of Laumon, Rapoport and Stuhler [LRS93], and their generalizations by L. Lafforgue [Laf97], Ngô [Ngô06] and Spieß [Spi10], who take 𝔊 as the group of units in a maximal (resp. hereditary) order 𝒟 of a central division algebra over Q, as well as the moduli spaces Ab-Sh_{H}^{r,d} of abelian τ-sheaves of the second author [Har05], who takes 𝔊 = GL_{r}, n = 2 and ω̃_{1} = (d, 0, …, 0) and ω̃_{2} = (0, …, 0, −d). Both [LRS93] and [Har05] require in addition that the characteristic section s_{2} : S → C factors through a fixed place ∞ ∈ C and that the associated local GL_{r}-shtuka at ∞ is isoclinic in the sense of [Har05], that is, basic in the sense of [HV11]. All these moduli spaces are special cases of our ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊).

Varshavsky’s stacks F Bun were recently used by V. Lafforgue [Laf18] to prove Langlands parameterization for split connected reductive groups over the function field Q. In the non-split case, Lafforgue [Laf18, § 12.3.2] chooses a parahoric model 𝔊 of the connected reductive group over Q and generalizes Varshavsky’s moduli stack to a moduli stack Cht_{N,I}^{(I_{1},...,I_{k})} which equals our ∇_{\underline{I}}ℋ_{D}^{1}(C, 𝔊) when N = D, and hence is an ind-DM-stack over C^{I} locally of ind-finite type by Theorem 3.15. Lafforgue [Laf18, § 12.1] truncates the stacks by bounding the degree of the 𝔊-bundles as follows. He fixes a representation θ: 𝔊 → → 𝔊^{ad} ↪ GL(V_{0}) and a dominant cocharacter μ = (μ_{1} ≥ … ≥ μ_{r}) of SL_{r}, where r = rk V_{0}, and considers the sub-stack ℋ ^{1}(C, 𝔊)^{≤μ} from Remark 2.9; see [Laf18, § 11.1 and (1.3)]. Since 𝔊^{ad} is semi-simple the representation θ factors through 𝔊 → → 𝔊^{ad} ↪ SL(V_{0}) ↪ GL(V_{0}), because its geometric generic fiber does; use [Hum75, 27.5 Theorem (d)]. Therefore the stacks ℋ ^{1}(C, 𝔊^{ad})^{≤μ} are of finite type over 𝔽_{q} by Remark 2.9.

4 LOOP GROUPS AND LOCAL ℙ-SHTUKAS 20

In addition, V. Lafforgue considers representations W of (^{L}𝔊)^{I}, where ^{L}𝔊 is the Langlands dual group of 𝔊_{Q} := 𝔊 ×_{C} Spec Q. For every such representation he defines the stack Cht_{N,I,W}^{\underline{I},≤μ} = ⋃\underline{ω} ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊)^{≤μ}, again with N = D, where ρ: 𝔊 ↪ SL(V) is a fixed faithful representation (as before Remark 3.5) and the union runs over a finite family of n-tuples \underline{ω} of coweights of SL_{rk V} related to W. More precisely, when 𝔊 is constant split the \underline{ω} are of the form \underline{ω} = ρ\underline{ω̃}, where the \underline{ω̃} run over those n-tuples of dominant coweights of 𝔊 whose Weyl modules occur in W. When 𝔊 is non-split this set is described in [Laf18, § 12.3.1]. In this case the perverse sheaves associated with W are not necessarily IC-sheaves. So Lafforgue does not need to care about the exact support of these sheaves and can use any bound to get Artin stacks locally of finite type. Our Theorem 3.15 (or Remark 2.9) shows that ∇\underline{ω} ; \underline{I}ℋ_{D}^{1}(C, 𝔊^{ad})^{≤μ} is a Deligne-Mumford stack of finite type and separated over (C ∖ D)^{I}. Finally Lafforgue fixes a cocompact lattice Ξ ⊂ Z(𝔸)/Z(Q), where Z is the center of 𝔊_{Q}, and shows that Cht_{N,I,W}^{\underline{I},≤μ} /Ξ is a Deligne-Mumford stack of finite type; see [Laf18, Lemma 12.19].

Remark 3.21. Varshavsky [Var04, Proposition 2.16] proves our Theorem 3.15 for a constant split reductive group 𝔊 = G ×_{𝔽_{q}} C by a different technique. He proceeds by augmenting the level D and introducing bounds on the degree of B-structures; see Remark 2.9. In this way he obtains open substacks of ℋ_{D}^{1}(C, 𝔊) which are representable by schemes, and he can prove that ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) is locally a quotient of a scheme by a finite group, so in particular a separated Deligne-Mumford stack. For general flat group schemes 𝔊 the existence of B-structures has been established so far only when 𝔊 is parahoric with quasi-split and semi-simple generic fiber; see Heinloth [Hei10, Corollary 26]. For this reason we use a different argument to prove Theorem 3.15.

Remark 3.22. Like in the case of Shimura varieties the question whether ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) is proper over (C ∖ D)^{I} depends on the group 𝔊, but unlike to that case it also depends on additional aspects. For example Drinfeld’s and Lafforgue’s moduli stack for 𝔊 = GL_{r} is not proper and this results in considerable efforts to compactify it; see [Laf02, Chapter III]. On the other hand the stack Eℓℓ_{C,𝒟,I} of Laumon, Rapoport and Stuhler, is proper outside the ramification locus of 𝒟; see [LRS93, Theorem 6.1]. Here 𝔊 is the group of units of a maximal order 𝒟 in a central division algebra, n = 2, and the bound on the relative position of G and σ^{∗}G under τ is non-trivial but as small as possible. If one allows larger bounds as considered by [Laf97, Ngô06, Lau04, Lau07] one has to increase the ramification of 𝒟 to retain the properness of the moduli space. Otherwise it can actually happen that the moduli spaces are not proper; see [Lau07, Theorem A] for the precise statement. This phenomenon is unknown from the theory of Shimura varieties.

4 Loop Groups and Local ℙ-Shtukas

In this chapter we let ℙ be a smooth affine group scheme over 𝔻 := Spec 𝔽[[z]]. We let 𝔻̇ := Spec 𝔽((z)) and P := ℙ ×_{𝔻} 𝔻̇ be the generic fiber of ℙ. 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}, assuming that 𝔊 is smooth above ν.

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)). Let ℋ ^{1}(Spec 𝔽, L^{+}ℙ) := [Spec 𝔽/L^{+}ℙ] (resp. ℋ ^{1}(Spec 𝔽, LP) := [Spec 𝔽/LP]) denote the classifying space of L^{+}ℙ-torsors (resp. 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) of S-valued points consists of all L^{+}ℙ-torsors (resp. LP -torsors) on S. The inclusion of sheaves L^{+}ℙ ⊂ LP gives rise to the natural morphism

L: ℋ ^{1}(Spec 𝔽, L^{+}ℙ) ⟶ ℋ ^{1}(Spec 𝔽, LP), L_{+} ↦ L := LL_{+} .  (4.1)

Definition 4.1. Let S ∈ Nilp_{𝔽[[ζ]]} and let σ̂ := σ̂_{S} be the 𝔽-Frobenius of S. 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 LP -torsors τ̂: σ̂^{∗}L → L from (4.1). A local ℙ-shtuka (L_{+}, τ̂) is called étale if τ̂ comes from an isomorphism of L^{+}ℙtorsors σ̂^{∗}L_{+} ^{∼} ⟶ L_{+}. We denote the stack fibered in groupoids over Nilp_{𝔽[[ζ]]} which classifies local ℙ-shtukas by Sht_{ℙ}^{𝔻}.

Definition 4.2. 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; see [AH14, Proposition 2.11]. Moreover, local ℙ-shtukas possess moduli spaces in the following sense. 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 underlying topological space T̄ = VT̂(ζ).

Definition 4.3. With a given local ℙ-shtuka \underline{𝕃} over an 𝔽-scheme T̄ we associate the functor

M\underline{𝕃}: NilpT̂ → 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}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 to an isomorphism \underline{L}′ → \underline{L}. The group QIsogT̄(\underline{𝕃}) of quasi-isogenies of \underline{𝕃} acts on the functor M\underline{𝕃} via j: (\underline{L}, δ̄) ↦ (\underline{L}, j ∘ δ̄) for j ∈ QIsogT̄(\underline{𝕃}).

We proved in [AH14, Theorem 4.4] that M\underline{𝕃} is representable by an ind-scheme, called an unbounded Rapoport-Zink space. For this recall that the affine flag variety Fℓ_{ℙ} 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. It is represented by an ind-scheme which is ind-quasi-projective over 𝔽, and hence ind-separated and of ind-finite type over 𝔽; see [PR08, Theorem 1.4]. If ℙ is parahoric in the sense of Bruhat and Tits, see [BT72, Définition 5.2.6] and [HR03], then Fℓ_{ℙ} is ind-projective over 𝔽 by [Ric13, Corollary 1.3]. Viewing 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]).

Theorem 4.4 ([AH14, Theorem 4.4]). The functor M\underline{𝕃} from Definition 4.3 is represented by an ind-scheme, ind-quasi-projective, hence ind-separated and of ind-finite type over T̂ = T̄ ˆ×_{𝔽} Spf 𝔽[[ζ]]. If ℙ is parahoric in the sense of Bruhat and Tits, see [BT72, Définition 5.2.6] and [HR03], then M\underline{𝕃} is ind-projective over T̂.

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

If one in addition imposes a boundedness condition on the Hodge polygon of the local ℙ-shtukas, and assumes T̄ = Spec k for a field k, then one even obtains a formal scheme locally formally of finite type over Spf k[[ζ]]. Here a formal scheme over Spf k[[ζ]] 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. To give more details we recall the following definition from [AH14, Definitions 4.5 and 4.8].

Definition 4.5. (a) We fix an algebraic closure 𝔽((ζ))^{alg} of 𝔽((ζ)), and consider pairs (R, Ẑ_{R}), where R/𝔽[[ζ]] is a finite extension of discrete valuation rings such that R ⊂ 𝔽((ζ))^{alg}, and where Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} := Fℓ_{ℙ} ˆ×_{𝔽} Spf R is a closed ind-subscheme. Two such pairs (R, Ẑ_{R}) and (R′, Ẑ′_{R′}) are equivalent if for some finite extension of discrete valuation rings R̃/𝔽[[ζ]] with R, R′ ⊂ R̃ the two closed indsubschemes Ẑ_{R} ˆ×_{Spf R} Spf R̃ and Ẑ′_{R′} ˆ×_{Spf R′} Spf R̃ of F̂ℓ_{ℙ,R̃} are equal. By [AH14, Remark 4.6] this then holds for all such rings R̃.

(b) A bound is an equivalence class Ẑ := [(R, Ẑ_{R})] of pairs (R, Ẑ_{R}) as above such that all Ẑ_{R} ⊂ F̂ℓ_{ℙ,R} are stable under the left L^{+}ℙ-action, and the special fiber Z_{R} := Ẑ_{R} ˆ×_{Spf R} Spec κ_{R} is a quasi-compact subscheme of Fℓ_{ℙ} ˆ×_{𝔽} κ_{R} where κ_{R} is the residue field of R. By [AH14, Remark 4.10] this implies that the Ẑ_{R} are formal schemes in the sense of [EGA, I_{new}].

(c) The reflex ring RẐ of a bound Ẑ = [(R, Ẑ_{R})] is the intersection of the fixed field of the group {γ ∈ Aut_{𝔽[[ζ]]}(𝔽((ζ))^{alg}): γ(Ẑ) = Ẑ } 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. We write RẐ = κ[[ξ]] and call its fraction field E := EẐ = κ((ξ)) the reflex field of Ẑ. We let R̆Ẑ := \overline{𝔽}[[ξ]] and Ĕ := ĔẐ := \overline{𝔽}((ξ)) be the completions of their maximal unramified extensions, where \overline{𝔽} is an algebraic closure of the finite field κ.

(d) Let Ẑ = [(R, Ẑ_{R})] be a bound with reflex ring RẐ. Let L_{+} and L′_{+} be L^{+}ℙ-torsors over a scheme S ∈ Nilp_{RẐ} and let δ: LL_{+} ^{∼} ⟶ LL′_{+} 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 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 \underline{L} = (L_{+}, τ̂) is bounded by Ẑ if τ̂ is bounded by Ẑ.

Remark 4.6. (a) The reflex field EẐ of Ẑ is always a finite extension of 𝔽((ζ)). For a detailed explanation of our definition of reflex ring and a comparison with the number field case see [AH14, Remark 4.7]. We do not know whether in general Ẑ has a representative over the reflex ring. In contrast, the equivalence class of the special fibers Z_{R} := Ẑ_{R} ˆ×_{Spf R} Spec κ_{R} always has a representative Z ⊂ Fℓ_{ℙ} ˆ×_{𝔽_{q}} Spec κ over the residue field κ of the reflex ring RẐ, because the Galois descent for closed ind-subschemes of Fℓ_{ℙ} is effective. We call Z the special fiber of Ẑ. It is a quasi-projective scheme over κ by [HV11, Lemma 5.4] because Fℓ_{ℙ} is ind-quasi-projective. It is even projective over κ if ℙ is parahoric, because then Fℓ_{ℙ} is ind-projective.

(b) The condition of Definition 4.5(d) is satisfied for all trivializations α and α′ and for all such finite extensions R of 𝔽_{q}[[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 [AH14, Remark 4.6].

We recall the following

Proposition 4.7 ([AH14, Proposition 4.11]). Let L_{+} and L′_{+} be L^{+}ℙ-torsors over a scheme S ∈ Nilp_{𝔽[[ζ]]} and let δ: L ^{∼} ⟶ L′ be an isomorphism of the associated LP -torsors. Let Ẑ be a bound. Then the condition that δ is bounded by Ẑ is represented by a closed subscheme of S.

To formulate the ind-representability in the bounded case we consider 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. Let \underline{𝕃} = ((L^{+}ℙ)_{k}, bσ̂^{∗}) be a trivialized local ℙ-shtuka over a field k in Nilp_{𝔽[[ζ]]} with b ∈ LP (k). Assume that b is decent with integer s in the sense of [AH14, Equation (4.4)], 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 [AH14, Remark 4.15]. So \underline{𝕃} is defined over T̄ = Spec ℓ and we may replace k by ℓ. Then M\underline{𝕃} is defined over Spf ℓ[[ζ]]. We remark that M\underline{𝕃} only depends on the isogeny class of \underline{𝕃}. Notice that if k is algebraically closed any local ℙ-shtuka over k is trivial and isogenous to a decent one by [AH14, Remark 4.15]. Also ℓ[[ξ]] is the finite unramified extension of RẐ with residue field ℓ. We recall the following Theorem 4.8 ([AH14, Theorem 4.18]). In the above situation if ℙ is a smooth affine group scheme over 𝔻 with connected reductive generic fiber, the functor

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

is ind-representable by a formal scheme over Spf ℓ[[ξ]] which is locally formally of finite type and separated. It is called a bounded Rapoport-Zink space for local ℙ-shtukas. Its underlying reduced subscheme equals the associated affine Deligne–Lusztig variety, which is the reduced closed ind-subscheme X_{Z}(\underline{𝕃}) ⊂ Fℓ_{ℙ} ˆ×_{𝔽} Spec ℓ whose K-valued points (for any field extension K of ℓ) are given by

X_{Z}(\underline{𝕃})(K) := X_{Z}(b)(K) := \{g ∈ Fℓ_{ℙ}(K): g^{−1} b σ̂^{∗}(g) ∈ Z(K)\}.  (4.2)

In particular X_{Z}(\underline{𝕃}) is a scheme locally of finite type over ℓ.

Remark 4.9. The group QIsog_{ℓ}(\underline{𝕃}) of quasi-isogenies of \underline{𝕃} acts on MẐ ; \underline{𝕃} via j: (\underline{L}, δ̄) ↦ (\underline{L}, j ∘ δ̄). Since \underline{𝕃} = ((L^{+}ℙ)_{ℓ}, bσ̂^{∗}) is trivialized and decent, QIsog_{ℓ}(\underline{𝕃}) = J_{b}(𝔽((z))) where J_{b} is the connected algebraic group over 𝔽((z)) which is defined by its functor of points that assigns to an 𝔽((z))-algebra R the group

J_{b}(R) := { j ∈ P (R ⊗_{𝔽((z))} ℓ((z))): j^{−1}bσ̂(j) = b } ,  (4.3)

see [AH14, Remark 4.16]. In particular, QIsog_{ℓ}(\underline{𝕃}) = J_{b}(𝔽((z))) ⊂ LP (ℓ) and this group acts on X_{Z}(b) via j: g ↦ j · g for j ∈ J_{b}(𝔽((z))).

5 Unbounded Uniformization

Analogously to the functor which assigns to an abelian variety A over a ℤ_{p}-scheme its p-divisible group A[p^{∞}] we introduced in [AH14, Section 5.2] a global-local functor which assigns to a global 𝔊-shtuka an n-tuple of local ℙ_{ν_{i}}-shtukas. The reason for the n-tuple is that our global 𝔊-shtukas have n characteristic places, whereas abelian varieties only have one characteristic place. We consider the following situation.

Definition 5.1. Fix a tuple \underline{ν} := (ν_{i})_{i=1...n} of places on C with ν_{i} ≠ ν_{j} for i ≠ j. We require throughout this chapter that 𝔊 is smooth above every ν_{i}. 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 finite 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 ind-DM-stack ∇_{n}ℋ ^{1}(C, 𝔊) along \underline{ν} ∈ C^{n}; see [Har05, Appendix A] for an explanation of this concept. It is an ind-DM-stack over Spf A\underline{ν}, that is, an ind-DM-stack fibered over the category Nilp_{A\underline{ν}}. It is ind-separated and locally of ind-finite type over 𝔽_{q} (and over Spf A\underline{ν}) by Theorem 3.15, with structure of ind-DM-stack given as follows. The formal spectrum Spf A\underline{ν} can be viewed as the indscheme lim_{⟶} Spec A\underline{ν}/(ζ_{1}, …, ζ_{n})^{j}. Then ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} is the limit of separated Deligne-Mumford stacks locally of finite type over 𝔽_{q}

∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} = lim_{⟶_{\underline{ω},j}} ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) ×_{C^{n}} Spec A\underline{ν}/(ζ_{1}, …, ζ_{n})^{j}.

Note that a stack fibered over Nilp_{A\underline{ν}} is the natural generalization of a scheme over Spf A\underline{ν}; see for example [Hak72].

The global-local functor Γ̂_{ν_{i}}: ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) → Sht_{ℙ_{ν_{i}}}^{Spec A_{ν_{i}}} (S) from [AH14, Definition 5.4] assigns to

a global 𝔊-shtuka \underline{G} over S its local ℙ_{ν_{i}}-shtuka Γ̂_{ν_{i}}(\underline{G}) at ν_{i}. The global-local functor

\underline{Γ̂} := ∏_{i} Γ̂_{ν_{i}}: ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) ⟶ ∏_{i} Sht_{ℙ_{ν_{i}}}^{Spec A_{ν_{i}}} (S)

assigns to \underline{G} the n-tuple (Γ̂_{ν_{i}}(\underline{G}))_{i}. Both functors also transform quasi-isogenies into quasi-isogenies.

Note that at a place ν outside the characteristic places ν_{i} we also associated with \underline{G} an étale local ℙ̃_{ν}-shtuka L_{ν}^{+}(\underline{G}) in [AH14, Remark 5.6]. Here ℙ̃_{ν} := Res_{𝔽_{ν}/𝔽_{q}} ℙ_{ν} is the Weil restriction. We proved the following result in [AH14, Proposition 5.7 and Remark 5.8].

Proposition 5.2. 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. If ν ∈ \underline{ν} consider a quasi-isogeny of local ℙ_{ν}-shtukas f: \underline{L}′_{ν} → Γ̂_{ν}(\underline{G}). If ν ∉ \underline{ν} consider a quasi-isogeny of local ℙ̃_{ν}-shtukas f: \underline{L}′_{ν} → L_{ν}^{+}(\underline{G}) and assume that \underline{L}′_{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 ℙ_{ν}-shtuka (resp. ℙ̃_{ν}-shtuka) associated with \underline{G}′ is \underline{L}′_{ν} and the quasi-isogeny of local ℙ_{ν}-shtukas (resp. ℙ̃_{ν}-shtukas) induced by g is f. We denote \underline{G}′ by f^{∗}\underline{G}.

As we shall see in the next theorem, the unbounded Rapoport-Zink space appears as the uniformization space for ∇_{n}ℋ ^{1}(C, 𝔊). In the present article this is only an auxiliary result for us, in which we construct the uniformization morphism. No bounds are needed for this purpose and the construction even works in the relative situation. This uniformization morphism will be used in our main result (Theorem 7.11) on the uniformization in the bounded case. We retain the notation of Definition 5.1.

Theorem 5.3. Let T̄ be a scheme over Spec A\underline{ν}/(ζ_{1}, …, ζ_{n}), and let \underline{G}_{0} ∈ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(T̄) be a global 𝔊-shtuka. Let (\underline{𝕃}_{i})_{i} := ˆ \underline{Γ}(\underline{G}_{0}) be the tuple of local ℙ_{ν_{i}}-shtukas over T̄ where ˆ \underline{Γ} is the global-local functor from Definition 5.1. Consider the unbounded Rapoport-Zink spaces M_{\underline{𝕃}_{i}} which are ind-quasi-projective indschemes over T̄ ˆ×_{𝔽_{ν_{i}}} Spf A_{ν_{i}} by Theorem 4.4 and the product ∏_{i} M_{\underline{𝕃}_{i}} := M_{\underline{𝕃}_{1}} ˆ×T̄ … ˆ×T̄ M_{\underline{𝕃}_{n}} which is an ind-quasi-projective ind-scheme over T̂ := T̄ ˆ×_{𝔽\underline{ν}} Spf A\underline{ν}. Then there is a natural morphism of ind-DM-stacks over T̂

Ψ_{\underline{G}_{0}} : ∏_{i} M_{\underline{𝕃}_{i}} ⟶ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{𝔽\underline{ν}} T̄ = ∇_{n}ℋ ^{1}(C, 𝔊) ˆ×_{C^{n}} T̂,  (5.1)
(\underline{L}_{i}, δ_{i}: \underline{L}_{i} → \underline{𝕃}_{i})_{i} ⟼ δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0}

which is ind-proper and formally étale.

Proof. By rigidity of quasi-isogenies [AH14, Proposition 2.11] the functors M_{\underline{𝕃}_{i}} are naturally isomorphic to the functors

S ⟼ \{Isomorphism classes of (\underline{L}_{i}, δ_{i}): where \underline{L}_{i} is a local ℙ_{ν_{i}}-shtuka over S
and δ_{i}: \underline{L}_{i} → \underline{𝕃}_{i,S} is a quasi-isogeny of local ℙ_{ν_{i}}-shtukas over S\}.

Let S → T̂ be a T̂-scheme. In particular via the morphism S → T̂ → T̄ ˆ×_{𝔽\underline{ν}} Spf 𝔽\underline{ν}[[ζ_{i}]] = T̄ ˆ×_{𝔽_{ν_{i}}} Spf A_{ν_{i}} the scheme S lies in Nilp_{T̄ ˆ×_{𝔽ν_{i}} Spf A_{ν_{i}}}. Let s_{i}: S → Spf A_{ν_{i}} → C be the induced characteristic morphism. Take an element (\underline{L}_{i}, δ_{i})_{i} of ∏_{i} M_{\underline{𝕃}_{i}}(S) and the global 𝔊-shtuka δ_{1}^{∗} \underline{G}_{0,S} from Proposition 5.2. Since ν_{i} ≠ ν_{j} for i ≠ j the local ℙ_{ν_{i}}-shtuka Γ̂_{ν_{i}}(δ_{1}^{∗} \underline{G}_{0,S}) equals \underline{𝕃}_{i} for i ≥ 2. We may therefore iterate this procedure for i = 2, …, n to obtain a global 𝔊-shtuka δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S} in ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S). It satisfies Γ̂_{ν_{i}}(δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S}) = \underline{L}_{i}. Sending (\underline{L}_{i}, δ_{i})_{i} to δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S} establishes the morphism Ψ_{\underline{G}_{0}}.

That Ψ_{\underline{G}_{0}} is formally étale is just another way of phrasing the rigidity of quasi-isogenies. We give more details. Let \underline{G} be an S-valued point of ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{𝔽\underline{ν}} T̄. Let S̄ be a closed subscheme of S defined by a locally nilpotent sheaf of ideals. Further assume that \underline{G}S̄ = δ̄_{n}^{∗} ∘ … ∘ δ̄_{1}^{∗} \underline{G}_{0,S̄} for quasi-isogenies δ̄_{i}: Γ̂_{ν_{i}}(\underline{G}S̄) → \underline{𝕃}_{i,S̄} as above, defined over S̄ and let ḡ: \underline{G}S̄ → \underline{G}_{0,S̄} be the corresponding quasi-isogeny from Proposition 5.2 with Γ̂_{ν_{i}}(ḡ) = δ̄_{i}. Now these quasi-isogenies lift uniquely to quasi-isogenies δ_{i}: Γ̂_{ν_{i}}(\underline{G}) → \underline{𝕃}_{i,S} and g: \underline{G} → \underline{G}_{0,S} over S by rigidity for local [AH14, Proposition 2.11], respectively global shtukas [AH14, Proposition 5.9]. By uniqueness we must have Γ̂_{ν_{i}}(g) = δ_{i}. Therefore the S-valued point δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S} of ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} is equal to \underline{G} by the uniqueness in Proposition 5.2.

6 TATE MODULES OF GLOBAL 𝔊-SHTUKAS 25

It remains to verify that Ψ_{\underline{G}_{0}} is ind-proper. Since ∏_{i} M_{\underline{𝕃}_{i}} is of ind-finite type over T̂ we can test the indproperness of Ψ_{\underline{G}_{0}} by the valuative criterion of properness; see [LM00, Theorem 7.3]. Let R be a complete discrete valuation ring with fraction field L and algebraically closed residue field. Let \underline{G} be an R valued point of ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{𝔽\underline{ν}} T̄ and set ˆ \underline{Γ}(\underline{G}) = (\underline{L}_{i})_{i}. Since R is strictly henselian, we may trivialize all local ℙ_{ν_{i}}-shtukas \underline{L}_{i} ≅ ((L^{+}ℙ_{ν_{i}})_{R}, b′_{i} σ̂_{ν_{i}}^{∗}) and (\underline{𝕃}_{i})_{R} ≅ ((L^{+}ℙ_{ν_{i}})_{R}, b_{i} σ̂_{ν_{i}}^{∗}) over R. Consider an L-valued point \underline{x} of ∏_{i} M_{\underline{𝕃}_{i}} which maps to \underline{G}_{L} under the morphism Ψ_{\underline{G}_{0}} and represent it as the tuple (((L^{+}ℙ_{ν_{i}})_{L}, b′_{i} σ̂_{ν_{i}}^{∗}), g_{i})_{i}, where g_{i} lies in LP_{ν_{i}}(L). We may take a faithful representation ρ: 𝔊 → GL(V), where V is a vector bundle over C of rank r; see Proposition 2.2(a). Let ρ_{∗}\underline{G}_{0} be the induced global GL(V)-shtuka over T̄. Note that we want to show that there is a unique morphism α̃ which fits into the following commutative diagram

[figure: Spec L; \underline{x}; ∏_{i} M_{\underline{𝕃}_{i}}; Ψ_{\underline{G}_{0}}; ρ_{∗}; ∏_{i} M_{ρ_{∗}\underline{𝕃}_{i}}; Ψ_{ρ∗\underline{G}_{0}}; Spec R; α̃; α; ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{𝔽\underline{ν}} T̄; ρ_{∗}; ∇_{n}ℋ ^{1}(C, GL(V))\underline{ν} ˆ×_{𝔽\underline{ν}} T̄; T̂; T̂.]

The horizontal arrows in the right commutative diagram are induced by the representation ρ. In addition the existence and uniqueness of the morphism α follows from the fact that ∏_{i} M_{ρ_{∗}\underline{𝕃}_{i}} is ind-proper over T̂ by Theorem 4.4 because GL(V) is reductive, and that ∇_{n}ℋ ^{1}(C, GL(V))\underline{ν} ˆ×_{𝔽\underline{ν}} T̄ is ind-separated over T̂ by Theorem 3.15. The R-valued point α is given by a tuple of the form ((ρ_{∗}(L^{+}ℙ_{ν_{i}})_{R}, ρ(b′_{i})σ̂_{ν_{i}}^{∗}) , g̃_{i})_{i}, where g̃_{i} lies in LGL_{r}(R). Since LP_{ν_{i}} is closed in LGL_{r} and g̃_{i} extends g_{i} over R the element g̃_{i} lies in fact in LP_{ν_{i}}(R). Thus g̃_{i} produces an extension of (((L^{+}ℙ_{ν_{i}})_{L}, b′_{i}σ̂_{ν_{i}}^{∗}), g_{i})_{i} over R. This gives the desired morphism α̃. Note that the commutativity of the diagram

\xymatrix{ & ∏_{i} M_{\underline{𝕃}_{i}} \ar[d]^{Ψ_{\underline{G}_{0}}} \\ Spec R \ar[r] \ar@{-->}[ur]^{α̃} & ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{𝔽\underline{ν}} T̄}

follows from the ind-separatedness of the stack ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{𝔽\underline{ν}} T̄; see Theorem 3.15.

Remark 5.4. We observe that the image of Ψ_{\underline{G}_{0}} lies inside the quasi-isogeny locus of \underline{G}_{0} in ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}. Indeed, by the above construction, starting with an S-valued point \underline{x} = (\underline{L}_{i}, δ_{i})_{i} of ∏_{i} M_{\underline{𝕃}_{i}}(S) there is a unique quasi-isogeny δ\underline{x}: Ψ_{\underline{G}_{0}}(\underline{x}) = δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S} → \underline{G}_{0,S} which is an isomorphism outside the ν_{i} with

(ˆ\underline{Γ}(Ψ_{\underline{G}_{0}}(\underline{x})), ˆ \underline{Γ}(δ\underline{x})) = \underline{x} by Proposition 5.2.

6 Tate Modules of Global 𝔊-Shtukas

From now on we assume that the group scheme 𝔊 is smooth over C and all its fibers are connected.

Fix an n-tuple \underline{ν} := (ν_{1}, …, ν_{n}) of pairwise different places on C and recall the notation from Definition 5.1. Let C′ := C ∖ \{ν_{1}, …, ν_{n}\}. Let 𝕆\underline{ν} := ∏_{ν∉\underline{ν}} A_{ν} be the ring of integral adeles of C outside \underline{ν} and 𝔸\underline{ν} := 𝕆\underline{ν} ⊗_{O_{C}} Q the ring of adeles of C outside \underline{ν}. The group 𝔊(𝕆\underline{ν}) acts through Hecke correspondences on the tower ∇_{n}ℋ_{D}^{1}(C, 𝔊)\underline{ν}. We want to extend this to an action of 𝔊(𝔸\underline{ν}); see Remark 7.12(b) below. For this purpose we generalize the notion of level structures on global 𝔊-shtukas in this chapter. Let S be a scheme in Nilp_{A\underline{ν}} and let \underline{G} ∈ ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) be a global 𝔊-shtuka over S. For a finite subscheme D of C set D_{S} := D ×_{𝔽_{q}} S and let \underline{G}|_{D_{S}} := \underline{G} ×_{C_{S}} D_{S} denote the pullback of \underline{G} to D_{S}. Let Rep_{𝕆\underline{ν}} 𝔊 be the category of representations of 𝔊 in finite free 𝕆\underline{ν}-modules V. More precisely, Rep_{𝕆\underline{ν}} 𝔊 is the category of 𝕆\underline{ν} -morphisms ρ: 𝔊 ×_{C} Spec 𝕆\underline{ν} → GL_{𝕆\underline{ν}}(V). Assume that S is connected, fix a geometric base point s̄ of S, and let F unct^{⊗}(Rep_{𝕆\underline{ν}} 𝔊, 𝔐od_{𝕆\underline{ν}[π_{1}^{ét}(S,s̄)]}) denote the category of tensor functors from Rep_{𝕆\underline{ν}} 𝔊 to the category 𝔐od_{𝕆\underline{ν}[π_{1}^{ét}(S,s̄)]} of 𝕆\underline{ν}[π_{1}^{ét}(S, s̄)]-modules. We define the (dual) Tate functors as follows

Ť_{−}: ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) ⟶ F unct^{⊗}(Rep_{𝕆\underline{ν}} 𝔊 , 𝔐od_{𝕆\underline{ν}[π_{1}^{ét}(S,s̄)]})  (6.1)
\underline{G} ⟼ (Ť\underline{G}: ρ ↦ lim_{⟵_{D⊂C′}}(ρ_{∗}\underline{G}|_{Ds̄})^{τ} ),
V̌_{−}: ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) ⟶ F unct^{⊗}(Rep_{𝕆\underline{ν}} 𝔊 , 𝔐od_{𝔸\underline{ν}[π_{1}^{ét}(S,s̄)]})  (6.2)
\underline{G} ⟼ (V̌\underline{G}: ρ ↦ lim_{⟵_{D⊂C′}}(ρ_{∗}\underline{G}|_{Ds̄})^{τ} ⊗_{𝕆\underline{ν}} 𝔸\underline{ν}).

Here Ds̄ is finite over s̄ = Spec k for an algebraically closed field k, and ρ_{∗}\underline{G}|_{Ds̄} is equivalent to (M, τ_{M}) where M is a free O_{Ds̄}-module of rank equal to dim ρ, and τ_{M}: σ^{∗}M ^{∼} ⟶ M is an isomorphism of O_{Ds̄}-modules. Then (ρ_{∗}\underline{G}|_{Ds̄})^{τ} := \{m ∈ M: τ_{M}(σ^{∗}m) = m\} denotes the τ-invariants. They form a free O_{D}-module of rank equal to dim ρ equipped with a continuous action of the étale fundamental group π_{1}^{ét}(S, s̄). This definition is independent of s̄ up to a change of base point. By [AH14, Remark 5.6] there is a canonical isomorphism Ť\underline{G}(ρ) = lim_{⟵_{D⊂C′}}(ρ_{∗}\underline{G}|_{Ds̄})^{τ} ≅ ∏_{ν∈C′} Ť_{L_{ν}^{+}(\underline{G})}(ρ_{ν}) where ρ = (ρ_{ν}) and where L_{ν}^{+}(\underline{G}) is the étale local

ℙ̃_{ν}-shtuka associated with \underline{G} = (G, τ) at a place ν ∈ C outside the characteristic places \underline{ν}. The functor V̌_{−} moreover transforms quasi-isogenies into isomorphisms. In fact, V̌\underline{G} extends to a tensor functor in F unct^{⊗}(Rep_{𝔸\underline{ν}} 𝔊 , 𝔐od_{𝔸\underline{ν}[π_{1}^{ét}(S,s̄)]}) by the usual arguments, see for example [HV17, Definition 7.1 and Remark 7.3], but we will not need this here.

Definition 6.1. Let ω_{𝕆\underline{ν}}^{∘}: Rep_{𝕆\underline{ν}} 𝔊 → 𝔐od_{𝕆\underline{ν}} and ω^{∘} := ω_{𝕆\underline{ν}}^{∘} ⊗_{𝕆\underline{ν}} 𝔸\underline{ν}: Rep_{𝕆\underline{ν}} 𝔊 → 𝔐od_{𝔸\underline{ν}} denote the forgetful functors. For a global 𝔊-shtuka \underline{G} over S let us consider the sets of isomorphisms of tensor functors Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}) and Isom^{⊗}(ω^{∘}, V̌\underline{G}). These sets are non-empty by Lemma 6.2 below. Note that 𝔊(𝔸\underline{ν}) = Aut^{⊗}(ω^{∘}) and 𝔊(𝕆\underline{ν}) = Aut^{⊗}(ω_{𝕆\underline{ν}}^{∘}) by the generalized Tannakian formalism [Wed04, Corollary 5.20], because 𝕆\underline{ν} is a Prüfer ring; see [LM71, Definition 10.17]. By the definition of the Tate functor, Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}) admits an action of π_{1}^{ét}(S, s̄) × 𝔊(𝕆\underline{ν}) where 𝔊(𝕆\underline{ν}) acts through ω_{𝕆\underline{ν}}^{∘} and π_{1}^{ét}(S, s̄) acts through Ť\underline{G}. Also Isom^{⊗}(ω^{∘}, V̌\underline{G}) admits an action of π_{1}^{ét}(S, s̄) × 𝔊(𝔸\underline{ν}).

Lemma 6.2. The sets Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}) and Isom^{⊗}(ω^{∘}, V̌\underline{G}) are non-empty. Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}), respectively Isom^{⊗}(ω^{∘}, V̌\underline{G}), is a principal homogeneous space under the group 𝔊(𝕆\underline{ν}), respectively 𝔊(𝔸\underline{ν}).

Proof. Using the injective map Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}) ⟶ Isom^{⊗}(ω^{∘}, V̌\underline{G}), γ ↦ γ_{Q} := γ ⊗_{𝕆\underline{ν}} 𝔸\underline{ν} it suffices to prove the non-emptiness of Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}). Since the question only depends on the pullback of \underline{G} to the geometric base point s̄ we may assume that S = Spec k for an algebraically closed field k. For each place ν ∈ C′ we consider the étale local ℙ̃_{ν}-shtuka L_{ν}^{+}(\underline{G}) associated with \underline{G} as in [AH14, Remark 5.6]. By our assumption that 𝔊 is smooth with connected fibers, [AH14, Corollary 2.9] yields an isomorphism L_{ν}^{+}(\underline{G}) ^{∼} ⟶ ((L^{+}ℙ̃_{ν})_{k}, 1·σ̂^{∗}) =: \underline{𝕃}_{ν}. For the local ℙ̃_{ν}-shtuka \underline{𝕃}_{ν} the Tate functor Ť_{\underline{𝕃}_{ν}} equals the forgetful functor Rep_{A_{ν}} ℙ_{ν} → 𝔐od_{A_{ν}}. Then ω_{𝕆\underline{ν}}^{∘} = ∏_{ν∈C′} Ť_{\underline{𝕃}_{ν}} ≅ ∏_{ν∈C′} Ť_{L_{ν}^{+}(\underline{G})} ≅ Ť\underline{G} provides an isomorphism in Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}). If γ, γ′ ∈ Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}) then γ^{−1} ∘ γ′ ∈ Aut^{⊗}(ω_{𝕆\underline{ν}}^{∘}) = 𝔊(𝕆\underline{ν}). This proves the lemma.

Corollary 6.3. Let m ∈ ℕ_{0} be a multiple of [𝔽\underline{ν}: 𝔽_{q}]. For every global 𝔊-shtuka \underline{G} ∈ (∇_{n}ℋ ^{1}(C, 𝔊) ×_{C^{n}} Spec 𝔽\underline{ν})(S) the map

τ_{G}^{m} := τ_{G} ∘ σ^{∗}τ_{G} ∘ … ∘ σ^{(m−1)∗}τ_{G}: σ^{m∗}\underline{G} ⟶ \underline{G}

is a quasi-isogeny, called the q^{m}-Frobenius isogeny of \underline{G}. If γ ∈ Isom^{⊗}(ω^{∘}, V̌\underline{G}) then σ^{m∗}(γ) = V̌_{τ_{G}^{m}}^{−1} ∘ γ in Isom^{⊗}(ω^{∘}, V̌_{σ^{m∗}\underline{G}}).

Proof. The global 𝔊-shtuka σ^{m∗}\underline{G} is obtained by pulling back \underline{G} under the absolute q^{m}-Frobenius σ^{m} = Frob_{q^{m},S}: S → S. Note that the characteristic sections s_{i} of \underline{G} satisfy s_{i} ∘ Frob_{q^{m},S} = Frob_{q^{m},𝔽\underline{ν}} ∘s_{i} = s_{i}, because (s_{1}, …, s_{n}): S → C^{n} factors through Spec 𝔽\underline{ν} ∈ C^{n} and Frob_{q^{m},𝔽\underline{ν}} = id_{𝔽\underline{ν}}. So \underline{G} and σ^{m∗}\underline{G} have the same characteristic and τ_{G}^{m} is a quasi-isogeny which is an isomorphism over (C ∖ \underline{ν})_{S} and satisfies τ_{G} ∘ σ^{∗}(τ_{G}^{m}) = τ_{G}^{m} ∘ τ_{σ^{m∗}G}.

The last equality can be proved as in Lemma 6.2. Namely, by the proof of the lemma γ arises as γ = (γ_{ν})_{ν} from a collection of quasi-isogenies γ_{ν}: \underline{𝕃}_{ν} := ((L^{+}ℙ̃_{ν})_{k}, 1·σ̂^{∗}) → L_{ν}^{+}(\underline{G}) of local ℙ̃_{ν}-shtukas at every place ν ∈ C ∖ \underline{ν}. Then σ^{m∗}γ_{ν}: \underline{𝕃}_{ν} = σ^{m∗}\underline{𝕃}_{ν} → L_{ν}^{+}(σ^{m∗}\underline{G}) satisfies L_{ν}^{+}(τ_{G}^{m}) ∘ σ^{m∗}γ_{ν} = γ_{ν} ∘ τ_{𝕃_{ν}}^{m} = γ_{ν}, and hence V̌_{τ_{G}^{m}} ∘ σ^{m∗}(γ) = γ.

Definition 6.4. (a) For a compact open subgroup H ⊆ 𝔊(𝔸\underline{ν}) we define a rational H-level structure γ̄ on a global 𝔊-shtuka \underline{G} over a connected scheme S ∈ Nilp_{A\underline{ν}} as a π_{1}^{ét}(S, s̄)-invariant H-orbit γ̄ = γH in Isom^{⊗}(ω^{∘}, V̌\underline{G}). For a non-connected scheme S we make a similar definition choosing a base point on each connected component and a rational H-level structure on the restriction to each connected component separately.

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

This definition of level structures generalizes our initial Definition 3.3 according to the following

Theorem 6.5. Let D ⊂ C be a finite subscheme disjoint from \underline{ν}, and consider the compact open subgroup H_{D} := ker(𝔊(𝕆\underline{ν}) → 𝔊(O_{D})) of 𝔊(𝔸\underline{ν}). Then there is a canonical isomorphism of stacks

∇_{n}ℋ_{D}^{1}(C, 𝔊)\underline{ν} ^{∼} ⟶ ∇_{n}^{H_{D}}ℋ ^{1}(C, 𝔊)\underline{ν}.

Proof. Let (\underline{G}, ψ) be an object in ∇_{n}ℋ_{D}^{1}(C, 𝔊)\underline{ν}(S) where ψ is a D-level structure; see Definition 2.4. For any representation ρ in Rep_{𝕆\underline{ν}} 𝔊 the isomorphism ψ: \underline{G}|_{D_{S}} ^{∼} ⟶ 𝔊 ×_{C} D_{S} with ψ ∘ τ|_{D_{S}} = σ^{∗}ψ induces an isomorphism ρ_{∗}ψ: ρ_{∗}\underline{G}|_{D_{S}} ^{∼} ⟶ GL_{dim ρ,D_{S}} with ρ_{∗}ψ^{−1} : ω_{𝕆\underline{ν}}^{∘}(ρ) ⊗_{𝕆\underline{ν}} O_{D} ^{∼} ⟶ Ť\underline{G}(ρ) ⊗_{𝕆\underline{ν}} O_{D} and consequently we obtain an isomorphism γ̄: ω_{𝕆\underline{ν}}^{∘} ⊗_{𝕆\underline{ν}} O_{D} ^{∼} ⟶ Ť\underline{G} ⊗_{𝕆\underline{ν}} O_{D} of tensor functors with γ̄(ρ) := ρ_{∗}ψ^{−1}. By Lemma 6.2 there exists a γ ∈ Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}) be a lift of γ̄ to 𝕆\underline{ν} and we let γ_{Q} ∈ Isom^{⊗}(ω^{∘}, V̌\underline{G}) be the isomorphism induced from γ. Then the coset γ_{Q}H_{D} only depends on ψ and we define the morphism

∇_{n}ℋ_{D}^{1}(C, 𝔊)\underline{ν}(S) ⟶ ∇_{n}^{H_{D}}ℋ ^{1}(C, 𝔊)\underline{ν}(S)  (6.3)

by sending (\underline{G}, ψ) to (\underline{G}, γ_{Q}H_{D}).

Let us show that this functor is essentially surjective. Let (\underline{G}, γ_{Q}H_{D}) be an object of the category ∇_{n}^{H_{D}}ℋ ^{1}(C, 𝔊)\underline{ν}. By Lemma 6.2 we may choose an isomorphism β: ω_{𝕆\underline{ν}}^{∘} ^{∼} ⟶ Ť\underline{G}. The automorphism β_{Q}^{−1}γ_{Q} ∈ Aut^{⊗}(ω^{∘}) corresponds to an element g ∈ 𝔊(𝔸\underline{ν}) by Lemma 6.2. We may write g := (g_{x_{1}}, …, g_{x_{r}}, g\underline{x}) ∈ 𝔊(Q_{x_{1}})×…×𝔊(Q_{x_{r}})×𝔊(𝕆^{\underline{ν},\underline{x}}) for suitable r and \underline{x} = (x_{i})_{i} ∈ (C′)^{r}. Then we set g′ := (1, …, 1, g\underline{x})^{−1} ·g = (g_{x_{1}}, …, g_{x_{r}}, 1) and β′ := β · (1, …, 1, g\underline{x}) ∈ Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť\underline{G}). For each x_{i} let L_{x_{i}}^{+}(\underline{G}) be the étale local ℙ̃_{x_{i}}- shtuka associated with \underline{G} by [AH14, Remark 5.6]. We consider the automorphism δ := (δ_{x_{1}}, …, δ_{x_{r}}, 1) := β′_{Q} · (g′)^{−1} · (β′_{Q})^{−1} = β′_{Q}γ_{Q}^{−1} ∈ Aut^{⊗}(V̌\underline{G}) and in particular its components δ_{x_{i}} ∈ Aut^{⊗}(V̌_{L_{x_{i}}^{+}(\underline{G})}). By [AH14, Proposition 3.6] there is a quasi-isogeny f_{i}: L_{x_{i}}^{+}(\underline{G}) → L_{x_{i}}^{+}(\underline{G}) with V̌_{L_{x_{i}}^{+}(\underline{G})}(f_{i}) = δ_{x_{i}}.

From Proposition 5.2 we obtain a global 𝔊-shtuka \underline{G}′ = f_{r}^{∗} ∘ … ∘ f_{1}^{∗} \underline{G} together with a quasi-isogeny f: \underline{G}′ → \underline{G} which is an isomorphism outside the x_{i}, such that the induced quasi-isogeny of the associated étale local ℙ̃_{x_{i}}-shtukas is f_{i}. Then the rational Tate functor V̌ from (6.2) takes f to

V̌_{f} = δ = β′_{Q} · (g′)^{−1} · (β′_{Q})^{−1} = β′_{Q}γ_{Q}^{−1}.

Consider the pair (\underline{G}′, β′_{Q}H_{D}) consisting of the global 𝔊-shtuka \underline{G}′ = (G′, τ′) together with the level structure β′: Ť_{\underline{G}}′ → ω_{𝕆\underline{ν}}^{∘} defined over 𝕆\underline{ν}. Note that this is quasi-isogenous to (\underline{G}, γ_{Q}H_{D}) under f and that f is an isomorphism outside the x_{i}.

We now show that (\underline{G}′, β′_{Q}H_{D}) = ((G′, τ′), β′_{Q}H_{D}) actually comes from a pair (\underline{G}′, ψ) in ∇_{n}ℋ_{D}^{1}(C, 𝔊)\underline{ν}. Consider the category Rep_{C} 𝔊 of representations of 𝔊 in finite locally free O_{C}-modules, the category Vect_{C} of finite locally free O_{C}-modules, and the natural forgetful functor ω_{C}^{∘}: Rep_{C} 𝔊 → Vect_{C}. The composition of the functor · ⊗_{O_{C}} 𝕆\underline{ν}: Rep_{C} 𝔊 → Rep_{𝕆\underline{ν}} 𝔊 followed by the fiber functor ω_{𝕆\underline{ν}}^{∘} equals ω_{C}^{∘} ⊗_{O_{C}} 𝕆\underline{ν}. Also consider the category 𝔉𝔐od_{D_{S}} of finite locally free sheaves on D_{S}, and the functor

N_{−}^{S}: ℋ ^{1}(C, 𝔊)(S) → F unct^{⊗}(Rep_{C} 𝔊, 𝔉𝔐od_{D_{S}}), G ⟼ (N_{G}^{S}: ρ ↦ ρ_{∗}G ⊗_{O_{C_{S}}} O_{D_{S}}).

In particular N_{𝔊×_{C}C_{S}}^{S} = ω_{C}^{∘} ⊗_{O_{C}} O_{D_{S}}. For a representation ρ ∈ Rep_{C} 𝔊 the “finite shtuka” ρ_{∗}\underline{G}′ ×_{C_{S}} D_{S} over S consists of the locally free sheaf N := ρ_{∗}G′ ⊗_{O_{C_{S}}} O_{D_{S}} = N_{G′}^{S}(ρ) on D_{S} together with the isomorphism ρ_{∗}τ′: σ^{∗}N ^{∼} ⟶ N. Let S̃ → S be the finite étale Galois covering corresponding to the kernel of π_{1}^{ét}(S, s̄) → Aut_{O_{D}}(Ť_{\underline{G}}′(ρ) ⊗_{𝕆\underline{ν}} O_{D}), g ↦ (β′)^{−1} ∘ g(β′). Then π_{1}^{ét}(S, s̄) acts on Ť_{\underline{G}}′(ρ) ⊗_{𝕆\underline{ν}} O_{D} through its quotient

Gal(S̃/S). As in the proof of [AH14, Proposition 3.4] there is an isomorphism N S̃ ; G′(ρ) := ρ_{∗}G′ ⊗_{O_{C_{S}}} O_{DS̃} ≅ (Ť_{\underline{G}}′(ρ)⊗_{𝕆\underline{ν}} O_{D})⊗_{𝔽_{q}} OS̃ which is compatible with the action of Gal(S̃/S) and the action of Frobenius through ρ_{∗}τ′ on the left and id ⊗σ on the right. Therefore the π_{1}^{ét}(S, s̄)-invariant coset β′_{Q}H_{D} ⊂ Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť_{\underline{G}}′)

induces an isomorphism N S̃ ; 𝔊×_{C}C_{S}(ρ) ^{∼} ⟶ N S̃ ; G′(ρ) which descends to S. In this way β′ induces an isomorphism of tensor functors η: N_{𝔊×_{C}C_{S}}^{S} ^{∼} ⟶ N_{G′}^{S}.

We want to show that this isomorphism comes from a level structure ψ. Since D is finite and 𝔊 is smooth there is an étale covering S′ → S and a trivialization α: (𝔊 ×_{C} D_{S′}, b · σ^{∗}) ^{∼} ⟶ \underline{G}′ ×_{C_{S}} D_{S′} for an element b ∈ 𝔊(D_{S′}), compare the proof of Theorem 3.15. The composition of tensor isomorphisms η_{S′}^{−1} ∘ N_{α}^{S′} ∈ Aut^{⊗}(N_{𝔊×_{C}C_{S}}^{S′}) = Aut^{⊗}(ω_{C}^{∘})(O_{D_{S′}}) = 𝔊(D_{S′}) is given by an element ψ′ ∈ 𝔊(D_{S′}) by [Wed04, Corollary 5.20]. The latter induces an isomorphism ψ′: (𝔊 ×_{C} D_{S′}, b · σ^{∗}) ^{∼} ⟶ (𝔊 ×_{C} D_{S′}, 1 · σ^{∗}). This isomorphism descends to the desired level structure ψ: \underline{G}′ ×_{C_{S}} D_{S} ^{∼} ⟶ (𝔊 ×_{C} D_{S}, 1 · σ^{∗}).

Analyzing this construction further also shows that the functor (6.3) is fully faithful. This proves the theorem.

Remark 6.6. (a) Let (\underline{G}, γH) ∈ ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν}(S). Then every choice of a representative γ ∈ Isom^{⊗}(ω^{∘}, V̌\underline{G}) of the H-level structure γH induces a representation of the étale fundamental group

ρ_{\underline{G},γ}: π_{1}^{ét}(S, s̄) ⟶ H , g ⟼ γ^{−1} ∘ g(γ) =: ρ_{\underline{G},γ}(g) .  (6.4)

Indeed, ρ_{\underline{G},γ}(gg′) = γ^{−1} ∘ g(γ) ∘ g(γ^{−1} ∘ g′(γ)) = ρ_{\underline{G},γ}(g) · ρ_{\underline{G},γ}(g′), because γ^{−1} ∘ g′(γ) lies in H on which π_{1}^{ét}(S, s̄) acts trivially. Replacing γ by γh for h ∈ H yields ρ_{\underline{G},γ} = Int_{h} ∘ρ_{\underline{G},γh}, where Int_{h} is conjugation by h on H.

(b) For any compact open subgroup H ⊆ 𝔊(𝔸\underline{ν}) and any element h ∈ 𝔊(𝔸\underline{ν}) there is an isomorphism ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} ^{∼} ⟶ ∇_{n}^{h^{−1}Hh}ℋ ^{1}(C, 𝔊)\underline{ν}, (\underline{G}, γH) ↦ (\underline{G}, γh(h^{−1}Hh)).

Theorem 6.7. (a) For any compact open subgroup H ⊆ 𝔊(𝔸\underline{ν}) the stack ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} is an ind-DM-stack, ind-separated and locally of ind-finite type over Spf A\underline{ν}.

(b) If H̃ ⊂ H ⊆ 𝔊(𝔸\underline{ν}) are compact open subgroups then the forgetful morphism

∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} ⟶ ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν}, (\underline{G}, γH̃) ⟼ (\underline{G}, γH)

is finite étale and surjective.

(c) If H̃ is a normal subgroup of H then the group H/H̃ acts on ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} from the right via hH̃: (\underline{G}, γH̃) ↦ (\underline{G}, γhH̃) for hH̃ ∈ H/H̃. The stack ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} is canonically isomorphic to the stack quotient [∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}/(H/H̃)] and ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} is a right H/H̃-torsor over ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} under the forgetful morphism.

Proof. (b) Let (\underline{G}, γH) ∈ ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν}(S) for a connected scheme S. Consider the finite π_{1}^{ét}(S, s̄)-set

\{ H̃-level structures γ̃H̃ on \underline{G} with γ̃ ∈ Isom^{⊗}(ω^{∘}, V̌\underline{G}) satisfying γ̃H = γH \} .  (6.5)

This set is finite, because after choosing a representative γ ∈ Isom^{⊗}(ω^{∘}, V̌\underline{G}), it becomes bijective to H/H̃ under the map h ↦ γhH̃ for h ∈ H/H̃; see Lemma 6.2. Let S′ → S be the finite étale covering space corresponding to (6.5), that is

F_{S,s̄}^{ét}(S′) := S′ ×_{S} s̄ = \{ H̃-level structures γ̃H̃ on \underline{G} satisfying γ̃H = γH \} ,

where F_{S,s̄}^{ét}: S′ ↦ S′ ×_{S} s̄ denotes the fiber functor from finite étale covering spaces of S to finite π_{1}^{ét}(S, s̄)- sets. The choice of a representative γ corresponds to an element γH̃ ∈ F_{S,s̄}^{ét}(S′), and hence to the choice of a base point s̄′ of S′ lifting s̄. Then γH̃ is π_{1}^{ét}(S′, s̄′)-invariant, and hence a rational H̃-level structure on \underline{G} over S′. This defines an S-morphism

S′ ⟶ ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} ×_{∇_{n}^{H}ℋ ^{1}(C,𝔊)\underline{ν}} S  (6.6)

which we will show to be an isomorphism. For this purpose let f: T → S be a connected scheme over S. A T -valued point of the fiber product in (6.6) is given by a rational H̃-level structure γ̃H̃ on f^{∗}\underline{G} lifting γH. This means that γ̃ ∈ Isom^{⊗}(ω^{∘}, V̌_{f^{∗}\underline{G},t̄}) and the H̃-orbit γ̃H̃ is π_{1}^{alg}(T, t̄)-invariant where t̄ is a geometric base point of T and V̌_{f^{∗}\underline{G},t̄} is the fiber over t̄. We must show that γ̃H̃ arises from a uniquely determined S-morphism T → S′. Moving the base point s̄ we may assume that f(t̄) = s̄, and hence V̌_{f^{∗}\underline{G},t̄} = V̌\underline{G}. Consider the finite étale covering space S′ ×_{S} T → T. Then

F_{T,t̄}^{ét}(S′ ×_{S} T) = F_{S,s̄}^{ét}(S′) = \{ H̃-level structures γ̃H̃ on \underline{G} satisfying γ̃H = γH \} .

The H̃-level structure γ̃H̃ on f^{∗}\underline{G} is an element of this set, because γ̃H̃ lifts γH, that is γ̃H = γH. In particular, this element γ̃H̃ defines a π_{1}^{ét}(T, t̄)-equivariant map from the one-element set {t̄} = F_{T,t̄}^{ét}(T) to F_{T,t̄}^{ét}(S′ ×_{S} T). By [SGA 1, Exposé V, § 7 and Théorème 4.1] this map corresponds to a uniquely determined T -morphism T → S′ ×_{S} T. The projection T → S′ onto the first component is the desired S-morphism which induces the rational H̃-level structure γ̃H̃ over T.

This proves that (6.6) is an isomorphism. And therefore the forgetful morphism ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} ⟶ ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} is finite étale and surjective.

(c) Let us next assume that H̃ ⊂ H is normal. Then Remark 6.6(b) defines a right action of H/H̃ on ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}. By definition the stack quotient [∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}/(H/H̃)] is the category fibered in groupoids over Nilp_{A\underline{ν}} whose S-valued points are pairs (S′ → S, (\underline{G}, γH̃)), where S′ → S is an H/H̃-torsor

and (\underline{G}, γH̃) ∈ ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}(S′) defines a morphism S′ → ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} which is equivariant for the action of H/H̃; see [StPr, Tag 04WL]. The isomorphism

∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} ^{∼} ⟶ [∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}/(H/H̃)] , (\underline{G}, γH) ⟼ (S′ → S, (\underline{G}, γH̃))  (6.7)

is given as follows. Let S′ := ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} ×_{∇_{n}^{H}ℋ ^{1}(C,𝔊)\underline{ν}} S. By the proof of (b) this means that S′ → S is the finite étale covering space corresponding to the π_{1}^{ét}(S, s̄)-set from (6.5). This set is in bijection with H/H̃ on which π_{1}^{ét}(S, s̄) acts via the representation ρ_{\underline{G},γ}: π_{1}^{ét}(S, s̄) → H → → H/H̃ from (6.4). In particular, S′ → S is a Galois covering with Galois group H/H̃. The projection S′ → ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} onto the first factor is H/H̃-equivariant and establishes the map (6.7). Its inverse is defined as follows. The object (\underline{G}, γH̃) ∈ ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}(S′) defines the object (\underline{G}, γH) ∈ ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν}(S′) which descends to S by étale descent for S′ → S. By construction the canonical morphism ∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν} ^{∼} ⟶ [∇H̃ ; nℋ ^{1}(C, 𝔊)\underline{ν}/(H/H̃)] is finite étale surjective and an H/H̃-torsor.

(a) The intersection H_{1} := H ∩ 𝔊(𝕆\underline{ν}) has finite index in H, because it is open and H is compact. Thus the intersection H_{2} := ⋂_{h∈H/H_{1}} hH_{1}h^{−1} ⊂ H_{1} ⊂ 𝔊(𝕆\underline{ν}) is compact open, normal in H, and of finite index in 𝔊(𝕆\underline{ν}). There is a proper closed subscheme D ⊂ C with H_{D} ⊂ H_{2}, and this is a normal subgroup because H_{D} is normal in 𝔊(𝕆\underline{ν}). Therefore, statement (a) holds for ∇_{n}^{H_{D}}ℋ ^{1}(C, 𝔊)\underline{ν} by Theorems 6.5 and 3.15, see Definition 5.1. Consequently also ∇_{n}^{H_{2}}ℋ ^{1}(C, 𝔊)\underline{ν} and ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} are ind-DM-stacks locally of ind-finite type over Spf A\underline{ν} by (c) because they are obtained as stack quotients by finite groups. They are ind-separated over Spf A\underline{ν}, because the forgetful morphisms in (c) are finite surjective with ind-separated source.

7 The Uniformization Theorem in the Bounded Case

We still assume that the group scheme 𝔊 is smooth over C and all its fibers are connected. In addition we assume that the generic fiber of 𝔊 is reductive. Our aim in this chapter is to study the morphism Θ from (1.1) in the introduction. We first describe its target.

Definition 7.1. Fix an n-tuple \underline{ν} = (ν_{i}) of places on the curve C with ν_{i} ≠ ν_{j} for i ≠ j. Let \underline{Ẑ} := (Ẑ_{i})_{i} be an n-tuple of bounds Ẑ_{i} in F̂ℓ_{ℙ_{ν_{i}}} in the sense of Definition 4.5 with reflex rings R_{i} := RẐ_{i} = κ_{i}[[ξ_{i}]]. Let κ be the compositum of the κ_{i} in an algebraic closure of 𝔽_{q}, and set R_{\underline{Ẑ}} := κ[[ξ_{1}, …, ξ_{n}]]. It is a finite extension of A\underline{ν}. Let \underline{G} be a global 𝔊-shtuka in ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν}(S) over a scheme S ∈ Nilp_{R_{\underline{Ẑ}}}. Then S is

a scheme in Nilp_{R_{i}} via the inclusion κ_{i}[[ξ_{i}]] ↪ R_{\underline{Ẑ}}. We say that \underline{G} is bounded by \underline{Ẑ} := (Ẑ_{i})_{i} if for every

i the associated local ℙ_{ν_{i}}-shtuka Γ̂_{ν_{i}}(\underline{G}) is bounded by Ẑ_{i}. We denote by ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} the substack of ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{A\underline{ν}} Spf R_{\underline{Ẑ}} consisting of global 𝔊-shtukas bounded by \underline{Ẑ}.

Proposition 7.2. The special fiber ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ×_{R_{\underline{Ẑ}}} Spec κ is a Deligne-Mumford stack locally of finite type and separated over Spec κ.

Proof. As in the proof of Theorem 6.7(a) it suffices to treat the case where H = H_{D} ⊂ 𝔊(𝔸\underline{ν}), because taking stack quotients by finite groups preserves the 2-category of Deligne-Mumford stacks locally of finite type and separated over Spec κ. We fix a representation ρ: 𝔊 ↪ SL(V_{0}) with rk V_{0} =: r as before Remark 3.5. Then we show that ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ×_{R_{\underline{Ẑ}}} Spec κ is a closed substack of ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊)×_{(C∖D)^{n}} Spec κ for a suitable tuple \underline{ω} = (ω_{i})_{i=1...n} of coweights of SL_{r}; see Definition 3.13. The representation ρ induces an immersion Fℓ_{ℙ_{ν_{i}}} ↪ Fℓ_{SL_{r}} of the associated affine flag varieties. Let z_{i} be a uniformizing parameter of A_{ν_{i}}. Since Z_{i} ⊂ Fℓ_{ℙ_{ν_{i}}} ×_{𝔽_{ν_{i}}} Spec κ_{i} is a quasi-compact scheme, there is an integer N_{i} ≥ 0 such that all entries of the universal matrix over L SL_{r} ×_{Fℓ_{_{SL}_{r}}} Z_{i} have pole order at most N_{i} with respect to z_{i}. We set ω_{i,ℓ} := −N_{i} for all ℓ = 2, …, r and ω_{i,1} := (r−1)N_{i}. Then every global 𝔊-shtuka \underline{G} over a κ-scheme S which is bounded by \underline{Ẑ} belongs to ∇\underline{ω} ; nℋ ^{1}(C, 𝔊)(S). By Proposition 4.7 (and Theorem 6.5) the stack ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ×_{R_{\underline{Ẑ}}} Spec κ is a closed substack of ∇\underline{ω} ; nℋ_{D}^{1}(C, 𝔊) ×_{(C∖D)^{n}} Spec κ, and hence a Deligne-Mumford stack locally of finite type and separated over Spec κ by Theorem 3.15.

Remark 7.3. Likewise, by Proposition 4.7 the entire stack ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} is a closed ind-substack of ∇_{n}^{H}ℋ ^{1}(C, 𝔊)\underline{ν}, and hence an ind-DM-stack over Spf R_{\underline{Ẑ}} which is ind-separated and locally of ind-finite

type by Theorem 6.7. In contrast to Proposition 7.2 we do not know whether ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} is the formal completion of a Deligne-Mumford stack (or at least a formal algebraic Deligne-Mumford stack over Spf R_{\underline{Ẑ}}) without further conditions on the Ẑ_{i}. If we fix a representation ρ: 𝔊 ↪ SL(V_{0}) with rk V_{0} =: r as before Remark 3.5 and let \underline{ω} = (ω_{i})_{i=1...n} be a tuple of coweights of SL_{r}, see Definition 3.13, we can consider the closed substack ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} of ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} on which ρ_{∗}\underline{G} is bounded by \underline{ω} as in (3.2). If H = H_{D} then

∇_{n}^{H_{D},\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} = ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ×_{∇_{n}ℋ ^{1}(C,𝔊)} ∇\underline{ω} ; nℋ ^{1}(C, 𝔊). By arguing as in Proposition 7.2 the

stack ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} is a locally noetherian, adic formal algebraic Deligne-Mumford stack over Spf R̆\underline{Ẑ};

use [Har05, Proposition A.14]. In particular, ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{Spf R̆\underline{Ẑ}} Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{j} is an (alge-

braic) Deligne-Mumford stack separated and locally of finite type over Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{j} for every j. So

similarly to Definition 5.1, the structure of ind-DM-stack on ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} is given as the limit

∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} = lim_{⟶_{\underline{ω},j}} ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{Spf R̆\underline{Ẑ}} Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{j}.

We do not need this here, but ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} will be a formal algebraic Deligne-Mumford stack over Spf R_{\underline{Ẑ}} if we impose a condition of the following form.

• There is a faithful representation ρ: 𝔊 ↪ SL(V_{0}) for a vector bundle V_{0} of rank r on C such that on Ẑ_{i} the universal matrix induced from the morphism Fℓ_{ℙ_{ν_{i}}} ↪ Fℓ_{SL_{r}} has poles with respect to Γ_{s_{i}} bounded by some integer N_{i} for every ν_{i} ∈ \underline{ν}.

Without such a condition on \underline{Ẑ} the problem is the following. ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} is the limit of an inductive system of algebraic stacks. One has to find a compatible inductive system of presentations for these algebraic stacks such that this inductive system forms an ind-scheme. Then one could argue as in [AH14, Remark 4.10] that this ind-scheme is in fact a formal scheme and consequently ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} is a formal algebraic stack of DM-type in the sense of [Har05, Appendix A].

7.4. We keep the notation of Definition 7.1 and let k be an algebraic closure of the residue field 𝔽\underline{ν} of \underline{ν} ∈ C^{n}. Then (s_{1}, …, s_{n}): Spec k → C^{n} has characteristic \underline{ν}. Let R̆_{\underline{Ẑ}} := k[[ξ_{1}, …, ξ_{n}]] be the completion of the maximal unramified extension of R_{\underline{Ẑ}}. Let \underline{G}_{0} be a global 𝔊-shtuka over k, and let I_{\underline{G}_{0}}(Q) denote the (abstract) group QIsog_{k}(\underline{G}_{0}) of quasi-isogenies of \underline{G}_{0}; see Definition 3.4.

Proposition 7.5. Let S be a connected non-empty scheme in NilpR̆_{\underline{Ẑ}}. Then the natural group homomorphism QIsog_{k}(\underline{G}_{0}) → QIsog_{S}(\underline{G}_{0,S}), g ↦ g_{S} is an isomorphism.

Proof. Since S is non-empty the group homomorphism is injective. To prove surjectivity we consider a quasi-isogeny f ∈ QIsog_{S}(\underline{G}_{0,S}). Since the question is local on S we may assume that S = Spec B is affine. Let D ⊂ C be a divisor such that f is an automorphism of G_{0,S}|_{C_{S}∖D_{S}}. We set C′ := C ∖(D ∪\underline{ν}) and choose a finite flat ring homomorphism π^{∗}: 𝔽_{q}[t] ↪ O_{C}(C′) of some degree d. We also choose a representation ρ: 𝔊 ↪ GL(V) for a locally free sheaf V on C of rank r as in Proposition 2.2(a). The restriction to C′_{k} of the global GL_{r}-shtuka ρ_{∗}\underline{G}_{0} corresponds to a locally free sheaf M of rank r on C′_{k} together with an isomorphism τ: σ^{∗}M ^{∼} ⟶ M, because C′ ∩ \underline{ν} = ∅. The quasi-isogeny ρ_{∗}f corresponds to an automorphism f: M ⊗_{k} B ^{∼} ⟶ M ⊗_{k} B with fτ = τσ^{∗}(f). Via π^{∗} we view M as a (locally) free k[t]-module of rank rd. We choose a basis and write τ and f as matrices T ∈ GL_{rd}(k[t]) and F ∈ GL_{rd}(B[t]) satisfying F T = T σ^{∗}(F). We expand T = ∑_{i} T_{i}t^{i} and T^{−1} = ∑_{i} T′_{i}t^{i} and F = ∑_{i} F_{i}t^{i} in powers of t. Then F T = T σ^{∗}(F) is equivalent to the equations F_{m} = ∑_{i+j+ℓ=m} T_{i}σ^{∗}(F_{j})T′_{ℓ} and σ^{∗}(F_{m}) = ∑_{i+j+ℓ=m} T′_{i}F_{j}T_{ℓ} for all m, where the matrix σ^{∗}(F_{m}) is obtained from F_{m} by raising all entries to the q-th power. This shows that the entries of the matrices F_{m} satisfy finite étale equations over k. As k is algebraically closed and S is connected we must have F ∈ GL_{rd}(k[t]) ⊂ GL_{rd}(B[t]). This implies that f = g_{S} for a quasi-isogeny g ∈ QIsog_{k}(\underline{G}_{0}).

7.6. We next describe the source of the morphism Θ from (1.1). Keep the situation of 7.4 and let (\underline{𝕃}_{i})_{i=1...n} := ˆ\underline{Γ}(\underline{G}_{0}) denote the associated tuple of local ℙ_{ν_{i}}-shtukas over k, where ˆ \underline{Γ} is the global-local functor from Definition 5.1. Let MẐ_{i} ; \underline{𝕃}_{i} denote the Rapoport-Zink space of \underline{𝕃}_{i} with underlying topological space X_{Z_{i}}(\underline{𝕃}_{i}),

see Theorem 4.8. Let M̆Ẑ_{i} ; \underline{𝕃}_{i} and X̆_{Z_{i}}(\underline{𝕃}_{i}) be their base changes to R̆_{i} := k[[ξ_{i}]], respectively to k. By

Theorem 4.8 the product ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} := M̆Ẑ_{1} ; \underline{𝕃}_{1} ˆ×_{k} … ˆ×_{k} M̆Ẑ_{n} ; \underline{𝕃}_{n} is a formal scheme locally formally of finite type over Spf R̆_{1} ˆ×_{k} … ˆ×_{k} Spf R̆_{n} = Spf k[[ξ_{1}, …, ξ_{n}]] = Spf R̆_{\underline{Ẑ}} and the product ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) := X̆_{Z_{1}}(\underline{𝕃}_{1}) ×_{k} … ×_{k} X̆_{Z_{n}}(\underline{𝕃}_{n}) is a scheme locally of finite type over Spec k.

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 M̆Ẑ_{i} ; \underline{𝕃}_{i} and

on X̆_{Z_{i}}(\underline{𝕃}_{i}); see Remark 4.9. Especially we see that the group I_{\underline{G}_{0}}(Q) acts on ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} and ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) via the natural group homomorphism

I_{\underline{G}_{0}}(Q) ⟶ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}), η ⟼ (η_{i})_{i} := \underline{Γ̂}(η) := (Γ̂_{ν_{i}}(η))_{i}  (7.1)

by sending (\underline{L}_{i}, δ_{i})_{i} to (\underline{L}_{i}, Γ̂_{ν_{i}}(η) ∘ δ_{i})_{i}.

The group I_{\underline{G}_{0}}(Q) also acts naturally on V̌_{\underline{G}_{0}} and Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}}) by sending γ ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}}) to V̌_{η} ∘γ for η ∈ I_{\underline{G}_{0}}(Q). After choosing an element γ_{0} ∈ Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť_{\underline{G}_{0}}), this defines a morphism

ε: I_{\underline{G}_{0}}(Q) ⟶ Aut^{⊗}(ω^{∘}) ≅ 𝔊(𝔸\underline{ν}), η ↦ γ_{0} ∘ V̌_{η} ∘ γ_{0}^{−1}.  (7.2)

To state the properties of the source of Θ we say that a formal algebraic Deligne-Mumford stack X over Spf R̆_{\underline{Ẑ}} (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.

Proposition 7.7. The quotient of ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H by the abstract group I_{\underline{G}_{0}}(Q) exists as a locally noetherian, adic formal algebraic Deligne-Mumford stack locally formally of finite type over Spf R̆_{\underline{Ẑ}} and is of the form

I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) ≅ ∐_{γ̄} Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} .  (7.3)

Here γ̄ := γH ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H runs through a set of representatives for the countable double coset I_{\underline{G}_{0}}(Q)\ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H ≅ ε(I_{\underline{G}_{0}}(Q))\𝔊(𝔸\underline{ν})/H, and

Γγ̄ := I_{\underline{G}_{0}}(Q) ∩ (∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) × γHγ^{−1}) ⊂ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}})

is a subgroup, which is discrete for the product of the ν_{i}-adic topologies, and separated in the profinite topology, that is, for every 1 ≠ g ∈ Γγ̄ there is a normal subgroup of finite index in Γγ̄ that does not contain g. In particular the closed substack of (7.3) defined by the largest ideal of definition is the Deligne-Mumford stack locally of finite type over Spec k given by

I_{\underline{G}_{0}}(Q)\(∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) ≅ ∐_{γ̄} Γγ̄\ ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) .  (7.4)

Proof. We first prove that the group homomorphisms Γ̂_{ν_{i}} and ε from (7.1) and (7.2) are injective. We consider a faithful representation ρ: 𝔊 ↪ GL(V) as in Proposition 2.2(a). Then ρ_{∗}η induces a quasi-isogeny of the vector bundle M associated with ρ_{∗}\underline{G}_{0}. If η lies in the kernel of ε then its restriction to C_{k} ∖ \underline{ν} is the identity on M because of [AH14, Proposition 3.4]. Therefore η must be the identity. This proves that ε is injective. On the other hand, Γ̂_{ν_{i}}(M) is the completion of M at the graph Γ_{s_{i}} of s_{i}: Spec k → C. Since this completion functor is faithful, also Γ̂_{ν_{i}}: I_{\underline{G}_{0}}(Q) → J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) is injective. Thus we get the following injective morphism

(\underline{Γ̂}, ε): I_{\underline{G}_{0}}(Q) ⟶ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) × 𝔊(𝔸\underline{ν})

and we identify I_{\underline{G}_{0}}(Q) with its image. We claim that I_{\underline{G}_{0}}(Q) is a discrete subgroup of ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) × 𝔊(𝔸\underline{ν}). To show this we take an open subgroup U ⊂ ∏_{i} Aut_{k}(\underline{𝕃}_{i}) and consider the open subgroup U × 𝔊(𝕆\underline{ν}) ⊂ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) × 𝔊(𝔸\underline{ν}). Since 𝔊(𝕆\underline{ν}) = γ_{0} Aut^{⊗}(Ť_{\underline{G}_{0}})γ_{0}^{−1} the elements of I_{\underline{G}_{0}}(Q) ∩ (U × 𝔊(𝕆\underline{ν})) give automorphisms of the global 𝔊-shtuka \underline{G}_{0} := (G, τ_{0}). Then the finiteness of I_{\underline{G}_{0}}(Q) ∩ (U × 𝔊(𝕆\underline{ν})) follows from Corollary 3.16.

Now choose an element γ̄ := γH ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H, set h := γ_{0}^{−1}γ ∈ 𝔊(𝔸\underline{ν}) and let

Γγ̄ := \underline{Γ̂}(I_{\underline{G}_{0}}(Q) ∩ (\underline{Γ̂}^{−1}(∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}})) × γHγ^{−1}))

:= \underline{Γ̂}(I_{\underline{G}_{0}}(Q) ∩ (\underline{Γ̂}, ε)^{−1}(∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) × hHh^{−1})) ⊂ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}})

be the image under the injective homomorphism \underline{Γ̂} from (7.1). Since the intersection 𝔊(𝕆\underline{ν}) ∩ hHh^{−1} has finite index in 𝔊(𝕆\underline{ν}) and in hHh^{−1}, the intersection I_{\underline{G}_{0}}(Q)∩(U ×hHh^{−1}) ≅ Γγ̄ ∩U is also finite, and hence the subgroup Γγ̄ ⊂ ∏_{i} J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) is discrete. The group Γγ̄ is separated, because for every element 1 ≠ η ∈ Γγ̄ there is a normal subgroup H̃ ⊂ H of finite index such that hH̃h^{−1} does not contain the element ε(η) ≠ 1. Therefore Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} and I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) are formal algebraic Deligne-Mumford stacks by [AH14, Proposition 4.27]. That they are Deligne-Mumford and the last assertion about (7.4) follow from the proof of [AH14, Proposition 4.27] where it is shown that (7.4) (respectively (7.3)) are locally the stack quotient of a (formal) scheme by a finite group.

7.8. There is a further group acting on the source and the target of the morphism Θ from (1.1), namely the group ∏_{i} Z(Q_{ν_{i}}) where Z ⊂ 𝔊 ×_{C} Spec Q is the center. Indeed, Z(Q_{ν_{i}}) lies in the center of 𝔊(Q_{ν_{i}}) = LP_{ν_{i}}(𝔽_{ν_{i}}). Writing \underline{𝕃}_{i} ≅ ((L^{+}ℙ_{ν_{i}})_{k}, b_{i}σ̂_{ν_{i}}^{∗}) with b_{i} ∈ LP_{ν_{i}}(k), there is an inclusion

Z(Q_{ν_{i}}) ⟶ J_{\underline{𝕃}_{i}}(Q_{ν_{i}}) = \{ j ∈ LP_{ν_{i}}(k): j b_{i} = b_{i} σ̂_{ν_{i}}(j) \} = QIsog_{k}(\underline{𝕃}_{i}) ,  (7.5)
c_{i} ⟼  c_{i} = σ̂_{ν_{i}}(c_{i})

through which c_{i} ∈ Z(Q_{ν_{i}}) acts on M̆Ẑ_{i} ; \underline{𝕃}_{i} via c_{i}: (\underline{L}_{i}, δ_{i}) ↦ (\underline{L}_{i}, c_{i}δ_{i}). This action can also be described in a different way. For any local ℙ_{ν_{i}}-shtuka \underline{L}_{i} over a scheme S ∈ Nilp_{A_{ν_{i}}} we claim that c_{i} induces an element c_{i} ∈ QIsog_{S}(\underline{L}_{i}) of the quasi-isogeny group of \underline{L}_{i}. Namely, over an étale covering S′ → S we can choose a trivialization α: \underline{L}_{i,S′} ^{∼} ⟶ ((L^{+}ℙ_{ν_{i}})_{S′}, b′_{i}σ̂_{ν_{i}}^{∗}) and then c_{i} = σ̂_{ν_{i}}(c_{i}) implies c_{i} b′_{i} = b′_{i} σ̂_{ν_{i}}(c_{i}), that is α^{−1} ∘ c_{i} ∘ α ∈ QIsog_{S′}(\underline{L}_{i}). To show that this quasi-isogeny descends to S let pr_{1}, pr_{2} : S′′ := S′ ×_{S} S′ → S′ be the projections onto the first and second factor and set g := pr_{2}^{∗}α ∘ pr_{1}^{∗}α^{−1} ∈ L^{+}ℙ_{ν_{i}}(S′′). Then g ∘ pr_{1}^{∗}c_{i} ∘ g^{−1} = pr_{1}^{∗}c_{i} = pr_{2}^{∗}c_{i} implies

pr_{1}^{∗}(α^{−1} ∘ c_{i} ∘ α) = pr_{2}^{∗}α^{−1} ∘ g ∘ pr_{1}^{∗}c_{i} ∘ g^{−1} ∘ pr_{2}^{∗}α = pr_{2}^{∗}(α^{−1} ∘ c_{i} ∘ α) ,

and this shows that α^{−1} ∘c_{i} ∘α descends to a quasi-isogeny of \underline{L}_{i} over S which we denote by c_{i} ∈ QIsog_{S}(\underline{L}_{i}). If moreover we are given a quasi-isogeny δ_{i}: \underline{L}_{i} → \underline{𝕃}_{i,S}, that is if (\underline{L}_{i}, δ_{i}) ∈ M_{\underline{𝕃}_{i}}(S), then δ_{i} is automatically compatible with the quasi-isogenies c_{i} ∈ QIsog_{S}(\underline{L}_{i}) and c_{i} ∈ QIsog_{k}(\underline{𝕃}_{i}), that is δ_{i} ∘ c_{i} = c_{i} ∘ δ_{i}. Indeed, using the trivialization α over S′ from above, δ_{i} corresponds to g_{i} := δ_{i} ∘ α^{−1} ∈ LP_{ν_{i}}(S′), and then

δ_{i} ∘ c_{i} := δ_{i} ∘ (α^{−1} ∘ c_{i} ∘ α) = g_{i} ∘ c_{i} ∘ α = c_{i} ∘ g_{i} ∘ α = c_{i} ∘ δ_{i} .

This shows that the action of c_{i} ∈ Z(Q_{ν_{i}}) on M̆Ẑ_{i} ; \underline{𝕃}_{i} is given as

c_{i}: M̆Ẑ_{i} ; \underline{𝕃}_{i} ⟶ M̆Ẑ_{i} ; \underline{𝕃}_{i} , (\underline{L}_{i}, δ_{i}) ⟼ (\underline{L}_{i}, c_{i} ∘ δ_{i}) = (\underline{L}_{i}, δ_{i} ∘ c_{i}) .  (7.6)

This action commutes with the action of η ∈ I_{\underline{G}_{0}}(Q), because η and (c_{i})_{i} act on ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} ×Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H by

(\underline{L}_{i}, δ_{i})_{i} × γH ⟼ (\underline{L}_{i}, Γ̂_{ν_{i}}(η) ∘ δ_{i} ∘ c_{i})_{i} × V̌_{η}γH .

On the other hand (c_{i})_{i} ∈ ∏_{i} Z(Q_{ν_{i}}) also acts on ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} as follows. Let (\underline{G}, γH) be in ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(S) and consider the associated local ℙ_{ν_{i}}-shtukas \underline{L}_{i} := Γ̂_{ν_{i}}(\underline{G}). We have seen above that c_{i} induces an element c_{i} ∈ QIsog_{S}(\underline{L}_{i}) of the quasi-isogeny group of \underline{L}_{i}. By Proposition 5.2 there is a uniquely determined global 𝔊-shtuka c_{n}^{∗} ∘ … ∘ c_{1}^{∗} \underline{G} and a quasi-isogeny c: c_{n}^{∗} ∘ … ∘ c_{1}^{∗} \underline{G} → \underline{G} which is an isomorphism outside the ν_{i} and satisfies Γ̂_{ν_{i}}(c) = c_{i}. We can now define the action of (c_{i})_{i} ∈ ∏_{i} Z(Q_{ν_{i}}) on (\underline{G}, γH) ∈ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(S) as

(c_{i})_{i}: (\underline{G}, γH) ⟼ (c_{n}^{∗} ∘ … ∘ c_{1}^{∗} \underline{G} , Ť_{c}^{−1}γH) .  (7.7)

7.9. Source and target of Θ carry a Weil-descent datum for the ring extension R_{\underline{Ẑ}} ⊂ R̆_{\underline{Ẑ}} = k[[ξ_{1}, …, ξ_{n}]], compare [RZ96, Definition 3.45]. We explain what this means. Consider the R_{\underline{Ẑ}}-automorphism λ of R̆_{\underline{Ẑ}} given by

λ: ξ_{i} ⟼ ξ_{i} and λ|_{k} = Frob_{#κ,k}: x ⟼ x^{#κ} for x ∈ k .

For a scheme (S, θ) ∈ NilpR̆_{\underline{Ẑ}} where θ: S → Spf R̆_{\underline{Ẑ}} denotes the structure morphism of the scheme S we

denote by S_{[λ]} ∈ NilpR̆_{\underline{Ẑ}} the pair (S, λ ∘ θ). For a stack H over Spf R̆_{\underline{Ẑ}} we consider the stack H^{λ} defined by

H^{λ}(S) := H(S_{[λ]}) .

Then a Weil-descent datum on H is an isomorphism of stacks H ^{∼} ⟶ H^{λ}, that is an equivalence H(S) ^{∼} ⟶ H(S_{[λ]}) for every S ∈ NilpR̆_{\underline{Ẑ}} compatible with morphisms in NilpR̆_{\underline{Ẑ}}.

On ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} the canonical Weil-descent datum is given by the identity

id: ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(S) ^{∼} ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(S_{[λ]}) , (\underline{G}, γH) ⟼ (\underline{G}, γH)  (7.8)

because under the inclusion NilpR̆_{\underline{Ẑ}} ↪ Nilp_{R_{\underline{Ẑ}}} we have S = S_{[λ]} in Nilp_{R_{\underline{Ẑ}}}.

On M̆Ẑ_{i} ; \underline{𝕃}_{i} we consider the Weil descent datum given by

M̆Ẑ_{i} ; \underline{𝕃}_{i}(S) ^{∼} ⟶ M̆Ẑ_{i} ; \underline{𝕃}_{i}(S_{[λ]}),  (7.9)
(\underline{L}_{i}, δ_{i}: \underline{L}_{i} → \underline{𝕃}_{i,S}) ⟼ (\underline{L}_{i}, θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ δ_{i}: \underline{L}_{i} → \underline{𝕃}_{i,S_{[λ]}}) .

Here we observe that 𝕃_{i,S} := θ^{∗}\underline{𝕃}_{i} and \underline{𝕃}_{i,S_{[λ]}} := (λ ∘ θ)^{∗}\underline{𝕃}_{i} = θ^{∗}λ^{∗}\underline{𝕃}_{i} = θ^{∗}σ^{[κ: 𝔽_{q}]∗}\underline{𝕃}_{i}, and that τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]} : \underline{𝕃}_{i} → σ^{[κ: 𝔽_{q}]∗}\underline{𝕃}_{i} is a quasi-isogeny.

We do not discuss the question whether these Weil descent data are effective. In the analogous situation for p-divisible groups this is true and proved by Rapoport and Zink in [RZ96, Theorem 3.49]. Their argument uses a morphism 𝔾_{m} → 𝔊, which might not exist in our setup.

Moreover, on ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H we consider the product of the Weil Descent data (7.9) with the identity on Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H. This Weil descent datum commutes with the action of η ∈ I_{\underline{G}_{0}}(Q) by the following diagram

[figure: (\underline{L}_{i}, δ_{i}); η; descent datum; (\underline{L}_{i}, θ^{∗}(Γ̂_{ν_{i}}(η)) ∘ δ_{i}); descent datum; (\underline{L}_{i}, θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ δ_{i}); η; (\underline{L}_{i}, θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ θ^{∗}(Γ̂_{ν_{i}}(η)) ∘ δ_{i}); (\underline{L}_{i}, (λθ)^{∗}(Γ̂_{ν_{i}}(η)) ∘ θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ δ_{i}) ,]

as θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]} ∘ Γ̂_{ν_{i}}(η)) = θ^{∗}(σ^{[κ: 𝔽_{q}]∗}(Γ̂_{ν_{i}}(η)) ∘ τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) = (λθ)^{∗}(Γ̂_{ν_{i}}(η)) ∘ θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}). This defines a Weil

descent datum on Y := I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) by

Y(S) ^{∼} ⟶ Y^{λ}(S) = Y(S_{[λ]})  (7.10)
(\underline{L}_{i}, δ_{i})_{i} × γH ⟼ (\underline{L}_{i}, θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ δ_{i})_{i} × γH .

7.10. For every multiple m ∈ ℕ_{0} of [κ_{i} : 𝔽_{q}] the special fiber M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k of M̆Ẑ_{i} ; \underline{𝕃}_{i} carries a Frobenius endomorphism Φ_{m} defined as follows. Consider the absolute q^{m}-Frobenius σ^{m} := Frob_{q^{m},S}: S → S on a k-scheme S and a pair (\underline{L}_{i}, δ_{i}) ∈ M̆Ẑ_{i} ; \underline{𝕃}_{i}(S). It induces the left horizontal morphisms in the diagram

[figure: S; _{Frob_{q^{m},S}}; (\underline{L}_{i}, δ_{i}); (σ^{m∗}\underline{L}_{i}, σ^{m∗}δ_{i}); M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k; _{Frob_{q^{m}}}; Spec k; _{Frob_{q^{m},k}}; S; (\underline{L}_{i}, δ_{i}); M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k; Spec k]

Let θ: S → Spec k be the structure morphism of S. Then the upper S viewed as a scheme over the lower Spec k has structure morphism Frob_{q^{m},k} ∘θ. So in terms of 7.9 with λ := λ_{m} := Frob_{q^{m},k} the upper S is S_{[λ_{m}]} over the lower Spec k. Since m is a multiple of [κ_{i}: 𝔽_{q}] and the reduced subscheme Z_{i} of the bound Ẑ_{i} is defined over κ_{i} the Frobenius τ_{σ^{m∗}L_{i}} = σ^{m∗}τ_{L_{i}} lies in σ^{m∗}Z_{i} = Z_{i}. This means that also σ^{m∗}\underline{L}_{i} is bounded by Ẑ_{i}, and therefore the diagonal arrow (σ^{m∗}\underline{L}_{i}, σ^{m∗}δ_{i}) lies in M̆Ẑ_{i} ; \underline{𝕃}_{i}(S_{[λ_{m}]}) = (M̆Ẑ_{i} ; \underline{𝕃}_{i})^{λ_{m}}(S). Via

the Weil descent datum (7.9) it is mapped to (σ^{m∗}\underline{L}_{i}, θ^{∗}(τ_{𝕃_{i}}^{m}) ∘ σ^{m∗}δ_{i}) which lies in M̆Ẑ_{i} ; \underline{𝕃}_{i}(S). We therefore

define the q^{m}-Frobenius endomorphism of M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k as

Φ_{m}: M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k ⟶ (M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k)  (7.11)
(\underline{L}_{i}, δ_{i}) ⟼ (σ^{m∗}\underline{L}_{i}, τ_{𝕃_{i}}^{m} ∘ σ^{m∗}δ_{i}) = (σ^{m∗}\underline{L}_{i}, δ_{i} ∘ τ_{L_{i}}^{m}) .

If m is a multiple of [κ: 𝔽_{q}] the product of the q^{m}-Frobenius morphisms Φ_{m} of the M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k from (7.11) with the identity on Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H gives a q^{m}-Frobenius morphism

Φ_{m}: (∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H ⟶  (7.12)

⟶ (∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H .

It directly follows that Φ_{m} commutes with the action of η ∈ I_{\underline{G}_{0}}(Q), because the composition is given by

(\underline{L}_{i}, δ_{i})_{i} × γH ⟼ (\underline{L}_{i}, Γ̂_{ν_{i}}(η) ∘ δ_{i} ∘ τ_{L_{i}}^{m})_{i} × V̌_{η}γH .

This defines the q^{m}-Frobenius endomorphism Φ_{m} of I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H).

Likewise for every multiple m ∈ ℕ_{0} of [κ: 𝔽_{q}] the stack ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spec k carries the relative q^{m}-Frobenius, which is an endomorphism, because this stack arises by base change from Spec κ. This q^{m}-Frobenius endomorphism is given

Φ_{m}: ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spec k ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spec k
(\underline{G}, γH) ⟼ (σ^{m∗}\underline{G}, σ^{m∗}(λ)H) .  (7.13)

Since m is a multiple of [κ_{i}: 𝔽_{q}] and the reduced subscheme Z_{i} of the bound Ẑ_{i} is defined over κ_{i} the Frobenius Γ̂_{ν_{i}}(τ_{σ^{m∗}G}) = σ^{m∗}Γ̂_{ν_{i}}(τ_{G}) lies in σ^{m∗}Z_{i} = Z_{i}. This means that also σ^{m∗}\underline{G} is bounded by \underline{Ẑ}.

Theorem 7.11. Keep the above notation and consider a compact open subgroup H ⊂ 𝔊(𝔸\underline{ν}).

(a) The morphism Ψ_{\underline{G}_{0}} from Theorem 5.3 induces an I_{\underline{G}_{0}}(Q)-invariant morphism

Θ′: ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} ,  (7.14)

where I_{\underline{G}_{0}}(Q) acts trivially on the target and diagonally on the source as described in 7.6. Furthermore, this morphism factors through a morphism

Θ: I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}  (7.15)

of ind-DM-stacks over Spf R̆_{\underline{Ẑ}}, which is a monomorphism in the sense that the functor Θ is fully faithful, or equivalently that its diagonal is an isomorphism. Both morphisms are ind-proper and formally étale.

(b) Let \{T_{j}\} be a set of representatives of I_{\underline{G}_{0}}(Q)-orbits of the irreducible components of the scheme ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H which is locally of finite type over k. Then the image Θ′(T_{j}) of T_{j} under Θ′ is a closed substack with the reduced structure and each Θ′(T_{j}) intersects only finitely many others. Let Z be the union of the Θ′(T_{j}). Its underlying set is the isogeny class of \underline{G}_{0}, that is the

set of all (\underline{G}, γH) for which \underline{G} is isogenous to \underline{G}_{0}. Let ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z be the formal completion of

∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} along Z; see Remark 7.12(a). Then Θ induces an isomorphism of locally noetherian, adic formal algebraic Deligne-Mumford stacks locally formally of finite type over Spf R̆_{\underline{Ẑ}}

Θ_{Z}: I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) ^{∼} ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z ,

and in particular of the underlying Deligne-Mumford stacks

Θ_{Z}: I_{\underline{G}_{0}}(Q)\(∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H) ^{∼} ⟶ Z

which are locally of finite type and separated over Spec k.

(c) The morphisms Θ′, Θ and Θ_{Z} are compatible with the following actions on source and target: the action of ∏_{i} Z(Q_{ν_{i}}) described in 7.8, the action of 𝔊(𝔸\underline{ν}) through Hecke-correspondences, see Remark 7.12(b) below, and the Weil descent data described in 7.9. For every multiple m ∈ ℕ_{0} of [κ: 𝔽_{q}] the base changes of Θ′, Θ and Θ_{Z} to Spec k are compatible with the q^{m}-Frobenius endomorphisms Φ_{m} from (7.12) and (7.13).

As an explanation of the theorem and a preparation for its proof we begin with a

Remark 7.12. (a) Notice that the T_{j} correspond bijectively to the irreducible components of the Deligne-Mumford stack (7.4) which is locally of finite type over Spec k. Since Θ is a monomorphism by part (a) of the theorem, each Θ′(T_{j}) intersects only finitely many others. The restriction of Θ′ to T_{j} is proper, because T_{j} is quasi-compact by [AH14, Corollary 4.26] and Θ′ is ind-proper by part (a) of the theorem. So the Θ′(T_{j}) are closed substacks. Reasoning as in [RZ96, 6.22] we may form the formal completion along their union Z. It is defined by requiring that its category ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z(S) of S-valued points is the full subcategory given by

\{ f: S → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}, such that f|_{S_{red}} factors through Z \} ,

where S_{red} is the underlying reduced closed subscheme. A priory this formal completion is only an ind-DM-stack over Spf R̆_{\underline{Ẑ}}, but it will follow from Theorem 7.11(b) that it is a locally noetherian, adic formal algebraic Deligne-Mumford stack locally formally of finite type over over Spf R̆_{\underline{Ẑ}}. Note that it follows immediately that the natural morphism

∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}

is a monomorphism and formally étale, because for an affine scheme S = Spec B ∈ NilpR̆_{\underline{Ẑ}} and an ideal

I ⊂ B with I^{2} = (0) one has S_{red} = (Spec B/I)_{red}.

(b) The action of h ∈ 𝔊(𝔸\underline{ν}) by Hecke correspondences is explicitly given as follows. Let H, H′ ⊂ 𝔊(𝔸\underline{ν}) be compact open subgroups. Then the Hecke correspondences π(h)_{H′,H} are given by the diagrams

\xymatrix{ & ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/(hHh^{−1} ∩ H′) \ar@{-->}[dl] \ar@{-->}[dr] \\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H &  & ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H′ \ar@{-->}[ll] \\  & (\underline{L}_{i}, δ_{i})_{i} × γ(hHh^{−1} ∩ H′) \ar@{|->}[dl] \ar[dr] \\ (\underline{L}_{i}, δ_{i})_{i} × γhH &  & (\underline{L}_{i}, δ_{i})_{i} × γH′}  (7.16)

and

\xymatrix{ & ∇_{n}^{(hHh^{−1}∩H′),\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} \ar[dl] \ar[dr] \\ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} &  & ∇_{n}^{H′,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} \ar@{-->}[ll] \\  & (\underline{G}, γ(hHh^{−1} ∩ H′)) \ar@{|->}[dl] \ar[dr] \\ (\underline{G}, γhH) &  & (\underline{G}, γH′)}  (7.17)

A special case for H′ ⊂ H and h = 1 are the forgetful morphisms

π(1)_{H′,H}: ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H′ → ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H

and π(1)_{H′,H}: ∇_{n}^{H′,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}, which are finite étale and surjective; see Theorem 6.7(b).

Proof of Theorem 7.11(a) and (c). For the empty closed subscheme D = ∅ ⊂ C recall that H_{∅} = 𝔊(𝕆\underline{ν}) and ∇_{n}^{H_{∅}}ℋ ^{1}(C, 𝔊)\underline{ν} ≅ ∇_{n}ℋ_{∅}^{1}(C, 𝔊)\underline{ν} = ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} by Theorem 6.5. Consider the following diagram of ind-DM-stacks in which the map Ψ_{\underline{G}_{0}} in the bottom row was introduced in Theorem 5.3

[figure: ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i}; Ψ_{\underline{G}_{0}}; ∇_{n}^{H_{∅},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}; _{∏_{i} M̆_{\underline{𝕃}_{i}}}; Ψ_{\underline{G}_{0}}; ∇_{n}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}; (\underline{L}_{i}, δ_{i}: \underline{L}_{i} → \underline{𝕃}_{i})_{i}; δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0} .]  (7.18)

Since δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0} is bounded by \underline{Ẑ} if and only if every \underline{L}_{i} ≅ Γ̂_{ν_{i}}(δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0}) is bounded by Ẑ_{i}, the diagram (7.18) is 2-cartesian. In particular the morphism in the upper row, which we again call Ψ_{\underline{G}_{0}}, is

ind-proper and formally étale. Consider an S-valued point (\underline{L}_{i}, δ_{i})_{i} of ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} and let \underline{G} := δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S} denote its image under Ψ_{\underline{G}_{0}}. By Proposition 5.2 there is a unique quasi-isogeny δ: \underline{G} → \underline{G}_{0,S} which is an isomorphism outside the ν_{i} and satisfies ˆ \underline{Γ}(δ) = (δ_{i})_{i}. This induces an isomorphism Ť_{δ}: Ť\underline{G} ^{∼} ⟶ Ť_{\underline{G}_{0}} of tensor functors, see (6.1). Since δ is defined over S, this isomorphism Ť_{δ} is equivariant for the action of π_{1}^{ét}(S, s̄) which acts on Ť_{\underline{G}_{0}} through the map π_{1}^{ét}(S, s̄) → π_{1}^{ét}(Spec k, s̄) = (1), that is trivially. In particular, the H-orbit Ť_{δ}^{−1}γH of the tensor isomorphism Ť_{δ}^{−1}γ: ω^{∘} ^{∼} ⟶ V̌\underline{G} is invariant under π_{1}^{ét}(S, s̄). Now sending (\underline{L}_{i}, δ_{i})_{i} × γH to (\underline{G}, Ť_{δ}^{−1}γH) defines the morphism

Θ′: ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H ⟶ ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}  (7.19)
(\underline{L}_{i}, δ_{i})_{i} × γH  ⟼ (\underline{G}, Ť_{δ}^{−1}γH) .

It sends the k-valued point (\underline{𝕃}_{i}, id)_{i}×γH to (\underline{G}_{0}, γH) and is obviously equivariant for the action of ∏_{i} Z(Q_{ν_{i}}) given in (7.6) and (7.7), and the action of 𝔊(𝔸\underline{ν}) through Hecke correspondences given in (7.16) and (7.17).

The morphism Θ′ is compatible with the Weil descent data (7.8) and (7.9), because for (S, θ) ∈ NilpR̆_{\underline{Ẑ}} and S_{[λ]} = (S, λθ) ∈ NilpR̆_{\underline{Ẑ}} the S-valued point (\underline{L}_{i}, δ_{i})_{i} × γH, respectively the S_{[λ]}-valued point

(\underline{L}_{i}, θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ δ_{i})_{i} × γH of the source of Θ′ are sent to (\underline{G}, Ť_{δ}^{−1}θ^{∗}(γ)H) and (\underline{G}′, Ť_{δ′}^{−1}(λθ)^{∗}(γ)H), respec-

tively, where \underline{G} := δ_{n}^{∗} ∘…∘δ_{1}^{∗} \underline{G}_{0,S} and \underline{G}′ := (θ^{∗}τ_{𝕃_{n}}^{−[κ: 𝔽_{q}]}δ_{n})^{∗} ∘…∘(θ^{∗}τ_{𝕃_{1}}^{−[κ: 𝔽_{q}]}δ_{1})^{∗} \underline{G}_{0,S_{[λ]}}, and δ: \underline{G} → \underline{G}_{0,S} :=

θ^{∗}\underline{G}_{0} and δ′: \underline{G}′ → \underline{G}_{0,S_{[λ]}} := (λ ∘ θ)^{∗}\underline{G}_{0} = θ^{∗}λ^{∗}\underline{G}_{0} = θ^{∗}(σ^{[κ: 𝔽_{q}]∗}\underline{G}_{0}) are the quasi-isogenies with Γ̂_{ν_{i}}(δ) = δ_{i}

and Γ̂_{ν_{i}}(δ′) = θ^{∗}(τ_{𝕃_{i}}^{−[κ: 𝔽_{q}]}) ∘ δ_{i}, which are isomorphisms outside \underline{ν}. Then φ := δ^{−1} ∘ θ^{∗}(τ_{G_{0}}^{[κ: 𝔽_{q}]}) ∘ δ′: \underline{G}′ → \underline{G} is a quasi-isogeny by Corollary 6.3 with Γ̂_{ν_{i}}(φ) = id_{\underline{L}_{i}} and V̌_{φ} ∘ Ť_{δ′}^{−1} ∘ θ^{∗}λ^{∗}(γ)H = Ť_{δ}^{−1}θ^{∗}(γ)H, because λ^{∗}(γ) = σ^{[κ: 𝔽_{q}]∗}(γ) and V̌_{τ_{G_{0}}^{[κ: 𝔽q]}} ∘ σ^{[κ: 𝔽_{q}]∗}(γ) = γ. Since the Weil descend datum on the source of Θ′

commutes with the action of I_{\underline{G}_{0}}(Q) this also proves the compatibility of Θ with the Weil descend data

(7.8) and (7.10). Finally, the target ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z carries the Weil descend datum (7.8) induced from

∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}. The reason is that if a morphism S_{red} → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} given

by (\underline{G}, γH) factors through Z = im(Θ′), then also the morphism (S_{[λ]})_{red} → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} given by (\underline{G}, γH) factors through Z = im(Θ′), because Θ′ commutes with the Weil descend data. This shows that also Θ_{Z} is compatible with the Weil descend data (7.8) and (7.10).

We also prove that Θ′ commutes with the q^{m}-Frobenius endomorphisms Φ_{m}. Let \underline{y} := (\underline{L}_{i}, δ_{i})_{i} ×

γH be an S-valued point of (∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} ˆ×_{R_{i}} Spec k) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H. The images of this point and of

Φ_{m}(\underline{y}) = (σ^{m∗}\underline{L}_{i}, τ_{𝕃_{i}}^{m} ∘ σ^{m∗}δ_{i})_{i} × γH in ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} are given by Θ′(\underline{y}) = (\underline{G}, Ť_{δ}^{−1}γH) and Θ′ ∘ Φ_{m}(\underline{y}) = (\underline{G}′, Ť_{δ′}^{−1}γH), respectively, where \underline{G} := δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S} and \underline{G}′ := (τ_{𝕃_{n}}^{m}σ^{m∗}δ_{n})^{∗} ∘ … ∘ (τ_{𝕃_{1}}^{m}σ^{m∗}δ_{1})^{∗} \underline{G}_{0,S}, and δ: \underline{G} → \underline{G}_{0,S} and δ′: \underline{G}′ → \underline{G}_{0,S} are the quasi-isogenies with Γ̂_{ν_{i}}(δ) = δ_{i} and Γ̂_{ν_{i}}(δ′) = τ_{𝕃_{i}}^{m} ∘ σ^{m∗}δ_{i}, which are isomorphisms outside \underline{ν}. We obtain for the image Φ_{m} ∘ Θ′(\underline{y}) = (σ^{m∗}\underline{G}, σ^{m∗}(Ť_{δ}^{−1}γ)H), which comes with the quasi-isogeny σ^{m∗}(δ): σ^{m∗}\underline{G} → σ^{m∗}\underline{G}_{0,S}. Then φ := σ^{m∗}(δ)^{−1} ∘ τ_{G_{0}}^{−m} ∘ δ′: \underline{G}′ → σ^{m∗}\underline{G} is a quasi-isogeny by Corollary 6.3 with Γ̂_{ν_{i}}(φ) = id_{σ^{m∗}\underline{L}_{i}} and V̌_{φ} ∘ Ť_{δ′}^{−1}γH = Ť_{σ^{m∗}δ}^{−1} ∘ V̌_{τ_{G_{0}}^{m}}^{−1} ∘ γH = σ^{m∗}(Ť_{δ}^{−1}γ)H, because

V̌_{τ_{G_{0}}^{m}}^{−1} ∘ γ = σ^{m∗}(γ). This proves Θ′ ∘ Φ_{m} = Φ_{m} ∘ Θ′. Since Φ_{m} commutes with the action of I_{\underline{G}_{0}}(Q) this

also proves that Θ commutes with the Φ_{m}. Finally, the target ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z carries the q^{m}-Frobenius

endomorphism Φ_{m} induced from (7.13) on ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}. The reason is that if a morphism

S_{red} → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} given by (\underline{G}, γH) factors through Z = im(Θ′), then also the morphism

S_{red} → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} given by (σ^{m∗}\underline{G}, σ^{m∗}(λ)H) factors through Z = im(Θ′), because Θ′ commutes with the Φ_{m}. This shows that also Θ_{Z} is compatible with the q^{m}-Frobenius endomorphisms Φ_{m} (7.13) and (7.12). So we have already proved (c).

The group I_{\underline{G}_{0}}(Q) acts on the source of the morphism Θ′ by sending an S-valued point (\underline{L}_{i}, δ_{i})_{i} × γH to (\underline{L}_{i}, Γ̂_{ν_{i}}(η)δ_{i})_{i} × V̌_{η}γH for η ∈ I_{\underline{G}_{0}}(Q). These two S-valued points are mapped under Θ′ to global 𝔊shtukas with H-level structure (\underline{G}, Ť_{δ}^{−1}γH) and (\underline{G̃}, Ť −1 ; δ̃V̌_{η}γH) over S, where δ: \underline{G} := δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0} → \underline{G}_{0} is the isogeny satisfying Γ̂_{ν_{i}}(δ) = δ_{i} and δ̃: \underline{G̃} := (Γ̂_{ν_{n}}(η)δ_{n})^{∗} ∘ … ∘ (Γ̂_{ν_{1}}(η)δ_{1})^{∗} \underline{G}_{0} → \underline{G}_{0} is the isogeny satisfying Γ̂_{ν_{i}}(δ̃) = Γ̂_{ν_{i}}(η)δ_{i}. Since V̌_{δ̃^{−1}ηδ} ∘ Ť_{δ}^{−1}γH = Ť −1 ; δ̃V̌_{η}γH these two global 𝔊-shtukas with H-level structure are isomorphic via the quasi-isogeny δ̃^{−1}ηδ: \underline{G} → \underline{G̃}, which is an isomorphism at the ν_{i}, because Γ̂_{ν_{i}}(δ̃^{−1}ηδ) = (Γ̂_{ν_{i}}(η)δ_{i})^{−1} ∘ Γ̂_{ν_{i}}(η)δ_{i} = id. In other words, Θ′ is invariant under the action of I_{\underline{G}_{0}}(Q) and factors through the morphism Θ from (7.15) of ind-DM-stacks. Then Theorem 7.11(a) follows from Lemmas 7.13 and 7.14 below.

Lemma 7.13. We use the abbreviations Y_{1} := ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H and Y_{2} := ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H. Then for j = 1 or j = 2 the action of I_{\underline{G}_{0}}(Q) on Y_{j} induces an isomorphism of stacks

I_{\underline{G}_{0}}(Q) × Y_{j} := ∐_{I_{\underline{G}_{0}}(Q)} Y_{j} ^{∼} ⟶ Y_{j} ×_{∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C,𝔊)\underline{ν} ˆ×_{R\underline{Ẑ}} Spf R̆_{\underline{Ẑ}}} Y_{j} ,

where the map to the first copy of Y_{j} is the identity and the map to the second copy is given by the action of I_{\underline{G}_{0}}(Q) on Y_{j}. In particular, Θ is a monomorphism in the sense stated in Theorem 7.11(a).

Proof. That the two definitions of a monomorphism given in Theorem 7.11(a) are equivalent follows from [StPr, Tag 04Z7]. The morphism is well defined by the I_{\underline{G}_{0}}(Q)-equivariance of Θ′. To describe its inverse, consider a connected scheme S ∈ NilpR̆\underline{Ẑ} and two S-valued points of Y_{j}

\underline{y} := ((\underline{L}_{i}, δ_{i})_{i}, γH) and \underline{y}′ := ((\underline{L}′_{i}, δ′_{i})_{i}, γ′H)

which under Θ′ are mapped to global 𝔊-shtukas (\underline{G}, Ť_{δ}^{−1}γH) and (\underline{G}′, Ť_{δ′}^{−1}γ′H) with H-level structures, where δ: \underline{G} → \underline{G}_{0,S} and δ′: \underline{G}′ → \underline{G}_{0,S} are the canonical quasi-isogenies which are isomorphisms outside \underline{ν} with ˆ \underline{Γ}(δ) = (δ_{i})_{i} and ˆ \underline{Γ}(δ′) = (δ′_{i})_{i}. Assume that (\underline{G}, Ť_{δ}^{−1}γH) and (\underline{G}′, Ť_{δ′}^{−1}γ′H) are isomorphic in

∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(S) via a quasi-isogeny φ: \underline{G} → \underline{G}′ which is an isomorphism at the ν_{i} and compatible with the H-level structures, that is V̌_{φ} ∘ Ť_{δ}^{−1}γH = Ť_{δ′}^{−1}γ′H; see Definition 6.4. Consider the quasi-isogeny η := δ′φδ^{−1} from \underline{G}_{0,S} to itself. By Proposition 7.5 we may view η as an element of I_{\underline{G}_{0}}(Q).

Between the associated local ℙ_{ν_{i}}-shtukas we consider the corresponding quasi-isogenies

\xymatrix{\underline{L}_{i} \ar[r]^{Γ̂_{ν_{i}}(φ)} \ar[d]_{δ_{i}} & \underline{L}′_{i} \ar[d]_{δ′_{i}} \\ _{\underline{𝕃}_{i,S}} \ar[r]^{Γ̂_{ν_{i}}(η)} & \underline{𝕃}_{i,S}.}

Since φ: \underline{G} → \underline{G}′ is an isomorphism at the ν_{i} the quasi-isogenies Γ̂_{ν_{i}}(φ) are isomorphisms. This shows

that η · (\underline{L}_{i}, δ_{i}) := (\underline{L}_{i}, Γ̂_{ν_{i}}(η) ∘ δ_{i}) ≅ (\underline{L}′_{i}, δ′_{i}) in M̆Ẑ_{i} ; \underline{𝕃}_{i}(S). Moreover, η sends γH ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H to

V̌_{η} ∘ γH = Ť_{δ′} V̌_{φ}Ť_{δ}^{−1} ∘ γH = γ′H. This proves that η · \underline{y} = \underline{y}′ and thus Θ is a monomorphism. Moreover, (η, \underline{y}) maps to (\underline{y}, \underline{y}′).

Lemma 7.14. The morphisms Θ′ and Θ from (7.14) and (7.15) are formally étale and ind-proper.

Proof. If H′ ⊂ H is a normal subgroup, then dividing out the action of H/H′ on source and target of the morphism Θ (respectively Θ′) for H′ yields the morphism Θ (respectively Θ′) for H, because Θ and Θ′ are compatible with the action of h ∈ H ⊂ 𝔊(𝔸\underline{ν}) by the Hecke correspondences π_{H′,H′}(h). Using the argument of Theorem 6.7(a), which produces normal subgroups H_{2} ⊂ H and H_{D} ⊂ H_{2}, it suffices to prove the lemma for H = H_{D}, where D ⊂ C is a proper closed subscheme. Fix an element γH_{D} ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H_{D} and

consider the component ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × \{γH_{D}\} of ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H_{D}. The morphism

Θ′: ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × \{γH_{D}\} ⟶ ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} ,  (7.20)
(\underline{L}_{i}, δ_{i})_{i} × γH_{D} ⟼ (\underline{G}, Ť_{δ}^{−1}γH_{D})

is formally étale, because its composition with the finite étale forgetful morphism π(1)_{H_{D},H_{∅}} from Re-

mark 7.12(b) is the morphism Ψ_{\underline{G}_{0}} : ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} ⟶ ∇_{n}^{H_{∅},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} from (7.18) which is formally étale. This proves that Θ′ is formally étale. For the same reason Θ′ satisfies the valuative criterion for properness [LM00, Théorème 7.3].

To prove that Θ is formally étale consider a component Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} of the source of Θ for \overline{γ} = γH_{D}, see Proposition 7.7, and a commutative diagram of solid arrows

[figure: \overline{T}; Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i}; Θ; T; ?; ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}}]  (7.21)

where \overline{T} ⊂ T is a closed subscheme defined by a sheaf of ideals I ⊂ O_{T} with I^{2} = (0). We use the notation of the proof of [AH14, Proposition 4.27]. There Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} is the union of open substacks (Γ′_{x}\Γγ̄)\V_{x} where Γ′_{x} ⊂ Γγ̄ is a normal subgroup of finite index, V_{x} = ⋃_{β∈Γ′_{x}\Γγ̄} β ·U_{x} is an open formal subscheme

of Γ′_{x}\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i}, and β ·U_{x} ⊂ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} is a formal open subscheme such that β ·U_{x} → Γ′_{x}\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} is an open immersion. Let \overline{T}_{x} be the preimage of (Γ′_{x}\Γγ̄)\V_{x} in \overline{T} and let T_{x} be the open subscheme of T with underlying topological space \overline{T}_{x}. Its base change

\overline{T} ′_{x} := \overline{T}_{x} ×_{(Γ′_{x}\Γγ̄)\V_{x}} V_{x} ⟶ \overline{T}

is finite étale Γ′_{x}\Γγ̄-Galois. By [SGA 1, Exposé I, Théorème 8.3 and Exposé IX, Proposition 2.4] there exists a finite étale Γ′_{x}\Γγ̄-Galois cover T′_{x} → T_{x} with T′_{x} ×_{T_{x}} \overline{T}_{x} = \overline{T} ′_{x}. Under the projection map \overline{T} ′_{x} → V_{x} we let \overline{T} ′_{x,β} be the preimage of β ·U_{x} ⊂ V_{x} and we let T′_{x,β} be the open subscheme of T′_{x} with underlying

topological space \overline{T} ′_{x,β}. Then we obtain a morphism \overline{T} ′_{x,β} → β ·U_{x} ⊂ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} = ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × \{γH_{D}\}, and since the morphism Θ′ from (7.20) is formally étale, it lifts uniquely to a morphism f_{β}: T′_{x,β} → β·U_{x} ↪ V_{x}. For two different β, β′ the two maps f_{β} and f_{β′} from T′_{x,β} ∩ T′_{x,β′} to β ·U_{x} ∩ β′ ·U_{x} ⊂ β ·U_{x} ⊂ V_{x} coincide by uniqueness, because their restrictions to \overline{T} ′_{x,β} ∩ \overline{T} ′_{x,β′} do. Thus the morphisms f_{β} for all β glue to give a uniquely determined morphism f: T′_{x} → V_{x}. This morphism is Γ′_{x}\Γγ̄-equivariant because its restriction to \overline{T} ′_{x} is. Dividing out Γ′_{x}\Γγ̄ yields a uniquely determined morphism T_{x} → Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i}. For all x the latter morphisms glue to produce the dashed arrow in diagram (7.21). This proves that Θ is formally étale.

To show that Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} → ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} satisfies the valuative criterion for proper-

ness [LM00, Théorème 7.3] we use that Γ′_{x}\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} → Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} is finite étale Γ′_{x}\Γγ̄-Galois. In the analogous diagram to (7.21) with \overline{T} replaced by Spec K and T replaced by Spec R for a valuation ring R with fraction field K, there are x, β and a finite field extension K′ of K such that the morphism Spec K → Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} lifts to Spec K′ → β · U_{x}. By the ind-properness of Θ′ from (7.20) this lifts to

Spec R′ → ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} → Γγ̄\ ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} and proves the valuative criterion for properness.

Finally Θ′ and Θ are ind-proper, because their sources ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} and Γγ̄\ ∏_{i} M̆_{\underline{𝕃}_{i}} are an ind-closed ind-

subscheme of the ind-quasi-projective ind-scheme ∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i}, respectively a quotient of that.

Proof of Theorem 7.11(b). We already started with the proof in Remark 7.12(a). Let us next prove that the underlying set of Z is the isogeny class of \underline{G}_{0}. Every K-valued point (\underline{G}, γH) of Z for a field K lies in the image of Θ′, and hence is of the form \underline{G} = δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,K} with an isogeny δ: \underline{G} → \underline{G}_{0,K}. This shows that Z is contained in the isogeny class of \underline{G}_{0}. Conversely, let (\underline{G}′, γ′H) be a K-valued point

of ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spec k in the isogeny class of \underline{G}_{0}, and let δ′: \underline{G}′ → \underline{G}_{0,K} be an isogeny. Let

\underline{L}_{i} := Γ̂_{ν_{i}}(\underline{G}′) and δ_{i} := Γ̂_{ν_{i}}(δ′): \underline{L}_{i} → \underline{𝕃}_{i} and γH := V̌_{δ′}γ′H ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H. Then (\underline{L}_{i}, δ_{i}) × γH is a K-valued point of the source of Θ′ which is mapped under Θ′ to (\underline{G}, Ť_{δ}^{−1}γH), where \underline{G} := δ_{n}^{∗}∘…∘δ_{1}^{∗} \underline{G}_{0,K} and δ: \underline{G} → \underline{G}_{0,K} is the isogeny with Γ̂_{ν_{i}}(δ) = δ_{i} which is an isomorphism outside \underline{ν}. The isogeny δ^{−1}δ′: \underline{G}′ → \underline{G}

satisfies Γ̂_{ν_{i}}(δ^{−1}δ′) = id and V̌_{δ^{−1}δ′} ∘ γ′H = Ť_{δ}^{−1}γH, and so (\underline{G}′, γ′H) ≅ (\underline{G}, γH) in ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν}(K). The point (\underline{L}_{i}, δ_{i}) × γH lies on an irreducible component of ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H belonging to the I_{\underline{G}_{0}}(Q)-orbit of some irreducible component T_{j}. By the I_{\underline{G}_{0}}(Q)-equivariance of Θ′ we can move the point (\underline{L}_{i}, δ_{i}) × γH to T_{j} and then its image (\underline{G}′, γ′H) under Θ′ lies in Θ′(T_{j}) ⊂ Z as desired.

To prove that Z is separated over Spec k we use the valuative criterion [LM00, Proposition 7.8]. Let R be a valuation ring containing k and consider two morphisms f_{1}, f_{2} : Spec R → Z whose restrictions f_{i,K} to the fraction field K of R are isomorphic in Z(K). We must show that f_{1} ≅ f_{2} in Z(R). The K-valued point f_{1,K} ≅ f_{2,K} lies on Θ′(T_{j}) ⊂ Z for some j. Since Θ′(T_{j}) is a closed substack of ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spec k,

also the two morphisms f_{1}, f_{2} factor through Θ′(T_{j}). Since ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spec k is separated over k, also Θ′(T_{j}) is separated over k, and so f_{1} ≅ f_{2}. Thus Z is separated over k.

Now we prove that Θ_{Z} is an isomorphism. The morphism Θ_{Z} is locally of ind-finite presentation because its source Y := I_{\underline{G}_{0}}(Q)\(∏_{i} M̆Ẑ_{i} ; \underline{𝕃}_{i} × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H), as a locally noetherian, adic formal algebraic Deligne-Mumford stack locally formally of finite type, is locally of ind-finite presentation over Spf R̆\underline{Ẑ}. Since ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} is a monomorphism of ind-DM-stacks by Remark 7.12(a), and Θ is formally étale and satisfies the valuative criterion for properness [LM00, Théorème 7.3] by Lemma 7.14, it follows that also Θ_{Z} is formally étale and satisfies the valuative criterion for properness.

We fix a representation ρ: 𝔊 ↪ SL(V) as before Remark 3.5 and a tuple \underline{ω} = (ω_{i})_{i=1...n} of coweights of SL_{r}, and consider the closed substacks ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} of ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} from Remark 7.3, which are locally noetherian, adic formal algebraic Deligne-Mumford stacks over Spf R̆\underline{Ẑ}. Their formal completion along Z

X \underline{ω} := ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z ×_{∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C,𝔊)\underline{ν}} ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν}

are closed substacks of ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z and likewise locally noetherian, adic formal algebraic Deligne-Mumford stacks over Spf R̆\underline{Ẑ} by [Har05, Proposition A.14]. We may write

X := ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ; /Z = lim_{⟶} X \underline{ω} .

The base change Y\underline{ω} := Y ×_{X} X \underline{ω} is a closed formal algebraic substack of Y. Since X \underline{ω} and Y\underline{ω} are locally noetherian, adic formal algebraic stacks they have maximal ideals of definition I\underline{ω} and J\underline{ω} containing (ξ_{1}, …, ξ_{n}). For positive integers m consider the algebraic substacks X \underline{ω} ; m := V(Im ; \underline{ω}) and Y\underline{ω} ; m := V(J m ; \underline{ω}) ⊂ Y\underline{ω}. They are Deligne-Mumford stacks locally of finite type over Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{m}, because Y\underline{ω} ; m ⊂ Y is closed,

and X \underline{ω} ; m is a closed substack of the Deligne-Mumford stack ∇_{n}^{H,\underline{Ẑ},\underline{ω}}ℋ ^{1}(C, 𝔊)\underline{ν} ×R̆\underline{Ẑ} Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{m}

which is locally of finite type over Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{m}. By definition of I\underline{ω} and J\underline{ω} the stacks X \underline{ω} ; 1 and Y\underline{ω} ; 1 are reduced. Moreover, Θ_{Z} induces a morphism Θ_{Z}: Y\underline{ω} ; 1 → X \underline{ω} ; 1, because if Y \underline{ω} → → Y\underline{ω} ; 1 is a presentation, then Y \underline{ω} is a reduced scheme, and hence Θ_{Z} induces a morphism Y \underline{ω} → (Z ×_{X} X \underline{ω})_{red} = X \underline{ω} ; 1 which descends to a morphism Θ_{Z}: Y\underline{ω} ; 1 → X \underline{ω} ; 1. In particular, Θ_{Z}^{∗}(I\underline{ω}) ⊂ J\underline{ω}. This shows that Θ_{Z} induces a morphism Y\underline{ω} ; m → X \underline{ω} ; m for every m, which is locally of finite presentation as a morphism between Deligne-Mumford stacks locally of finite type over Spec R̆\underline{Ẑ}/(ξ_{1}, …, ξ_{n})^{m}.

Now part (b) will follow from Lemmas 7.15 and 7.16 below. More precisely, by Lemma 7.16 we have Θ_{Z}^{∗}(I\underline{ω}) = J\underline{ω}. Then Y\underline{ω} ; m → X \underline{ω} ; m is obtained from Y → X by base change, and hence is a formally étale morphism locally of finite presentation of algebraic stacks. Since Θ is a monomorphism by part (a) and X → ∇_{n}^{H,\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} is a monomorphism by Remark 7.12(a), also Y → X is a monomorphism. In particular, it is relatively representable by an étale monomorphism of schemes; see [LM00, Corollaire 8.1.3 and Théorème A.2]. In addition Y\underline{ω} ; m → X \underline{ω} ; m is surjective by Lemma 7.15, hence an isomorphism by [EGA, IV_{4}, Théorème 17.9.1]. As this holds for all m and all \underline{ω} we conclude that Θ_{Z} : Y → X is an isomorphism of stacks.

Lemma 7.15. The induced morphism Θ_{Z}: Y\underline{ω} ; m → X \underline{ω} ; m is quasi-compact and surjective.

Proof. The assertion only depends on the underlying topological spaces |X \underline{ω} ; m| = |X \underline{ω} ; 1| and |Y\underline{ω} ; m| = |Y\underline{ω} ; 1|; see [LM00, Chapitre 5]. So we may assume that m = 1. The morphism Θ_{Z}: Y\underline{ω} ; 1 → X \underline{ω} ; 1 = (Z ×_{X} X \underline{ω})_{red} is surjective by definition of Z as the image of Θ′.

We prove that the morphism is quasi-compact. For this purpose we may pass to finer level. Namely, we choose a proper closed subscheme D ⊂ C with H_{D} ⊂ H and use on source and target of Θ_{Z} the finite étale and surjective morphisms π(1)_{H_{D},H} from Remark 7.12(b). Then the morphism Θ_{Z}: Y\underline{ω} ; 1 → X \underline{ω} ; 1 for H_{D} is obtained by base change from the one for H. So we may assume H = H_{D} from now on. Let S be a quasi-compact scheme and f: S → X \underline{ω} ; 1 be a morphism given by a global 𝔊-shtuka (\underline{G}′, β′_{Q}H_{D}) with H_{D}-level structure over S, where β′ ∈ Isom^{⊗}(ω_{𝕆\underline{ν}}^{∘}, Ť_{\underline{G}}′); see the proof of Theorem 6.5. We must show that S ×_{X \underline{ω} ; 1} Y\underline{ω} ; 1 is quasi-compact. Consider the topological space |Z| underlying Z and the set \{T_{j}\}_{j∈J} of representatives of I_{\underline{G}_{0}}(Q)-orbits of the irreducible components of the scheme ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i})×Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H_{D} from Theorem 7.11(b). Every point z ∈ |Z| lies only on finitely many Θ′(T_{j}), see Remark 7.12(a). Let J(z) ⊂ J be the set with j ∈ J(z) if and only if z ∈ Θ′(T_{j}). The open substack

U_{z} := ⋃_{j∈J(z)} Θ′(T_{j}) ∖ ⋃_{j∉J(z)} Θ′(T_{j}) ⊂ Z

contains z, and hence all the U_{z} cover Z. Let s ∈ S be a point and set z = f(s) ∈ |Z|. The preimage f^{−1}(U_{f(s)}) of U_{f(s)} in S contains s. We choose an affine open neighborhood S_{s} of s in S which is contained in f^{−1}(U_{f(s)}). Then the S_{s} cover S, and hence already S = S_{s_{1}} ∪ … ∪ S_{s_{r}} for finitely many points s_{l} ∈ S, because S is quasi-compact. The scheme S′ defined as the finite disjoint union

S′ := ∐_{l=1}^{r} ∐_{j∈J(f(s_{l}))} S_{s_{l}} ×_{f,Z,Θ′} T_{j}

is quasi-compact, because Θ′: T_{j} → Z is proper by Remark 7.12(a) and so S_{s_{l}} ×_{Z} T_{j} is proper over the affine scheme S_{s_{l}}. The projection S′ → S is surjective, because every point s ∈ S lies in one S_{s_{l}}, and then f(s) ∈ U_{f(s_{l})} ⊂ ⋃_{j∈J(f(s_{l}))} Θ′(T_{j}) has a preimage in one of the T_{j}. On every component S′_{l,j} := S_{s_{l}} ×_{Z} T_{j} of

S′ the projection onto T_{j} defines a tuple (\underline{L}_{i}, δ_{i}) ∈ M̆Ẑ_{i} ; \underline{𝕃}_{i}(S_{l,j}) for i = 1, …, n. Let \underline{G} := δ_{n}^{∗} ∘ … ∘ δ_{1}^{∗} \underline{G}_{0,S_{l,j}}

and let δ: \underline{G} → \underline{G}_{0,S_{l,j}} be the induced quasi-isogeny. By definition of Θ′ via the morphism Ψ_{\underline{G}_{0}}, there is an

isomorphism α: \underline{G}′ ^{∼} ⟶ \underline{G} of global 𝔊-shtukas over S_{l,j}. Then γ_{l,j}H_{D} := Ť_{δα} ∘ β′_{Q}H_{D} ∈ Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H_{D}. The projection S_{l,j} → T_{j} together with the element γ_{l,j}H_{D} defines the upper horizontal morphism in the commutative diagram

[figure: S′; ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H_{D}; Θ′; S; f; X \underline{ω} ; 1; ∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C, 𝔊)\underline{ν} ˆ×_{R_{\underline{Ẑ}}} Spf R̆_{\underline{Ẑ}} .]

Then we can consider the surjective morphisms

[figure: I_{\underline{G}_{0}}(Q) × S′; ∼; S′ ×_{∇_{n}^{H_{D},\underline{Ẑ}}ℋ ^{1}(C,𝔊)\underline{ν} ˆ×_{R\underline{Ẑ}} Spf R̆_{\underline{Ẑ}}} ∏_{i} X̆_{Z_{i}}(\underline{𝕃}_{i}) × Isom^{⊗}(ω^{∘}, V̌_{\underline{G}_{0}})/H_{D}; S ×_{X \underline{ω} ; 1} Y\underline{ω} ; 1; S′ ×_{X \underline{ω} ; 1} Y\underline{ω} ; 1]

REFERENCES 43

in which the isomorphism in the upper row comes from Lemma 7.13. By the I_{\underline{G}_{0}}(Q)-equivariance of Θ′ we obtain a surjective map S′ → → S ×_{X \underline{ω} ; 1} Y\underline{ω} ; 1, and hence S ×_{X \underline{ω} ; 1} Y\underline{ω} ; 1 is quasi-compact by [StPr, Tag 04YC] as desired. Lemma 7.16. For every \underline{ω} we have Θ_{Z}^{∗}(I\underline{ω}) = J\underline{ω}. In particular the morphism Θ_{Z}: Y\underline{ω} → X \underline{ω} of formal algebraic stacks is adic. Proof. Let X\underline{ω} → → X \underline{ω} ; 1 = V(I\underline{ω}) be a presentation. By Lemma 7.13 and our arguments for the proof of Theorem 7.11(b) given above, we see that Θ_{Z,m}: Y\underline{ω} ; m ×_{X \underline{ω}} X\underline{ω} = Y\underline{ω} ; m ×_{X \underline{ω} ; m} X\underline{ω} → X\underline{ω} is a monomorphism locally of finite presentation satisfying the valuative criterion for properness. Since it is quasi-compact by Lemma 7.15 it is a proper monomorphism, hence a closed immersion of schemes by [LM00, Corollaire A.2.2]. Since Θ_{Z,m} is surjective by Lemma 7.15 and X\underline{ω} is reduced, it must be an isomorphism for all m. Therefore

Y\underline{ω} ; m ×_{X \underline{ω}} X\underline{ω} = X\underline{ω} = Y\underline{ω} ; 1 ×_{X \underline{ω}} X\underline{ω} and Y\underline{ω} ×_{X \underline{ω}} X\underline{ω} = lim_{⟶} Y\underline{ω} ; m ×_{X \underline{ω}} X\underline{ω} = Y\underline{ω} ; 1 ×_{X \underline{ω}} X\underline{ω}. This shows that

V(Θ_{Z}^{∗}I\underline{ω}) = Y\underline{ω} ×_{X \underline{ω}} X \underline{ω} ; 1 = Y\underline{ω} ; 1 ×_{X \underline{ω}} X \underline{ω} ; 1 ⊂ Y\underline{ω} ; 1 = V(J\underline{ω}) as closed substacks of Y\underline{ω}. Therefore J\underline{ω} ⊂ Θ_{Z}^{∗}(I\underline{ω}). With the opposite inclusion established above J\underline{ω} = Θ_{Z}^{∗}(I\underline{ω}), and hence Θ_{Z} is adic.

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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/