Functoriality of Moduli Spaces of Global $\mathbb G$-Shtukas
Paul Breutmann


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		Helical Twisting Number and Braiding Linkage Number
		of Solar Coronal Loops
            Markus J. Aschwanden^1
    December 30, 2023
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	Moduli spaces of global 𝔾-shtukas play a crucial role in the Langlands program for function fields. We analyze their functoriality properties following a change of the curve and a change of the group scheme 𝔾 under various aspects. In particular, we prove two finiteness results which are of interest in the study of stratifications of these moduli spaces and which potentially allow the formulation of an analog of the André-Oort conjecture for global 𝔾-shtukas.






§ INTRODUCTION


Global 𝔾-shtukas are the function field analogue of abelian varieties. Their moduli spaces play a crucial role in the Langlands program for function fields. This article is concerned about functoriality properties of these moduli spaces in various aspects. Let us look in more detail. 
We choose a smooth, projective, geometrically irreducible curve C over a finite field _q with q elements. Let 𝔾 be a smooth, affine group scheme over C and denote by σ the _q-Frobenius of a scheme S over _q. Then a global 𝔾-shtuka =(, s_1, …, s_n, τ_) over S consists of a 𝔾-torsor  over C_S:=C×__qS, n sections s_i:S→ C called legs and an isomorphism τ_:σ^⋆|_C_S\∪_iΓ_s_i→|_C_S\∪_iΓ_s_i outside the union of the graphs Γ_s_i of the s_i. 
The precise definitions of all the notations used in this article are given in the second section. The stack whose S-valued points parametrize the global 𝔾-shtukas over S with n legs is denoted by ∇_nℋ^1(C,𝔾). Once we fix n closed points (_1,…, _n)= in C to which we refer as characteristic places we can introduce boundedness conditions Z_ for all ∈ and H-level structures. Here a bound Z_ is roughly a L^+G_-invariant closed subscheme of the affine flag variety _𝔾_ (see <ref> for a precise definition) and H is an open, compact subgroup of 𝔾(^), where ^ denotes the ring of the adeles outside . Then we denote by ∇_n^Ẑ_, Hℋ^1(C,𝔾) the stack which parametrize 𝔾-shtukas  over S bounded by (Ẑ_)_∈ together with an H-level structure. At the beginning of the third section we define all the parameters (C,𝔾,,Z_,H) as a shtuka datum. This definition results in the natural question of if an appropriate change of this shtuka datum induces a morphism of the corresponding moduli spaces and if so, which properties it has.

In subsection 3.1 we define a morphism of shtuka data and clarify what an appropriate change of the shtuka datum should be. Roughly a morphism from (C,𝔾,,Ẑ_, H) to (C',𝔾',w, Ẑ'_w, H') is a pair (π, f), where π:C→ C' is a finite morphism and f is a morphism of group schemes from the Weil restriction π_⋆𝔾 to 𝔾' such that w, Ẑ'_w and H' satisfy certain conditions.

In the following subsections we then answer the questions about the functoriality of ∇_n^Ẑ_, Hℋ^1(C,𝔾). More precisely, we first consider in subsection <ref> the case that we basically only change the curve C, which yields the following main result of that subsection.

cf. Theorem <ref>
Let (C,𝔾, , Ẑ_, H) be a shtuka datum and π:C→ C' a finite morphism of smooth, projective, geometrically irreducible curves over _q with _i:=π(_i) and :=(_1,…,_n). Then the morphism (π,id_π_⋆𝔾):(C,𝔾, , Ẑ_, H)→ (C',π_⋆𝔾, , π_⋆Ẑ_, π_⋆ H) of shtuka data (see definition <ref> and remark <ref>) induces a finite morphism of the moduli stacks

    π_⋆: ∇_n^Ẑ_,Hℋ^1(C,𝔾)→∇_n^π_⋆Ẑ_,π_⋆ Hℋ^1(C',π_⋆𝔾)
    .



The construction of this morphism and the proof of the theorem relies on a lemma in subsection <ref> that states an equivalence of categories between 𝔾-torsors over C and π_⋆𝔾 over C'.
 

The next subsection <ref> addresses the questions about functoriality in the case that we only change the group scheme 𝔾. While we construct  a morphism 

    ∇_n^Ẑ_,Hℋ^1(C,𝔾)→∇_n^Ẑ'_,H'ℋ^1(C,𝔾')

for all morphisms (id_C,f) of shtuka data, we need to make different assumptions to state different results on the properties of this morphism. Assuming that f:𝔾→𝔾' is generically an isomorphism, we get the following first main result of this subsection.

cf. Theorem <ref>
Let w=(w_1,…,w_m) be a finite set of closed points in C and let (id_C,f):(C,𝔾,,Ẑ_,H)→ (C,𝔾',,Ẑ'_,H) be a morphism of shtuka data, where f:𝔾→𝔾' is an isomorphism over C\w. Then the morphism 

    f_⋆: ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ'_, Hℋ^1(C,𝔾'),    (,γ H) ↦ (f_⋆,γ H)

is schematic and quasi-projective. In the case that 𝔾 is a parahoric Bruhat-Tits group scheme this morphism is projective.
For any morphism (', γ' H):S→∇_n^Ẑ'_, Hℋ^1(C,𝔾') 

    S×_∇_n^Ẑ'_, Hℋ^1(C,𝔾')∇_n^Ẑ'_, Hℋ^1(C,𝔾)


    S×__q((L^+_w_1(')/ L^+𝔾_w_1)×__q…×__q(L^+_w_m(')/ L^+𝔾_w_m)).

If Ẑ_ arises as a base change of Ẑ'_ for all ∈, the morphism f_⋆ is surjective.



This result will again be important for the verification of the axioms on the moduli space ∇_n^Ẑ_, Hℋ^1(C,𝔾) in <cit.>.
If we assume that f:𝔾→𝔾' is a closed immersion instead of a generic isomorphism we get the second main result of this subsection.

cf. Theorem <ref>
Let f:𝔾→𝔾' be a closed immersion of smooth, affine group schemes over C. Then the induced morphism f_⋆: ∇_nℋ^1(C, 𝔾)→∇_nℋ^1(C,𝔾') is unramified and schematic.


Assuming additionally that 𝔾 is a parahoric Bruhat-Tits group scheme, we prove as well:
cf. Theorem <ref>
Let 𝔾 be a parahoric Bruhat-Tits group scheme and f:𝔾→𝔾' be a closed immersion of smooth, affine group schemes and =(_1,…,_n) be a set of closed points in C. Then the induced morphism

    f_⋆: ∇_nℋ^1(C, 𝔾)^→∇_nℋ^1(C,𝔾')^



Apart from the interest of these morphisms in the study of ∇_n^Ẑ_, Hℋ^1(C,𝔾) in general, there are two other motivations. The first one is that the finiteness results in theorem <ref> and theorem <ref> potentially enable us to formulate 

an analog of the André-Oort conjecture for moduli spaces of global 𝔾-shtukas, as explained in more detail in remark <ref>. The second motivation arises from the study of Newton and Kottwitz-Rapoport stratifications of ∇_n^Ẑ_, Hℋ^1(C,𝔾) in <cit.>, where the above results are again needed.

We remark that the way to choose a bound Ẑ_v as we do in this article is quite general. In <cit.> we will restrict our interest to moduli spaces of global 𝔾-shtukas that are bounded by a cocharacter of a maximal torus of 𝔾, which can also be formulated using Grassmanians and seems more natural when working with local models.


Acknowledgements. I would like to thank my advisor Urs Hartl, for all his helpful discussions. I am grateful to Timo Richarz and Johannes Anschütz for their comments and remarks. During the work of this project, the author was supported by the SFB 878 "Groups, Geometry & Actions" of the German Science Foundation (DFG), the CNRS and the ERC Advanced Grant 742608 "GeoLocLang". 













































§ PRELIMINARIES

Before we start with the functoriality of ∇_nℋ^1(C,𝔾), we introduce the basic objects and notations that we use in this article. 

Most of the notations introduced in this section, can also be found in <cit.> and <cit.>. 
Let q be a power of some prime number p. 
Preliminaries
We start with a smooth, projective, geometrically irreducible curve C
CCsmooth projective irreducible curve over _q
 over the field _q with q elements.
We denote by Q:=_q(C) 
QQfunction field of C
its function field. For a closed point v∈ C we denote by A_v the completion of the stalk _C,v and by Q_v the fraction field of A_v. Furthermore we choose a uniformizer z_v in A_v, denote the residue field of v by _v and set deg v=[_v:_q].

Let G𝔾smooth, affine group scheme over C 
𝔾 be a smooth, affine group scheme over C and 𝐆:=𝔾×_C Q its generic fiber. 
G𝐆generic fiber of 𝔾
We write 𝔾_:=𝔾×_CSpec A_ and 𝐆_:=𝔾×_CSpec Q_=𝔾_×_A_Spec Q_ for the appropriate base changes.

For an _q-scheme S we denote by σ_S:S→ S the absolute _q-Frobenius, which acts as the q-power map on the structure sheaf. Further we define σ as the endomorphism id_C×σ_S of C_S:=C×__qS. For a morphism s:S→ C we denote as usual by Γ_s:S→ C_S the graph of s, which is a closed immersion.

Let 𝒳 be a site with a final object x and G a sheaf of groups on 𝒳. Then a (right) G-torsor is a sheaf 𝒢 on 𝒳 with a right action of G on 𝒢 such that G×𝒢≃𝒢×𝒢,  (g,h)↦ (hg,h) is an isomorphism and 𝒢(U)≠∅ for some covering U→ X. When we speak about a torsor, we always mean a right torsor and if nothing else is mentioned we mean a sheaf on the big étale site of a scheme. For any scheme S we write S_Ét for the big étale site of this scheme. We denote by ℋ^1(C,𝔾) the stack fibered over (_q)_Ét, whose fiber category ℋ^1(C,𝔾)(S) is given by the category of 𝔾-torsors over C_S.




  
Global 𝔾-Shtukas:

Let S be an _q-scheme. A global 𝔾-shtuka over S is a tuple
𝒢=(𝒢, s_1,…, s_n, τ_𝒢), where

 
  * 𝒢 is a 𝔾-torsor over C_S, 
 
  * s_1,…, s_n are morphisms S→ C and
 
  * τ_𝒢:σ^⋆𝒢|_C_S\(i⋃Γ_s_i)→𝒢|_C_S\(i⋃Γ_s_i) is an isomorphism of the 𝔾-torsors σ^⋆𝒢 and 𝒢 restricted to C_S\(Γ_s_1⋃…⋃Γ_s_n).


We take the notation  from <cit.> for the stack fibered over (_q)_Ét whose S-valued points for a scheme S are given by 𝔾-shtukas 𝒢 over S. Morphisms from (𝒢, s_1,…, s_n,τ) to (𝒢', s'_1,…, s'_n,τ') in the fiber category (S) only exist if s_i=s'_i and are given by morphisms f:𝒢→𝒢' of 𝔾-torsors over C_S such that f∘τ=τ'∘σ^⋆ f.
Given two 𝔾-shtukas 𝒢 and 𝒢' over S with s_i=s'_i, we also define a quasi-isogeny from 𝒢 to 𝒢' to be an isomorphism f:𝒢|_C_S\ D_S→𝒢'|_C_S\ D_S of 𝔾-torsors satisfying f∘τ=τ'∘σ^⋆ f, where D is some effective divisor on C.
The moduli space  is an ind-algebraic stack that is ind-separated and locally of ind-finite type <cit.>.

 


  
Loop Groups: 

Let  be a finite field and ℍ be a smooth, affine group scheme over 𝔻:=Spec z, with generic fiber 𝐇:=ℍ×_𝔻𝔻̇ where 𝔻̇:=Spec z. We are mainly interested in the case that 𝔻≃ Spec A_ and ℍ=𝔾_ for some closed point ∈ C.

We recall that the sheaf of groups L^+ℍ on _Ét, whose R-valued points for an -algebra R are given by 

    L^+ℍ(R):=ℍ(Rz):=ℍ(𝔻_R):=Hom_𝔻(𝔻_R,ℍ)   𝔻_R:=Spec Rz,

is an infinite-dimensional affine group scheme over . It is called the group of positive loops associated with ℍ.
The group of loops associated with 𝐇 is the sheaf of groups L𝐇 on _Ét, whose R-valued points are defined by 

    L𝐇(R):=𝐇(Rz):=𝐇(𝔻̇_R):=Hom_𝔻̇(𝔻̇_R,𝐇) ,

where we write Rz=Rz[1/z] and 𝔻̇_R=Spec Rz. The loop group L𝐇 is an ind-scheme of ind-finite type over .




  
Torsors for Loop Groups: 

We write ℋ^1(, L^+ℍ) for the stack fibered over ()_Ét whose fiber category ℋ^1(, L^+ℍ)(S) is the category of L^+ℍ-torsors over S. In the same way ℋ^1(, L𝐇) denotes the stack fibered over ()_Ét whose fiber category ℋ^1(, L𝐇)(S) is the category of L𝐇-torsors over S. There is a natural 1-morphism 

    L:ℋ^1(, L^+ℍ)→ℋ^1(, L𝐇),  ℒ^+↦ℒ

induced by the inclusion of sheaves L^+ℍ⊂ L𝐇.

We now consider also the z-adic completions of 𝔻 and ℍ and denote them by 𝔻̂:=Spf z and ℍ̂:=ℍ×_𝔻𝔻̂. Later when we pass from global 𝔾-shtukas to local 𝔾_-shtukas we often need to know that L^+ℍ-torsors are equivalent to formal ℍ̂-torsors. So we recall that for an -scheme S a z-adic formal scheme ℋ over 𝔻̂_S:=𝔻̂×̂_ S together with an action ℍ̂×̂_𝔻̂ℋ→ℋ of ℍ̂ is called a formal ℍ̂-torsor if there is an étale covering S'→ S and an ℍ̂-equivariant isomorphism
ℋ×̂_𝔻̂_S𝔻̂_S'⟶ℍ̂×_𝔻̂𝔻̂_S', where ℍ̂ is acting on itself by right multiplication.

We denote by ℋ^1(𝔻̂, ℍ̂) the category fibered in groupoids over ()_Ét whose fiber category ℋ^1(𝔻̂, ℍ̂)(S) is the groupoid of formal ℍ̂-torsors over S. We remark that Arasteh Rad and Hartl proved in <cit.> that there is a natural isomorphism of stacks ℋ^1(𝔻̂, ℍ̂)⟶ℋ^1(,L^+ℍ).
It sends a formal ℍ̂-torsor ℋ to the sheaf

    ℒ^+:S_Ét→𝐒𝐞𝐭𝐬,   T↦Hom_𝔻̂_S(𝔻̂_T,ℋ)

which becomes a L^+ℍ-torsor under the action of L^+ℍ(T)=Hom_𝔻̂(𝔻̂_T,ℍ̂).



  
Local ℍ-Shtukas: 

Let S be a -scheme and σ̂ its absolut -Frobenius. If  equals _q or _ we will write σ and σ_ respectively, instead of σ̂. 
oσ, σ_, σ̂absolut _q (resp. _, ) Frobenius of a scheme S over these fields
With the previous notations a local ℍ-shtuka over S is a pair (ℒ^+,τ_ℒ) where


  * ℒ^+ is a L^+ℍ-torsor over S and

  * τ_ℒ:σ̂^⋆ℒ⟶ℒ is an isomorphism of the associated loop group torsors from (<ref>) in <ref>.

A morphism from ℒ=(ℒ^+,τ_ℒ) to ℒ'=(ℒ^+',τ'_ℒ') of two local ℍ-shtukas over S is a morphism f:ℒ^+→ℒ^+' of L^+ℍ-torsors over S satisfying τ_ℒ∘ f=τ_ℒ'∘σ̂^⋆ f.
A quasi-isogeny from ℒ=(ℒ^+,τ_ℒ) to ℒ'=(ℒ^+',τ'_ℒ') is an isomorphism f:ℒ→ℒ' of the associated L𝐇-torsors satisfying τ_ℒ∘ f=τ_ℒ'∘σ̂^⋆ f.

A local ℍ-shtuka (ℒ^+,τ_ℒ) is called étale if τ_ℒ:σ̂^⋆ℒ→ℒ comes already from an isomorphism  τ_ℒ:σ̂^⋆ℒ^+→ℒ^+ of the L^+ℍ-torsors.
We denote the category of local ℍ-shtukas over S by Sht_ℍ(S) and the category of étale local ℍ-shtukas over S by ÉtSht_ℍ(S).

 We recall the Corollary <cit.> that states that if ℍ has a connected special fiber, then any étale local shtuka over an separably closed field k is already isomorphic to (L^+ℍ_k, 1·σ̂^⋆).

Let ζ be the power series ring over  in a variable ζ. We denote by 𝒩ilp_ζ the category of schemes over Spec ζ on which ζ is locally nilpotent in the structure sheaf. Therefore 𝒩ilp_ζ is the full subcategory of formal schemes over Spf ζ consisting of ordinary schemes. We will define boundedness conditions only for local shtukas over a scheme S in 𝒩ilp_ζ.




  
The Affine Flag Variety:

Let ℍ be a smooth, affine group scheme over Spec z as before. Then the affine flag variety ℱ𝑙_ℍ is defined as the quotient sheaf L𝐇/L^+ℍ on _Ét, that is the sheaf associated to the pre-sheaf 

    T↦ L𝐇(T)/L^+ℍ(T).

By <cit.> ℱ𝑙_ℍ is represented by an ind-scheme which is ind-quasi-projective and in particular ind-separated and of ind-finite type over . By <cit.> _ℍ is ind-projective if and only if ℍ is a Bruhat-Tits group scheme over z in the sense of <cit.>. We also remark that L^+ℍ acts from the left on ℱ𝑙_ℍ.



  
Bounds in ℱ̂𝑙̂_ℍ

We fix an algebraic closure ζ^alg of ζ. For a finite extension R of discrete valuation rings ζ⊂ R⊂ζ^alg with residue field κ_R we denote similar as before with 𝒩ilp_R the category of R-schemes on which ζ is locally nilpotent. Furthermore we set ℱ̂𝑙̂_ℍ,R:=_ℍ×̂_ Spf R as well as _ℍ:=_ℍ,ζ.

Now let R and R' be two such finite extensions of discrete valuation rings and let Ẑ_R⊂_ℍ,R and Ẑ'̂_R'⊂_ℍ,R'
be two closed ind-subschemes. We call Ẑ_R and Ẑ'̂_R' equivalent if there is a finite extension R of discrete valuation rings as above containing R and R' such that Ẑ_R×_Spf RSpf R=Ẑ'̂_R'×_Spf R'Spf R as closed ind-subschemes of _ℍ,R.

Now a bound is defined (compare <cit.> and <cit.>) as an equivalence class Ẑ=[Ẑ_R] of closed ind-subschemes Ẑ_R⊂_ℍ,R satisfying



  * firstly, that all subschemes Ẑ_R are stable under the left action of L^+ℍ on _ℍ,R, and


  * secondly, that all the special fibers Z_R:=Ẑ_R×_Spf RSpec κ_R are quasi-compact and connected subschemes of _ℍ×̂_κ_R.

We remark that in <cit.> and <cit.> the definition of a bound does not require the special fibers to be connected. We make this assumption because it does not change the theory and simplifies the formulation of certain statements. In fact, if Ẑ is the disjoint union of two bounds Ẑ_1∐Ẑ_2 then the moduli space ∇_n^Ẑℋ^1(C,𝔾) that will be defined in paragraph <ref> is the disjoint union of the moduli spaces ∇_n^Ẑ_1ℋ^1(C,𝔾) and ∇_n^Ẑ_2ℋ^1(C,𝔾). 



  
The Reflex Ring: 

For an equivalence class Ẑ=[Ẑ_R] as above we set G_Ẑ:={γ∈ Aut_ζ(ζ^alg) | γ(Ẑ)=Ẑ}. 
The ring R_Ẑ is defined as the intersection of the fixed field of G_Ẑ in ζ^alg with all the finite extensions R over which a representative Ẑ_R of Ẑ exists. In the case that Ẑ is a bound, we call R_Ẑ the reflex ring of Ẑ.

It is not always clear if there exists a representative of Ẑ over R_Ẑ. We write κ_Ẑ and κ_R for the residue fields of R_Ẑ and R respectively. Then the special fiber Z_R:=Ẑ_R×_Spf Rκ_R arises from a unique closed subscheme Z⊂_ℍ×_κ_Ẑ. This follows from Galois descent for closed ind-subschemes of _ℍ, which is effective. The subscheme Z is called the special fiber of Ẑ.




  
Boundedness of Local ℍ-Shtukas: 

Let Ẑ be a bound with reflex ring R_Ẑ. Furthermore let ℒ^+ and ℒ^+' be two L^+ℍ-torsors over a scheme S in 𝒩ilp_R_Ẑ and δ:ℒ→ℒ' an isomorphism of the associated L𝐇-torsors. We choose a covering S'→ S such that there are trivializations α:ℒ^+ L^+ℍ_S' and α':ℒ^+' L^+ℍ_S'. Then the automorphism α'∘δ∘α^-1:L𝐇_S' L𝐇_S' defines a morphism S'→ L𝐇. 

For any finite extension R of R_Ẑ we have an induced morphism 

    S'×_Spf R_ẐSpf R⟶
    L𝐇×̂_Spf R⟶_ℍ,R.

Then δ is said to be bounded by Ẑ if for all trivializations α and α' and all finite extensions R of R_Ẑ with a representative Ẑ_R, this morphism (<ref>) factors through Ẑ_R.
By <cit.> δ is bounded if and only if this condition is satisfied for one trivialization and for one such extension R.
By definition a local ℍ-shtuka (ℒ^+,τ_ℒ) is bounded by Ẑ if τ_ℒ is bounded by Ẑ.



  
A Version of the Theorem of Beauville-Laszlo: 

Let ∈ C be a closed point and set C^:=C\{} 
 as well as C^_S:=C^×__qS. We define ℋ_e^1 (C^, 𝔾) as the category fibered in groupoids over (_q)_Ét whose fiber category ℋ_e^1(C^, 𝔾)(S) consists of those 𝔾-torsors over C^_S that can be extended to a 𝔾-torsor over C_S. By restricting a 𝔾-torsor  over C_S to C_S^ we get a morphism ℋ^1(C,𝔾)→ℋ_e^1(C^,𝔾).
 We further introduce the notation 𝔾_=Res_A_/ _qz_𝔾_ and 𝐆_:=𝔾_×__qz__qz_. For ∈ℋ^1(C,𝔾) the base change _:=×_C_S(Spf A_×__qS) defines a formal 𝔾_-torsor over Spf A_×̂__qS. Its Weil restriction Res_A_/ _qz__ defines a formal 𝔾_-torsor over Spf _qz_×̂__qS. Using the category equivalence in <ref> it corresponds to an object in ℋ^1(_q,L^+𝔾_)(S) that we denote by L^+_(𝒢) which defines a functor 
 
    L^+_:ℋ^1(C,𝔾)⟶ℋ^1(_q, L^+𝔾_),  𝒢↦ L^+_(𝒢).

 
 Furthermore we have the functor
 
    L_: ℋ^1_e(C^,𝔾)→ℋ^1(_q,L𝐆_),  |_C^_S↦ L(L^+_())=L_()

which is independent of the extension  of |_C^_S. Now a version of the theorem of Beauville-Laszlo, that is proven in <cit.>, states that the following diagram is cartesian.

    ℋ^1(C,𝔾)[r] _L^+_[d]   ℋ^1_e(C^,𝔾)[d]_L_
    ℋ^1(_q, L^+𝔾_) [r]    ℋ^1(_q, L𝐆_)

 



  
The Global-Local Functor: 

Now we fix n closed points ={_1,…,_n} of C. Then define 
A_ as the completion of the local ring 𝒪_C^n, at the closed point . We set


    ∇_nℋ^1(C,𝔾)^ := ∇_nℋ^1(C,𝔾)×_C^nSpf A_.


An S-valued point of ∇_nℋ^1(C,𝔾)^ is therefore given by a global 𝔾-shtuka =(,s_1,…, s_n,τ_) such that s_i:S→ C factors through Spf A__i. We now want to associate with such a global 𝔾-shtuka a local 𝔾__i-shtuka for all _i∈. 
We write 𝔻__i:=Spec A__i and 𝔻̂__i:=Spf A__i as well as 𝔻̂__i,S:=𝔻__i×̂___iS. Then we have:

    𝔻̂__i×̂__qS=∐_l∈/deg _i V(𝔞__i,l)=∐_l∈/deg _i𝔻̂__i,S

 where 𝔞__i,l:=⟨ a⊗ 1-1⊗ s^⋆(a)^q^l | a∈__i⟩ and V(𝔞__i,l) is the closed subscheme given by this ideal. We remark that σ cyclically permutes these components and that the __i-Frobenius σ^deg _i leaves all these components V(𝔞__i,l) stable.
 For ∈∇_nℋ^1(C,𝔾)^ the base change 
 
    __i:=×̂_C_S(Spf A__i×̂__q S)=∐_l∈/ deg _i×̂_C_SV(𝔞_v_i,l)

defines a formal 𝔾̂__i-torsor over ∐_l∈/deg _i𝔻̂__i,S which is an object in ℋ^1(𝔻̂__i,𝔾̂__i)(∐_l∈/deg _i S).
Each component ×_C_SV(𝔞__i,l) defines a formal 𝔾̂__i-torsor. Similar to the notation in  <ref> we denote by ℒ_i,0^+ the L^+𝔾__i-torsor associated by <cit.> with the formal 𝔾̂__i-torsor ×_C_SV(𝔞__i,0). Then (ℒ^+_i,0,τ^deg _i) is a local 𝔾__i-shtuka, where 
τ^deg _i:(σ^deg _i)^⋆ℒ_i,0ℒ_i,0 is the isomorphism of L𝐆__i-torsors induced by τ_𝒢
 (compare also <cit.>). More precisely, it can be written as τ^deg _i=τ∘σ^⋆τ∘…∘(σ^deg _i-2)^⋆τ∘(σ^deg _i-1)^⋆τ.

 This now defines the global-local functor:
 
    Γ__i:∇_nℋ^1(C,𝔾)^(S)   ⟶ Sht_𝔾__i
     (S)
    =(, s_1,…,s_n,τ)    ⟼ (ℒ^+_i,0 , τ^deg _i)=Γ__i().

We remark that this functor transforms by <cit.> quasi-isogenies into quasi-isogenies.
If v∉v the component V(𝔞_v,0) exists only if S is an 𝔽_v-scheme.

If we do not restrict 𝒢×_C_S(Spf A__i×__qS) to the component V(𝔞_v_i,0) but consider its Weil restriction L^+__i() we get in a similar way a local 𝔾__i-shtuka (L__i^+(𝒢),τ__i) where τ__i:=L__i(τ_):σ^⋆ L__i()→ L__i().
We denote this local 𝔾__i-shtuka by L^+__i()=(L__i^+(𝒢),τ__i). We remark that L^+_() does not only exist for ∈ but also for other places ∈ C. In the case that ∉,   the local shtuka L^+_() is étale.



  
Boundedness of Global 𝔾-Shtukas: 

Recall that we fixed n closed points =(_1,…,_n) in C. If the group scheme 𝔾 is fixed we write for each of these points __i:=_𝔾__i for the corresponding affine flag variety over __i and __i,R=_𝔾__i,R=__i×̂___iSpf R for a finite extension A__i⊂ R.
In each of these affine flag varietes __i=__i,A__i we choose a bound Ẑ__i=[Ẑ__i,R] with reflex ring R_Ẑ__i and we write Ẑ_ for the tuple (Ẑ__1,…, Ẑ__n). Choosing a uniformizer π__i in R_Ẑ__i and defining _R as the compositum of all the residue fields R_Ẑ__i/(π__i), we set R_Ẑ_:=_Rπ__1…, π__n.

In particular the morphism Spf R_Ẑ_→ C^n factors through Spf A_. This means that every point =(, s_1,…,s_n, τ_) in ∇_nℋ^1(C,𝔾)×_C^nSpf R_Ẑ_(S) also defines an S-valued point in ∇ _nℋ^1(C,𝔾)^ so that we write Γ__i() for its associated local 𝔾__i-shtuka over S. The fact that S∈ Nilp_R_Ẑ__i allows us to ask if Γ__i() is bounded by Ẑ__i.

We define ∇_n^Ẑ_ℋ^1(C,𝔾) to be the stack consisting of these bounded global 𝔾-shtukas. That means the fiber category ∇_n^Ẑ_ℋ^1(C,𝔾)(S) is the full subcategory of ∇_nℋ^1(C,𝔾)×_C^nSpf R_Ẑ_(S) that consists of those global 𝔾-shtukas  over S that are bounded by Ẑ_. By <cit.> the moduli space ∇_n^Ẑ_ℋ^1(C,𝔾) is a closed ind-substack of ∇_nℋ^1(C,𝔾)×_C^nSpf R_Ẑ_. Moreover we 

denote by ∇_n^Ẑ_ℋ^1(C,𝔾)__R:=∇_n^Ẑ_ℋ^1(C,𝔾)×_Spf R_Ẑ__R  the special fiber of ∇_n^Ẑ_ℋ^1(C,𝔾).




  
D-Level Structures: 

Let D be a proper closed subscheme of C and let D_S:=D×__qS for some _q-scheme S and  a 𝔾-torsor on C_S. By <cit.> a D-level structure on  is a trivialization Ψ:×_C_SD_S→𝔾×_C D_S and ℋ^1_D(C,𝔾) denotes the stack fibered over (_q)_Ét whose fiber category ℋ^1_D(C,𝔾)(S) consists of pairs (, Ψ) where ∈ℋ^1(C,𝔾)(S) and Ψ is a D-level structure. A morphism from (,Ψ) to (',Ψ') in this fiber category is given by an isomorphism f:→' of 𝔾-torsors such that Ψ=Ψ'∘(f× id_D_S). The moduli stack of global 𝔾-shtukas with D-level structure is denoted by ∇_nℋ^1_D(C,𝔾). Its fiber category over S is given by tuples (, Ψ)=(,s_1,…, s_n,τ_,Ψ) where ∈∇_nℋ^1(C,𝔾)×_C^n(C\ D)^n(S) (i.e. s_i:S→ C factors through C\ D) and Ψ is a S-level structure on  satisfying Ψ∘ (τ× id_D_S)=σ^⋆(Ψ). A morphism from (,Ψ) to (', Ψ') in this fiber category is a morphism f∈∇_nℋ^1(C,𝔾)(S) (in particular an isomorphism f:→' of 𝔾-torsors) satisfying Ψ=Ψ'∘(f× id_D_S).

If D=∅ we have ∇_nℋ_D^1(C,𝔾)=∇_nℋ^1(C,𝔾). If _1,…,_n∉ D and Ẑ_ is a bound as before, we use the intuitive notations ∇_nℋ_D^1(C,𝔾)^ for the base change ∇_nℋ_D^1(C,𝔾)×_C^nSpf A_ and ∇_n^Ẑ_ℋ_D^1(C,𝔾) for the stack of 𝔾-shtukas  in ∇_n^Ẑ_ℋ^1(C,𝔾) with a D-level structure.




  
Local Shtukas and Local GL_r-Shtukas: 

The category of local GL_r-shtukas over an F_q-scheme S   

can be defined more explicitly. We briefly describe this here since it is useful for the definition of the Tate functors.

We denote by _Sz the sheaf of _S-algebras on S_Ét which associates with every S-scheme Y the ring _Sz(Y):=Γ(Y,_Y)z. Now every sheaf of _Sz-modules that is fqqc-locally free of rank r is by <cit.> already Zariski locally free of rank r. We call these locally free sheafes of _Sz-modules of rank r. For a commutative ring R we set Rz:=Rz[1/z]. This leads to the intuitive notation _Sz for the sheaf on S_Ét associated to the pre-sheaf Y↦Γ(Y,_Y)z. The absolut  Frobenius was denoted σ̂ and we use the same notation for the endomorphism of _Sz and _Sz that acts as σ̂ on sections of _S and as the identity on z. For a sheaf M of _Sz-modules we can consider the pullback σ̂^⋆ M:=M⊗__Sz,σ̂_Sz. Now by <cit.> a local shtuka of rank r over S is a pair (M, τ_M) consisting of a locally free sheaf M of _Sz-modules of rank r and an isomorphism 

    τ_M:σ̂^⋆ M ⊗__Sz_Sz→ M⊗__Sz_Sz.

The local shtuka (M,τ_M) is called étale if τ_M arises from an isomorphism σ̂^⋆ M→ M of _Sz-modules. A morphism from (M,τ_M) to (M',τ_M') between two local shtukas over S is a morphism f:M→ M' of _Sz-modules satisfying τ_M'∘σ̂^⋆ f=f∘τ_M. 
A quasi-isogeny from (M,τ_M) to (M',τ_M') is a morphism f:M⊗__Sz_Sz→ M'⊗__Sz_Sz of _Sz-modules satisfying τ_M'∘σ̂^⋆ f=f∘τ_M.
We denote the category of local shtukas over S by Sht__q(S) and the category of étale local shtukas over S by ÉtSht__q(S).


Now there is a category equivalence between local GL_r-shtukas as defined in <ref> and the category of local shtukas of rank r over S with isomorphisms as the only morphisms. It is naturally induced by the category equivalence <cit.> of ℋ^1(,L^+GL_r)(S) and the category of locally free sheaves of _Sz-modules of rank r with isomorphisms as morphisms.






  
Tate Functors on Local ℍ-Shtukas: 

Now let 

S be a connected _q-scheme with geometric base point s∈ S and algebraic fundamental group π_1(S,s). We denote by 𝔉𝔐𝔬𝔡_z[π_1(S,s)] (resp. 𝔉𝔐𝔬𝔡_z[π_1(S,s)] ) the category of finite and free _qz-modules (resp. z vector spaces) equipped with a continuous action of π_1(S,s). Then the dual Tate functor Ť on étale local shtukas is defined as 

    Ť_-:ÉtSht__q(S)→𝔉𝔐𝔬𝔡_z[π_1(S,s)]    M:=(M,τ_M)↦Ť_M:=(M⊗__Szκ(s)z)^τ_M,

where the superscript τ_M denotes the τ_M invariants. 

The rational dual Tate functor  is defined by 

    V̌_-:ÉtSht__q(S)→𝔉𝔐𝔬𝔡_z[π_1(S,s)]    M:=(M,τ_M)↦V̌_M:=Ť_M⊗_zz.


We also need Tate functors for local ℍ-shtukas. To define these we denote by Rep_zℍ the category of representations ρ:ℍ→GL(V), where V is a finite-free z-module and ρ a morphism of algebraic groups over z. 
Any such ρ naturally induces, as described in <cit.>, a functor ρ_⋆:ÉtSht_ℍ(S)→ÉtSht_(S) that is compatible with quasi-isogenies.

Let Funct^⊗(Rep_zℍ, 𝔉𝔐𝔬𝔡_z[π_1(S,s̅)] ) and Funct^⊗(Rep_zℍ, 𝔉𝔐𝔬𝔡_z[π_1(S,s̅)]) be the categories of the appropriate tensor functors whose morphisms are isomorphisms of functors. Now the dual Tate functor 𝒯̌_- and the rational dual Tate functor 𝒱̌_- are defined by 

    𝒯̌_-:    ÉtSht_ℍ(S)⟶ Funct^⊗(Rep_zℍ, 𝔉𝔐𝔬𝔡_z[π_1(S,s̅)] )   ℒ↦ (𝒯̌_ℒ:ρ↦Ť_ρ_⋆ℒ)
    
    𝒱̌_-:    ÉtSht_ℍ(S)⟶ Funct^⊗(Rep_zℍ, 𝔉𝔐𝔬𝔡_z[π_1(S,s̅)])     ℒ↦ (𝒱̌_ℒ:ρ↦V̌_ρ_⋆ℒ).






  
Tate Functors on Global 𝔾-Shtukas: 

Now we assume that the tuple v=(_1,…, _n) is given by n pairwise different places on C and set C=C\{_1,…,_n}. We denote by ^:=∏_∈ (C\) A_ the integral adeles of C outside  and by 𝔸^:=^⊗__CQ=∏'_∈ (C\)Q_ the adeles of C outside .
Let Rep_^𝔾 be the category of representations ρ:𝔾×_CSpec ^→GL_^(V)
where V is a finite free ^-module and ρ a morphism of group schemes over ^.
Let S be a connected scheme over Spf A_ with a fixed geometric base point s̅. We denote by 𝔐𝔬𝔡_^[π_1(S,s̅)] (resp. 𝔐𝔬𝔡_𝔸^[π_1(S,s̅)]) the category of ^-modules (resp. 𝔸^-modules) with a continuous π_1(S,s̅) action. For a finite subscheme D⊂ C we set D_s̅=D×__qs̅ as well as |_D_s̅=×_C_SD_s̅. Then the dual Tate functor 𝒯̌_- and the rational dual Tate functor 𝒱̌_- on global 𝔾-shtukas are defined by

    𝒯̌_-: ∇_nℋ^1(C,𝔾)^(S)
       ⟶ Funct^⊗(Rep_^𝔾, 𝔐𝔬𝔡_^[π_1(S,s̅)] ) 
       ↦(𝒯̌_:ρ↦_D⊂C (ρ_⋆|_D_s̅)^τ_)
    
    𝒱̌_-: ∇_nℋ^1(C,𝔾)^(S)
       ⟶ Funct^⊗(Rep_^𝔾, 𝔐𝔬𝔡_^[π_1(S,s̅)] ) 
       ↦(𝒱̌_:ρ↦_D⊂C (ρ_⋆|_D_s̅)^τ_⊗_^𝔸^).



We remark that the functor 𝒱̌ transforms by 
<cit.> quasi-isogenies into isomorphisms. Besides it is useful to know that there is a natural isomorphism _D⊂ C(ρ_⋆|_D_s̅)^τ_≃∏_∈ C\𝒯̌_L^+_()(ρ_) writing ρ=(ρ_)_∈C with ρ_:=ρ× id_A_. Here L^+_() is the étale local 𝔾_-shtuka and 𝒯̌_L^+_()(ρ_):=𝒯̌_L^+_()(ρ_) where ρ_ is the representation of 𝔾_ induced from ρ_ by Weil restriction (see <cit.>). 





  
H-Level Structures: 

Let H be an open, compact subgroup of 𝔾(𝔸^). In this paragraph we define H-level structures which are a generalization of the previous D-level structures. We denote by 

    ω_^^∘: Rep_^𝔾⟶𝔐𝔬𝔡_^ ,    ω_𝔸^^∘: Rep_𝔸^𝔾⟶𝔐𝔬𝔡_𝔸^

the forgetful functors and by Isom^⊗(ω_^^∘,𝒯̌_) and Isom^⊗(ω_𝔸^^∘,𝒱̌_) the sets of isomorphisms of tensor functors 
which are defined for every global  𝔾-shtuka  over S, where S is as before a scheme over Spf A_ with geometric base point s̅∈ S. By the definition of the Tate functor π_1(S,s̅) acts on 𝒯̌_ and 
𝔾(^) (resp. 𝔾(𝔸^)) acts on ω^∘_^ (resp. ω^∘_𝔸^)
since we have 𝔾(^)=Aut^⊗(ω^∘_^) by the generalized tannakian formalism <cit.>.
This induces an action of 𝔾(^)×π_1(S,s̅) on Isom^⊗(ω_^^∘,𝒯̌_) and of 𝔾(𝔸^)×π_1(S,s̅) on Isom^⊗(ω_𝔸^^∘,𝒱̌_).
Now by <cit.> a rational H-level structure γ̅ on a global 𝔾-shtuka  in ∇_nℋ^1(C,𝔾)^(S) is defined as a π_1(S,s̅)-invariant H-orbit γ̅=γ H in Isom^⊗( ω_^^∘,𝒱̌_ ). We denote by ∇_n^Hℋ^1(C,𝔾)^ the category fibered in groupoids over (_q)_Ét with the following fiber categories.
An object in ∇_n^Hℋ^1(C,𝔾)^(S) is a tuple (,γ), where ∈∇_nℋ^1(C,𝔾)^(S) and γ̅ is a H-level structure on . A morphism from (,γ̅) to (',γ̅') over S is a quasi-isogeny f:→' that is an isomorphism at the characteristic places _i and that satisfies 𝒱̌_f∘γ H=γ'H. 
(So f:|_C_S\ T_S'|_C_S\ T_S for a finite subscheme T⊂ C with _1,…,_n∉ T.)

Now let D be  a finite subscheme D⊂ C with _1,…,_n∉ D and let H_D be the open and compact subgroup ker(𝔾(^)→𝔾(_D)) of 𝔾(^). Then we remark hat by <cit.> there is a canonical isomorphism of stacks 

    ∇_nℋ_D^1(C,𝔾)^∇^H_D_nℋ^1(C,𝔾)^


Furthermore we note that for the conjugated group gHg^-1 with g∈𝐆(^) there is by <cit.> a natural isomorphism ∇_n^Hℋ^1(C,𝔾)^∇_n^gHg^-1ℋ^1(C,𝔾)^ sending (,γ H) to (, γ  g^-1(gHg^-1)). 

In addition we remark that by <cit.> for a  open, compact subgroup H⊂𝔾(^) contained in H we have a natural finite étale morphism

    ∇_n^Hℋ^1(C,𝔾)^∇_n^Hℋ^1(C,𝔾)^,  (,γH)↦ (, γ  H)





If we have additionally given a bound Ẑ_ at all places v∈ we denote by ∇_n^Ẑ_,Hℋ^1(C,𝔾) the closed substack of ∇^H_nℋ^1(C,𝔾)^×_Spf A_Spf R_Ẑ_ that consists of those points (,γ̅) such that  is bounded by Ẑ_. 




  
Parahoric Bruhat-Tits Group Schemes

At some points, mainly in theorem <ref>, we will assume 𝔾 to be a parahoric Bruhat-Tits group scheme, where we call a smooth, affine group scheme 𝔾 over C a parahoric Bruhat-Tits group scheme, if


  * all fibers are connected,

  * the generic fiber 𝐆 is a connected reductive group over Q and

  * for all ∈ |C| the group 𝔾(A_)⊂𝔾(Q_)=𝐆(Q_) is a parahoric subgroup in the sense of <cit.>.



For each parahoric subgroup in 𝐆_(Q_) there is a unique smooth, affine group scheme ℍ over A_ with connected special fiber, with generic fiber equal to 𝐆_ and with ℍ(A_) equal to this parahoric subgroup. Since this group scheme is exactly given by 𝔾_ our definition of parahoric  Bruhat-Tits group scheme coincides with the one in <cit.>.

Now Bruhat-Tits group schemes can be constructed as follows. We start with a reductive group scheme 𝐆 over the function field Q, which has a reductive model G over an open subscheme C∖{w_1,…, w_m} of C. For each of the pairwise different closed points w∈w:={w_1,…, w_m} we choose furthermore a parahoric subgroup H_w∈𝔾(Q_). Then H_ corresponds as explained above to a smooth, affine group scheme ℍ_ over A_ with generic fiber 𝐆×_QQ_. Consequently (∐_∈wℍ_) ∐ G is a group scheme over (∐_∈w Spec A_) ∐ C\{_1,…,_n}. 
Using the theorem of Beauville-Laszlo <cit.> the identification ℍ_×_A_Q_=G×_C∖{w}Q_ allows us to glue this group scheme to a group scheme 𝔾 over C.



This group scheme is by <cit.> smooth and by <cit.> affine over C. Therefore 𝔾 is by construction a parahoric Bruhat-Tits group scheme satisfying 𝔾_=ℍ_ and 𝔾×_C Q=𝐐.

Further we remark that if π:C→ C is a generically étale covering of C and 𝔾 is a parahoric Bruhat-Tits group scheme over C then by <cit.> the Weil restriction π_⋆𝔾 (see also lemma <ref>) of 𝔾 along π is again a parahoric Bruhat-Tits group scheme.
In addition we remark that parahoric Bruhat-Tits group schemes give an interesting class of smooth, affine group schemes over C since moduli spaces of global 𝔾-shtukas for such parahoric Bruhat-Tits group schemes 𝔾 are used by Lafforgue to establish in <cit.> and <cit.> the Langlands-parametrization over the function field Q. 














§ FUNCTORIALITY OF ∇_N^Ẑ_V,Hℋ^1(C,𝔾)


In this section we establish and analyze morphisms between moduli spaces of global 𝔾-shtukas, which are functorial in changing the curve C and the group scheme 𝔾. As mentioned in the introduction, apart from the general interest of these morphisms in the study of ∇_n^Ẑ_, Hℋ^1(C,𝔾), there are two other motivations. The first motivation concerns a potential formulation of an André-Oort conjecture for moduli spaces of global 𝔾-shtukas using the finiteness results in theorem <ref> and theorem <ref>. This potential formulation is exlained in more detail in remark <ref>.
The second motivation arises from the study of stratifications of ∇_n^Ẑ_, Hℋ^1(C,𝔾) in <cit.>, where the results of this third section are needed again.

The third section is divided into three subsections. In the first subsection we define a shtuka datum and morphisms of these. A shtuka datum contains all the necessary parameters to define a moduli space of 𝔾-shtukas. Then a morphism is defined in such a way that it exactly satisfies the properties to induce a morphism of the corresponding moduli spaces. The fact that ∇_n^Ẑ_, Hℋ^1(C,𝔾) is indeed functorial in the shtuka datum is then shown in the following two subsections. The second subsection discusses the case where we only change the curve C in the shtuka datum. The induced morphism is constructed and is proven in theorem <ref>  to be finite. 


In the third subsection a change of the group scheme 𝔾 by f:𝔾→𝔾' is analyzed. Before making any assumptions on f the morphism ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ'_, H'ℋ^1(C,𝔾') is constructed in general. Then, assuming that f is generically an isomorphism, we prove in theorem <ref> a projectivity and a surjectivity result. 
Afterwards, we consider closed immersions of group schemes. In this situation of a closed immersion we prove ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ'_, H'ℋ^1(C,𝔾') to be unramified (theorem <ref>) and even finite if 𝔾 is a parahoric Bruhat-Tits group scheme (theorem <ref>). 








 §.§ The Shtuka Datum

In this subsection, we define the category of Shtuka-data. While we can easily define the objects, we need some further 
Shtuka-datum
explanations to define the morphisms.



A Shtuka-datum is a tuple (C,𝔾, , Z_, H) where 


  * C is a smooth, projective, geometrically irreducible curve over _q,

  * 𝔾 is a smooth, affine group scheme over C, 

  * =(_1,…,_n) is a tuple of n closed points in C (not necessarily disjoint),

  * Ẑ_ is a bound in the sense of <ref>,

  * H is an open, compact subgroup of 𝔾(𝔸^).




Before we can define morphisms, we need the following lemmas. Let π:X→ Y be a morphism of schemes. We recall that for any functor
F: (𝐒𝐜𝐡/X)^op⟶𝐒𝐞𝐭
the push forward π_⋆F:(𝐒𝐜𝐡/Y)^op⟶𝐒𝐞𝐭 with respect to π is defined by (T→ Y)⟼ F(T×_YX).
In the case that F is a scheme (i.e. representable) and π_⋆ F is also representable, we call π_⋆ F the Weil restriction ℜ_X/Y(F) of F.
The basic properties and some conditions for the existence of Weil restrictions are discussed and developed in <cit.> and <cit.>. 
We have the following lemma, where we call a morphism of schemes finite locally free, if it is finite, flat and of finite presentation.


Let π:X→ Y be a surjective, finite locally free morphism of schemes, let 𝔾 be a smooth, affine group scheme over X and let  be a 𝔾-torsor on the big étale site of X, then 


  * π_⋆𝔾 is a smooth, affine group scheme over Y

  * π_⋆ is a π_⋆𝔾-torsor on the big étale site of Y.




Since π is finite and faithfully flat we can apply theorem 4 in <cit.> to see that the Weil restriction π_⋆𝔾 exists indeed as a scheme. 
Let U,V,W∈ Fun((𝐒𝐜𝐡/X)^op,𝐒𝐞𝐭) be arbitrary with natural transformations f_1:U→ W and f_2:V→ W, then for S∈ (𝐒𝐜𝐡/Y) we have 

    π_⋆(U×_W V)(S)   =   Hom_X(S×_Y X, U×_W V)
       =   {(f,g) |f∈ Hom(S×_Y X, U), g∈ Hom(S×_Y X, V), f_1∘ f=f_2∘ g}
       =   (π_⋆ U×_π_⋆ Wπ_⋆ V)(S).


This shows that π_⋆ commutes with fiber products and it follows that π_⋆𝔾 becomes a group scheme over Y. 

Let U⊂ Y be an affine open. Then π_⋆(X×_YU)=U 
and the compatibility with the fiber product implies π_⋆(𝔾×_X X×_Y U)=π_⋆𝔾×_Y U. Now π_⋆𝔾×_Y U is affine because 𝔾 is affine over X and π is finite. Since the Weil restriction of an affine scheme is by construction affine we conclude that π_⋆𝔾 is affine over Y. Furthermore we know by <cit.> that π_⋆𝔾 is again of finite type and smooth over Y, which proves the first part.

Now let  be a 𝔾-torsor over X. Since 𝔾 is smooth and affine,  is represented by a smooth, affine scheme over X, by faithfully flat descent,  <cit.> and <cit.>.
<cit.> and <cit.> tell us again, that π_⋆ is a smooth scheme over Y. Using once more the compatibility of the fiber product with the Weil restriction, the action of 𝔾 on  induces an action of π_⋆𝔾 on π_⋆ and additionally the isomorphism 𝔾×_X ≃×_X yields an isomorphism π_⋆𝔾×_Yπ_⋆≃π_⋆×_Yπ_⋆. It remains to show that π_⋆ has étale locally on Y a section to π_⋆. Since π_⋆→ Y is smooth and surjective
 this is content of proposition <cit.>. 



Now morphisms between 𝔾-torsors are sent by π_⋆ to morphisms of π_⋆𝔾-torsors and in fact we have the following lemma.


Let π:X→ Y be a surjective, finite locally free morphism and 𝔾 a smooth, affine group scheme over X. Then the functor 

        π_⋆:{ 𝔾X}   ⟶{π_⋆𝔾Y} ,

   ⟼π_⋆
induced by lemma <ref>, is an equivalence of categories.
The inverse functor sends some π_⋆𝔾-torsor  to π^⋆×^π^⋆π_⋆𝔾𝔾.



First we prove that π_⋆ is fully faithful. So let , be two 𝔾-torsors over X and f':π_⋆→π_⋆ be a morphism of π_⋆𝔾-torsors. We choose an étale covering U'→ Y with π_⋆(U')≠∅≠π_⋆(U'). Now this implies automatically that U:=U'×_Y X→ X is an étale covering with (U)≠∅≠(U). We choose two sections u∈(U)=π_⋆(U') and u∈(U)=π_⋆(U'), which determine trivializations

α':    π_⋆×_Y U'→π_⋆𝔾×_Y U'           α'   :π_⋆×_Y U'→π_⋆𝔾×_Y U' 

α:    ×_X U→𝔾×_X U           α   : ×_X U→𝔾×_X U

with α'^-1(1)=α^-1(1)=u and α'^-1(1)=α^-1(1)=u. Now we consider the following diagrams, where U_2:=U×_XU, U_2':=U'×_YU' with projections p_1, p_2, p_1' and p_2' and h:=α'∘(f'× id_U')∘α'^-1. Note that since h is π_⋆𝔾-equivariant h is determined by h:=h(1)∈π_⋆𝔾(U')=𝔾(U).
This same h defines then a morphism of h:𝔾×_X U→𝔾×_X U of 𝔾-torsors on U and we set f× id_U:=α^-1∘ h∘α:×_XU→×_X U.


    π_⋆𝔾( U') [rr]^(0,4)h      π_⋆𝔾(U')  
    
    π_⋆( U')   [ru]^α'[rr]_(0,3)f'× id_U'      π_⋆( U') [ru]^α'   
    
       π_⋆𝔾( U_2') @<0.7ex>[rr]^(0,3)p_1'^⋆h@<-0.7ex>[rr]_(0,3)p_2'^⋆h@<-@<0.7ex>[uu]^(0,7)p_1'@<-@<-0.7ex>[uu]_(0,7)p_2'      π_⋆𝔾( U_2') @<-@<0.7ex>[uu]^p_1'@<-@<-0.7ex>[uu]_p_2'
    
    π_⋆( U_2') @<-@<0.7ex>[uu]^p_1'@<-@<-0.7ex>[uu]_p_2'@<0.7ex>[ru]^p_1'^⋆α'@<-0.7ex>[ru]_p_2'^⋆α'[rr]_f'× id_U_2'      π_⋆( U_2')  @<-@<0.7ex>[uu]^(0,7)p_1'@<-@<-0.7ex>[uu]_(0,7)p_2'@<0.7ex>[ru]^p_1'^⋆α'@<-0.7ex>[ru]_p_2'^⋆α'   
    
       𝔾( U) [rr]^(0,4)h      𝔾(U)  
    
    ( U )  [ru]^α[rr]_(0,3)f× id_U      ( U )[ru]^α   
    
       𝔾( U_2 ) @<0.7ex>[rr]^(0,3)p_1^⋆h@<-0.7ex>[rr]_(0,3)p_2^⋆h@<-@<0.7ex>[uu]^(0,7)p_1@<-@<-0.7ex>[uu]_(0,7)p_2      𝔾( U_2) @<-@<0.7ex>[uu]^p_1@<-@<-0.7ex>[uu]_p_2
    
     ( U_2 )@<-@<0.7ex>[uu]^p_1@<-@<-0.7ex>[uu]_p_2@<0.7ex>[ru]^p_1^⋆α@<-0.7ex>[ru]_p_2^⋆α[rr]_f× id_U_2      ( U_2 ) @<-@<0.7ex>[uu]^(0,7)p_1@<-@<-0.7ex>[uu]_(0,7)p_2@<0.7ex>[ru]^p_1^⋆α@<-0.7ex>[ru]_p_2^⋆α

 
Now by definition of the Weil restriction we have equal sets at the corresponding vertices of the two cubes and the maps p_1', p_2' coincide with p_1 and p_2.
Furthermore by definition of α',h,α' these maps coincide with the maps α, h,α in the right hand cube. 
The morphisms p_1'^⋆α':π_⋆×_Y U_2'→π_⋆𝔾×_Y U_2' and p_1^⋆α are uniquely determined by the preimage of 1∈π_⋆𝔾(U_2')=𝔾(U_2). But this preimage is in both cases given as p_1^⋆(u)=p_1(α^-1(1))=p'_1(α'^-1(1))∈π_⋆(U_2')=(U_2). Hence the maps p_1'^⋆α' and p_1^⋆α coincide and equally p_1'^⋆ h, p_1'^⋆α', p_2'^⋆α', p_2'^⋆ h, p_2'^⋆α' coincide with p_1^⋆ h, p_1^⋆α, p_2^⋆α, p_2^⋆ h and p_2^⋆α respectively.


We further denote g':=p_2'^⋆α'∘ p_1'^⋆α'^-1(1)∈π_⋆𝔾(U_2') and g':=p_2'^⋆α'∘ p_1'^⋆α'^-1(1)∈π_⋆𝔾(U_2'). So that we have g'=g:=p_2^⋆α∘ p_1^⋆α^-1(1)∈𝔾(U_2) and g'=g:=p_2^⋆α∘ p_1^⋆α^-1(1)∈𝔾(U_2). 
With these notations we get the following bijections:

    Hom_𝔾(,) [rrr]_1:1^f ↦ h:=(α∘(f× id_U)∘α^-1)(1)@–>[d]                  {h∈𝔾(U) | g∘ p_1^⋆ h= p_2^⋆ h∘ g}@=[d]
    
    Hom_π_⋆𝔾(π_⋆,π_⋆) [rrr]_1:1^f' ↦ h':=(α'∘(f'× id_U')∘α'^-1)(1)         {h'∈π_⋆ G(U') | g'∘ p_1'^⋆ h= p_2'^⋆ h∘ g'}.

Here the horizontal bijections are due to faithfully flat descent <cit.> and the fact that the condition g∘ p_1^⋆ h=p_2^⋆ h∘ g is equivalent by definition of g,g to p_1^⋆(α^-1∘ h∘α)=p_2^⋆(α^-1∘ h∘α) and for g'∘ p_1'^⋆ h'=p_2'^⋆ h'∘ g' respectively. The equality on the right follows from the identifications in the above cubes. To prove the fully faithfulness it remains to show that the bijective dashed arrow is given by π_⋆. By definition of π_⋆ f the following diagram commutes 

    Hom(U',π_⋆) [r]^π_⋆ f@=[d]     Hom(U',π_⋆) @=[d]
    
    Hom(U'×_Y X, ) [r]^f     Hom(U'×_Y X,π_⋆)

which shows that f and π_⋆ f map to the same h on the right hand side.

It remains to show that π_⋆ is essentially surjective. So let  be a π_⋆𝔾-torsor over Y and choose again an étale covering U'→ Y and a trivialization α': ×_Y U'π_⋆𝔾×_Y U'.
Let U_2':=U'×_Y U' and g':=p_2'^⋆α'∘ p_1'^⋆α'^-1(1)∈π_⋆𝔾(U_2'), where p'^⋆_2α'∘ p_1'^⋆α'^-1:π_⋆𝔾×_Y U_2'π_⋆𝔾×_Y U_2'. 
So the descent datum of  is isomorphic to (π_⋆𝔾×_Y U', g').
Now U:=U'×_YX→ X is an ´étale covering and we set _U:=𝔾×_XU as well as U_2:=U×_X U=U_2'×_YX with projections p_1,p_2. Let g∈𝔾(U_2) be equal to g' using 𝔾(U_2)=π_⋆𝔾(U_2'). Then (_U, g) is a descent datum that comes by <cit.> from a 𝔾-torsor  on X. Now it is clear that (π_⋆_U, π_⋆ g)=(π_⋆𝔾×_YU',g'). Therefore we have π_⋆≃ which proves that π_⋆ is essentially surjective.
We only need to prove that for every 𝔾-torsor  on X the torsor π^⋆π_⋆×^π^⋆π_⋆𝔾𝔾 is isomorphic to . 
With the same notation as above  is given by the (𝔾×_X U, g) and π_⋆ is given by the descent datum (π_⋆𝔾×_Y U', g'). Restricting the latter torsor to X we get the descent datum (π^⋆(π_⋆𝔾×_YU'), g× id_X)=(π_⋆𝔾×_YU'×_YX, g× id_X). Using the adjunction Hom_Y(π_⋆𝔾,π_⋆𝔾)=Hom_X(π^⋆π_⋆𝔾,𝔾) we denote by φ:π^⋆π_⋆𝔾→𝔾 the morphism  corresponding to id_π_⋆𝔾. Now applying the functor ×^π^⋆π_⋆𝔾, φ𝔾 gives us the descent data (π^⋆(π_⋆𝔾×_YU')×^π^⋆π_⋆𝔾𝔾, (g'× id_X, 1_𝔾)). Since φ maps (g'× id_X) to g this descent data is isomorphic to (𝔾×_X U, g), which proves the lemma. 







Now let π:C→ C' be a finite morphism from C to some other smooth, projective, geometrically irreducible curve C'. This morphism is then automatically faithfully flat <cit.>. Let further (C,𝔾, , Z_, H) be a shtuka datum as in definition <ref>. Since 𝔾 is a smooth, affine group scheme over C, this allows us to apply lemma <ref> and <ref> in this situation.

 We denote by ℋ^1(C,𝔾) the category fibered in groupoids, whose S-valued points for some _q-scheme S are given by isomorphy classes of 𝔾-torsors over C_S. By lemma <ref> π_⋆ induces an isomorphism ℋ^1(C,𝔾)⟶ℋ^1(C',π_⋆𝔾).























Let w_i=π(v_i) and w=(w_1,…, w_n). Our next goal is to define the bound π_⋆ Z_v=(Z_w_i)_i at the points w_i. We need the following lemma and general remark, where w is a closed point in C', A'_w is the completion of the local ring _C',w and π_w=π× id_Spec A'_w:C×_C'Spec A_w=∐_v|wSpec A_v→ Spec A'_w.


In the next lemma, we need the following general fact about Weil restrictions. Let X,Y,S be schemes over some base scheme Z, π:X→ Y a Z-morphism, M an X-scheme and Y_S=Y×_ZS, X_S=X×_ZS and M_S=M×_ZS the appropriate base changes. 
Then we have (π× id_S)_⋆(M×_Z S)=π_⋆ M×_ZS. This is easily seen by the equation for T∈𝐒𝐜𝐡/Y_S:

    (π_⋆ M×_Z S)(T)    =    (π_⋆ M×_Y Y_S)(T) = 
     Hom_Y(T,π_⋆ M)
    = Hom_X(T×_Y X,M)
       =    Hom_X_S(T×_YX,M_S)=(π× id_S)_⋆(M×_Z S)(T).




We have (π_)_⋆(∐_v∈π^-1()𝔾_)
=(π_⋆𝔾)_ as a group scheme over Spec A'_.

This follows formally from remark <ref> with M=𝔾, X=C, Y=Z=C' and S=Spec A_ since we have

    (π_w)_⋆(∐_v|w𝔾_v)=(π_w)_⋆(𝔾×_C∐_v|w Spec A_v)=(π_w)_⋆(𝔾×_C' Spec A_w)
    =π_⋆𝔾×_C'Spec A_=(π_⋆𝔾)_.



The notation 𝔾_ was introduced in <ref> and denotes the Weil restriction of the group scheme 𝔾_ along Spec A_→ Spec _qz_. The lemma has the following corollary.


We have ∏_| L^+𝔾_=L^+ (π_⋆𝔾)_ as group schemes over _q.

Let R be a connected _q-algebra,  
then we have: 

    L^+(π_⋆𝔾)_(R)
    
       =(π_⋆𝔾)_(Rz_w)
    
    =Hom_Spec A_(Spec Rz_w⊗__q_, (π_⋆𝔾)_)
    
    
       =Hom_Spec A_(Spec R⊗̂__qA_, (π_⋆𝔾)_)
       =Hom_Spec A_(Spf R⊗̂__qA_, (π_w)_⋆(∐_v|w𝔾_v))
    
    
       = Hom_∐_|Spec A_(Spec R⊗̂__qA_⊗_A_∏_|A_, ∐_|𝔾_)
       =∏_|Hom_Spec A_(Spec Rz_⊗__q_, 𝔾_)
    =∏_|𝔾_(Rz_)
    
       =∏_|L^+𝔾_(R).



We have the following 2-cartesian diagrams:

    |∏ℱ l_𝔾_[r] [d]    _q [d]    ℱ l_(π_⋆𝔾)_[d] [r]    _q [d] 
    
    |∏ℋ^1(_q,L^+𝔾_) [r]    |∏ℋ^1(_q,L𝔾_)    ℋ^1(_q, L^+(π_⋆𝔾)_) [r]    ℋ^1(_q, L
    (π_⋆𝔾)_)


By corollary <ref> the lower stacks in the diagrams are isomorphic, so that we get an isomorphism ∏_|ℱ l_𝔾_ℱ l_(π_⋆𝔾)_ and by the base change with the compositum  of the finite fields _ for all | we get an isomorphism ∏_|∏_l∈/deg ℱ l_𝔾_×__≃∏_l∈/deg ℱ l_(π_⋆𝔾)_×__. Since σ^deg invariant components are mapped to σ^deg invariant components, it restricts to an isomorphism

    ∏_| ∏_ deg |ll∈/deg ℱ l_𝔾_×__≃
    ℱ l_(π_⋆𝔾)_×__.



Now let R be a DVR with ⊂ R and such that there exists a representative Ẑ_v, R of Ẑ_ v for all v∈v. Consider the ind-closed subscheme 

    ∏_| ∏_ deg |ll∈/deg Ẑ_,l⊂∏_| ∏_ deg |ll∈/deg ℱ l_𝔾_×__ Spf R

 where Ẑ_,l is always the closed stratum 𝒮(1)×__Spf R except for v∈v and l=0, where we set Ẑ__i,0=Ẑ__i, R. 
Here 𝒮(1) denotes the closed Schubert cell 1· L^+𝔾_v in ℱ l_𝔾_.
Via the previous isomorphism this defines an ind-closed subscheme in ℱ l_(π_⋆𝔾)_×__Spf R. This defines a bound Ẑ_w in the sense of <ref> in ℱ l_(π_⋆𝔾)_ and we set  π_⋆Ẑ_v:=Ẑ_w:=(Ẑ_w_i)_i.



Next we define π_⋆ H. We recall that H was an open, compact subgroup of 𝔾(𝔸^). 
Since ⊂π^-1()⊂ |C| we have a quotient map of topological rings 𝔸^⟶^π^-1().

Since this map is open, it induces by <cit.> an open continuous group homomorphism 𝔾(^)⟶𝔾(^π^-1()). We have ^×_C'C=^×_η'η=^π^-1() where η and η' are the generic points of C and C'. This gives us with the definition of the Weil restriction π_⋆𝔾(^)=𝔾(^π^-1()), where both groups carry the same topology by <cit.>. 
Now the image of H under this morphism gives us an open, compact subgroup in π_⋆𝔾(^) that we denote by π_⋆ H.





We have seen in <ref> that
there is the possibility to define level structures using finite closed subschemes D of C and  in <ref> we have remarked that D-level structures of a 𝔾-shtuka correspond bijectively to H_D-level structures, where H_D=ker(𝔾(𝕆^)→𝔾(𝒪_D)). Now we can also consider the Weil restriction π_⋆ D of D. It is a closed finite subset of C' consisting of the points {∈ |C'| | ×_C'C⊂ D}. And with a D-level structure of some 𝔾-shtuka 𝒢 we could associate a π_⋆ D-level structure of the corresponding π_⋆𝔾-shtuka π_⋆𝒢 (which will be defined in proposition <ref>). But compared to the associated π_⋆ H-level structure that we will define in theorem <ref> we would lose some information at the points D\(π_⋆ D×_C'C), which is seen in the following way. Since we have π_⋆ D×_C'C⊂ D we have H_D⊂ H_π_⋆ D×_C'C and hence π_⋆ H_D⊂π_⋆ H_π_⋆ D×_C'C. Now 

H_π_⋆D   =ker(π_⋆𝔾(𝕆^)→π_⋆𝔾(𝒪_π_⋆D))=ker(𝔾(𝕆^π_-1)→𝔾(𝒪_π_⋆D×_C'C))

   =H_π_⋆D×_C'C∩𝔾(^π^-1())=im(H_π_⋆D×_C'C→𝔾(^π^-1(w)))=:π_⋆H_π_⋆D×_C'C⊃π_⋆H_D

shows that π_⋆ H_D is in general a finer level than π_⋆ D (or equivalently H_π_⋆ D) and the previous equation shows that the information is lost exactly at the points D\ (π_⋆ D×_C' C).


All these previous explanations concerned the case that we change the curve in the shtuka datum but we can also change the group scheme in this datum. Let f:𝔾→𝔾' be any morphism of smooth, affine group schemes over C and  a closed point in C.  Firstly this induces a morphism L^+𝔾_→ L^+𝔾'_ of the positive loop groups as well as a morphism L𝐆_→ L𝐆'_ of the loop groups. Consequently we also get a morphism ℱ𝑙_𝔾_→ℱ𝑙_𝔾'_ of the affine flag varieties.
Secondly such a morphism induces a morphism f_^:𝔾(^)→𝔾'(^) of locally compact Hausdorf spaces by <cit.>.
Now we can define morphisms of shtuka data.


A morphism between two shtuka data (C,𝔾, , Ẑ_, H) and (C',𝔾', , Ẑ'_, H') is a pair (π,f) such that:

 
  * π:C→ C' is a finite morphism with π(_i)=_i 

  * f:π_⋆𝔾→𝔾' is a morphism of smooth, affine group schemes over C'

  * The morphism (π_⋆ Z_)_R→∏_∈_(π_⋆𝔾)_,R→∏_∈_𝔾'_,R factores through Ẑ'_,R, where R is a DVR such that there exists representatives (π_⋆Ẑ_)_R and Ẑ'_,R of the corresponding bounds


  * f_^(π_⋆ H)⊂ H' 





With this definition we have reached the goal of this subsection. In the next two subsections we will prove that such a morphism induces a morphism of the corresponding moduli stacks and determine some of its properties. But before we give some remarks.




 
  * Let π:C→ C' be a finite morphism and =π(). With the definition of π_⋆ H and π_⋆ Z_ on page defpiz it is clear that (π,id_π_⋆𝔾):(C,𝔾, , Z_, H)→ (C',π_⋆𝔾, , π_⋆ Z_, π_⋆ H) defines a morphism of shtuka data

  * Every morphism (π,f) of shtuka data factorizes as (id_C,f)∘(π,id_π_⋆𝔾). 

  * If f:π_⋆𝔾→𝔾' is an isomorphism in the generic fiber we have π_⋆𝔾(^)= 𝔾'(^) so that we can naturally choose H=H'.

  * If f:π_⋆𝔾→𝔾' is smooth in the generic fiber, then f_^:π_⋆𝔾(^)→𝔾'(^) is an open map by <cit.> so that we can naturally choose H'=f_^(π_⋆ H)

  * If f:π_⋆𝔾→𝔾' is proper in the generic fiber, then f_^:π_⋆𝔾(^)→𝔾'(^) is a topologically proper map by <cit.> so that we can naturally choose H=f_^^-1(H') 





























 §.§ Changing the Coefficients


In this subsection we prove that a morphism of shtuka data  (π, id), where we only change the curve, induces a finite morphism of the corresponding moduli stacks. We firstly prove this for the moduli stack , where the characteristic sections are not fixed and no boundedness condition or level structures are imposed. For this purpose we need the following lemma:


Let S be an _q-scheme with n morphisms s_i:S→ C for i=1,…, n. Then the scheme theoretic image of C_S:=C_S\⋃_i Γ_s_i in C_S equals C_S.


Since D:=⋃_i Γ_s_i is an effective Cartier-Divisor on C_S over S, we find an affine covering (U_j)_j∈ J of C_S with U_j:=Spec B_j such that D is the vanishing locus of an element f_j∈ B_j that can be written as f_j=a_j/b_j with two regular elements a_j, b_j∈ B_j (see <cit.>). Now the ring homomorphism B_j→Γ(U_j\ D, 𝒪) is injective, which is seen as follows. An element x∈ B_j is send to 0 if and only if f_j^mx=0 for some m∈. The latter condition implies a_j^mx=0 and since a_j is a non-zero divisor this means x=0 so that Spec B_j\ D is schematically dense in Spec B_j (compare also <cit.>).
Now gluing all the U_j shows that for every affine open V⊂ C_S the ring homomorphism Γ(V,𝒪)→Γ(V\ D, 𝒪) is injective and we conclude that C_S is schematically dense in C_S (see <cit.>).





Let π:C→ C' be a finite morphism of smooth, projective, geometrically irreducible curves over _q and let 𝔾 be a smooth, affine group scheme over C. This induces a finite morphism of the moduli stacks 

    π_⋆ : →∇_nℋ^1(C',π_⋆𝔾) .

 which factors through a closed immersion
 ∇_nℋ^1(C,𝔾)↪∇_nℋ^1(C',π_⋆𝔾)×_C'^nC^n.


 Let S be an _q-scheme and (,s_1,…, s_n, τ_)∈∇_nℋ^1(C,𝔾)(S). We describe its image (',s_1',…,s_n',τ_') to define the morphism. The torsor ' is given by (π_S)_⋆ and the sections s_i:S→ C are mapped to the composition s_i':=π∘ s_i:S→ C'. This implies π_S(⋃_i Γ_s_i)⊂⋃_i Γ_s'_i⊂ C'_S. Let C'_S=C'_S\ (∪_iΓ_s'_i) and C_S=C_S\ (∪_iΓ_s_i). 

Then U:=C×_C'C'_S=C_S×_C'_SC'_S is open in C_S. We denote by π_U:=π×_id_C'id_C'_S:U→C'_S and we have (π_U)_⋆(𝔾×_C U)=π_⋆𝔾×_C'C'_S. Now we restrict τ_:σ^⋆|_C_S→|_C_S to U_S and apply lemma <ref> to π_U. The category equivalence gives us the desired morphism τ_':(π_U)_⋆(σ^⋆×_C'_SC_S')=σ^⋆'|_C_S'→ (π_U)_⋆(×_C'_SC'_S)= '|_C_S'. This defines a global π_⋆𝔾-shtuka (',s_1',…,s_n', τ_') over S and therefore the morphism of the moduli stacks. 

We now show that this morphism is representable and finite.
Let S be again an arbitrary scheme over _q and ':S→∇_nℋ^1(C',π_⋆𝔾) be given by '=(',τ', s_1',…, s_n'). Then ∇_n ℋ^1(C,𝔾)×_∇_nℋ^1(C',π_⋆𝔾)S is the category fibered in groupoids over 𝐒𝐜𝐡/_q whose fiber category over an _q-scheme T is given by

    {
    (,g:T→ S,β)  |  =(,τ_,s_1,…,s_n)∈∇_n ℋ^1(C, 𝔾)(T)β:g^⋆'π_⋆}.


Using the n sections s_1',…,s_n':S→ C' and the morphism π:C→ C' we set S:=S×_C'^nC^n. 
Since S ×_ℋ^1 (C',π_⋆𝔾)ℋ^1(C,𝔾)=S by remark <ref> we know that Hom__q(T,S) is in bijection with the tuples { g:T→ S, :T→ℋ^1(C,𝔾), α:g^⋆'π_⋆}.
Consequently the _q-morphisms T→S are in bijection with the tuples (, s_1,…, s_n,g,α), where (,g,α)∈ S(T) as before and s_1,…,s_n:T→ C are morphisms making the following diagram commutative for all i=1,…,n

    T [d]_s_i[r]^g    [d]^s_i' S
    
    C [r]^π    C'
     .


We claim that we get a morphism ∇_nℋ^1(C, 𝔾)×_∇ℋ^1(C',π_⋆𝔾)S→S that is injective on T-valued points (hence a monomorphism) and satisfies the valuative criterion for properness. Then this implies by <cit.> that ∇_nℋ^1(C, 𝔾)×_∇ℋ^1(C',π_⋆𝔾)S is a closed subscheme of S. So first of all a given object (,τ_, s_1,…, s_n, g, α) in (∇_nℋ^1(C, 𝔾)×_∇ℋ^1(C',π_⋆𝔾)S)(T) is sent to (, s_1,…,s_n,g,α). Since α:g^⋆'π_⋆ is not only an isomorphism of torsors, but also of π_⋆𝔾-shtukas the n-sections g∘ s_i' of g^⋆' and (s_i∘π) of π_⋆ have to coincide, so that (,s_1,…,s_n,g,α) is a well defined object in S(T). This induces the morphism ∇_nℋ^1(C, 𝔾)×_∇ℋ^1(C',π_⋆𝔾)S→S. Further it was claimed, that this morphism is injective on T-valued points. So given two points (,τ_, s_1,…,s_n,g, α) and (,τ_,s_1,…,s_n,g,α) we need to show that this implies τ_=τ_. Since α is an isomorphism of π_⋆𝔾-shtukas, we have α^-1∘π_⋆τ_=g^⋆τ'∘σ^⋆α^-1=α^-1∘π_⋆τ_:σ^⋆(π_⋆)|_C_T' g^⋆'|_C_T', where we write again C'_T:=C_T'\⋃_i Γ_s_i'. This implies π_⋆τ_=π_⋆τ_ and using lemma <ref> applied to π× id_C_T'×_C'C we see τ_|_C_T'×_C'C=τ_|_C_T'×_C'C. 
We even need to know that τ_=τ'_. In the following diagram the restriction of (τ_, τ'_) to C'_T×_C'C factors by the previous observation through the diagonal Δ. 

    σ^⋆|_C_T[rr]^(τ_, τ'_)      |_C_T×_C_T|_C_T
    σ^⋆|_C_T'×_C'C[rr] [u]       |_C_T[u]_Δ   .

Since  is separated over C_T the diagonal is a closed immersion and (τ_, τ'_) factors already over C_T through the diagonal if the scheme theoretic image of σ^⋆|_C_T'×_C'C in σ^⋆|_C_T equals σ^⋆|_C_T. Since taking the scheme theoretic image is stable under flat base change by <cit.>, this is the case if the scheme theoretic image of C_T'×_C'C in C_T equals C_T. By the same argument this follows if the scheme theoretic image of C'_T in C'_T equals C_T'.
Now this is content of lemma <ref> so that we can conclude as desired τ_=τ'_.


Next we claimed that the morphism satisfies the valuative criterion for properness.
So let 

    Spec K [d]_j [rrrr]^(ℋ,τ_ℋ, r_1,…, r_n, f,β)               S×_∇_n ℋ^1(C',π_⋆𝔾)∇_nℋ^1(C,𝔾) [d]    
    
    Spec R @–>[rrrru] [rrrr]_(,s_1,…,s_n,g,α)            S[r]_finite     S

be a commutative diagram, where R is a complete discrete valuation ring with fraction field K, maximal ideal 𝔪 and algebraically closed residue field κ_R=R/𝔪. Note that R is a κ_R-algebra. We have to prove that there exists a unique dashed arrow making everything commutative.
The commutativity of the square shows ℋ=j^⋆, s_i∘ j=r_i:Spec K→ C, f=g∘ j:Spec K→ S and j^⋆α=β. To define this dashed arrow we have to extend (,s_1,…, s_n) to a 𝔾-shtuka (,τ_,s_1,…, s_n) over R such that α extends to an isomorphism α:g^⋆'→π_⋆ and j^⋆τ_=τ_ℋ. So we define this isomorphism τ_:σ^⋆|_C_R→|_C_R. Since ℋ^1(C'_R,π_⋆𝔾) and ℋ^1(C'_R×_C'C,𝔾) are isomorphic, τ_|_C'_R×_C'C is defined by α∘ g^⋆τ_'∘σ^⋆α^-1. Furthermore we know that τ_ is defined on the generic fiber C_K⊂C_R by τ_|_C_K=j^⋆τ_=τ_ℋ. So let p∈C_R\(C'_R×_C'C∪C_K), i.e. p∈((⋃_iΓ_s_i'×_C'C)\⋃_iΓ_s_i)⋂ C_R/𝔪. It remains to show, that τ_ extends to p. Since p is closed we choose an open V⊂C_R/𝔪 with V⋂(⋃_iΓ_s_i'×_C'C)=p and set V=V\ p. Then we consider the 2-cartesian diagram of stacks fibered over κ_R_Ét (compare <cit.>):

    ℋ^1(V,𝔾) [r] [d]_L^+_p   ℋ^1_e(V, 𝔾) [d]_L_p
    ℋ^1(κ_R, L^+𝐆_p) [r]    ℋ^1(κ_R, L𝔾_p) .

Here ℋ^1_e(V, 𝔾)(X) is the full subcategory of ℋ^1(V, 𝔾)(X) consisting of those 𝔾-torsors over V_X:=V×_κ_R X that can be extended to a 𝔾-torsor over V_X.

Now σ^⋆|_V_R and |_V_R define two R-valued points in ℋ^1(V,𝔾) and τ_|_V_R is an isomorphism in ℋ^1_e(V,𝔾)(R) that is already defined. Since R has algebraically closed residue field, we can choose trivializations α_1:L_p^+(σ^⋆)→ L^+𝔾_p and α_2:L_p^+()→ L^+𝔾_p (<cit.>). Then α_2^-1∘τ_∘α_1:L𝐆_p,R→ L𝐆_p,R is an isomorphism in ℋ^1(κ_R, L𝐆)(R) given by an element h∈ L𝐆_p(R). We know by assumption that the pull back of h to K is given by an element h_K∈ L^+𝔾_p(K), since τ_ is generically already an isomorphism over V. But since L^+𝔾_p is closed in L𝔾_p it follows that h∈ L^+𝔾_p(R). This implies that the isomorphism τ_|_V comes from an isomorphism in ℋ^1(V,𝔾)(R). So τ_ extends uniquely to p and the valuative criterion is proved. This proves that ∇_nℋ^1(C, 𝔾)×_∇ℋ^1(C',π_⋆𝔾)S is a closed subscheme of S and since S is finite over S it proves as well that  →∇_nℋ^1(C',π_⋆𝔾) is a finite morphism.







The next goal is to prove that for any shtuka datum (C,𝔾, v, Ẑ_, H) the morphism π:C→ C' induces also a finite morphism ∇_n^Ẑ_v, Hℋ^1(C, 𝔾)→∇_n^π_⋆Ẑ_, π_⋆ Hℋ^1(C', π_⋆𝔾). For this we need the following lemma that concerns the boundedness condition. Given a global 𝔾-shtuka  in  over S, we recall that we introduced in <ref> the global-local functor Γ__i that associates with it a local 𝔾__i-shtuka Γ__i() over S. On the other hand we explained (compare also <cit.>) that base change with Spf A__i×__qS≃∏_l∈/deg v_iV(𝔞_v_i,l) gives a local 𝔾_v_i-shtuka L_v_i^+() over S. Here 𝔾_v_i denotes the Weil restriction Res_A_v_i/_qz_v_i𝔾_v_i.

Now let Z_v_i be a bound in ℱ̂ ̂l̂_𝔾_v_i and R an DVR over A__i=__iz__i with a representative Ẑ__i,R⊂ℱ̂ ̂l̂_𝔾__i,R. We have ℱ̂ ̂l̂_𝔾__i×̂__qz__iSpf R=∏_/deg v_iℱ̂ ̂l̂_𝔾_v_i,R. Let 𝒮(1):= L^+__i𝔾__i· 1⊂ℱ l_𝔾_v_i

be the closed Schubert variety 
and 𝒮(1)_R=𝒮(1)×__q Spf R
  then Ẑ__i,R×𝒮(1)_R×…×𝒮(1)_R defines a bound in ℱ̂ ̂l̂_𝔾_ that we also denote by Z__i×𝒮(1)×…×𝒮(1). We have the following lemma.



Let ∈(S) as before. The local 𝔾_v_i-shtuka Γ_v_i() is bounded by Ẑ_v_i if and only if the local 𝔾_v_i-shtuka L^+_() is bounded by Ẑ_v_i×𝒮(1)×…×𝒮(1).



We choose an étale covering S' of S that trivializes L^+_v_i() as well as σ^⋆ L^+_v_i(). In particular S' trivializes also Γ_v_i() and σ^d⋆Γ_v_i(). We fix such trivializations and call them α:L^+_v_i()_S'⟶ L^+𝔾_v_i_S', α':σ^⋆ L^+_v_i()_S'⟶ L^+𝔾_v_i_S',  α:Γ_v_i()_S'⟶𝔾×_CV(𝔞_v_i,0) and α':σ^d⋆Γ_v_i()_S'⟶𝔾×_CV(𝔞_v_i,0).

Denote by τ_j the Frobenius morphism τ_ restricted to V(𝔞_v_i,j) for j=0,…, d-1, where d=deg v_i=[_v_i:_q]. So the local shtuka Γ_v_i() is given by (×_C V(𝔞_v_i,0),τ_0∘σ^⋆τ_1∘…∘σ^(d-1)⋆τ_d-1) and α∘τ_0∘σ^⋆τ_1∘…∘σ^(d-1)⋆τ_d-1∘α'^-1:L𝐆_v_i,S' L𝐆_v_i,S' computed in ℋ^1(_v_i, L𝐆_v_i)(S') defines a morphism S'→ L𝐆_v_i. Now Γ_v_i() is bounded by Z_v_i if and only if the morphism 
S'×_R_Z__iSpf R→ L𝐆_v_i×__v_iSpf R→ℱ̂ ̂l̂_𝔾_v_i,R factors through Z_v_i,R. Since τ is an isomorphism outside the graphs of s_i, τ_1, …, τ_d-1 are isomorphisms. Hence σ^⋆τ_1∘…∘σ^(d-1)⋆τ_d-1∘α'^-1 comes from some other trivialization β'^-1:𝔾×_C V(𝔞_v_i, 0)→σ^⋆Γ_v_i(). This shows that τ_0∘σ^⋆τ_1∘…∘σ^(d-1)⋆τ_d-1 is bounded by Z_v_i if and only if τ_0 is bounded by Z_v_i. Now α∘τ∘(α')^-1:L𝐆_v_i_,S' L𝐆_v_i_,S' computed in ℋ^1(_q,L𝐆_v_i)(S') defines in the same way a morphism S'→ L𝐆_v_i which induces a morphism S'×_R_Z__iSpf R→ℱ l_𝔾_v_i×__qSpf R=∏_l∈/deg v_iℱ l_𝔾_v_i×___iSpf R. Note that 

the morphism in the j-th component of ∏ℱ l_𝔾_v_i×__v_iSpf R is exactly defined by α∘τ_j∘ (α')^-1. Since τ_1,…, τ_d-1 are isomorphisms the morphism into the j-th component with j 1 always factors through 𝒮(1)_R.  This implies that τ is bounded by Z_v_i×𝒮(1)×…×𝒮(1) if and only if τ_0 is bounded by Z_v_i.




Now we can prove:


Let (C,𝔾, , Ẑ_, H) be a shtuka datum and π:C→ C' a finite morphism of smooth, projective, geometrically irreducible curves over _q with _i:=π(_i) and :=(_1,…,_n). Then the morphism (π,id_π_⋆𝔾):(C,𝔾, , Ẑ_, H)→ (C',π_⋆𝔾, , π_⋆Ẑ_, π_⋆ H) of shtuka data (see definition <ref> and remark <ref>) induces a finite morphism of the moduli stacks

    π_⋆: ∇_n^Ẑ_,Hℋ^1(C,𝔾)→∇_n^π_⋆Ẑ_,π_⋆ Hℋ^1(C',π_⋆𝔾)
    
    .



Let S be an _q-scheme and (,s_1,…, s_n, τ_, γ)∈∇_n^Ẑ_,Hℋ^1(C,𝔾)(S). We describe again its image (',s_1',…,s_n',τ_', γ') in ∇_n^Ẑ_,π_⋆ Hℋ^1(C',π_⋆𝔾)(S) to define the morphism. 
The π_⋆𝔾-torsor '=(',s_1', …, s_n', τ_') is already defined by the morphism in proposition <ref>, but we have to prove that it lies indeed in ∇_n^Ẑ_wℋ^1(C, π_⋆𝔾)(S). We will do this first and then define the π_⋆ H-level structure γ'. Since the section s_i:S→ C is mapped to s_i':=π∘ s_i and s_i is required to factor through Spf A_v_i, we easily see with  π(v_i)=w_i that s_i' factors through Spf A_w_i. Furthermore this shows that C_S'\⋃_iΓ_s_i'⊃ C'^⋆×__qS and C_S\⋃_iΓ_s_i⊃ C^⋆×__qS where we use the notation 
C^⋆=C\{_1,…,_n} and C'^⋆=C'\{_1,…_n}. It remains to show that τ_' is bounded by Z_w to see '∈∇_n^Ẑ_wℋ^1(C',π_⋆𝔾)(S).


Now by assumption τ_ is bounded by Z_v_i, which means by definition that the local shtukas Γ_v_i()

are bounded by Z_v_i for all i. 
By lemma <ref> this is equivalent to the fact that L^+_v_i()

is bounded by Z_v_i×𝒮(1)×…×𝒮(1). Now consider the following 2-cartesian diagram (compare <ref> and <cit.>) where we set U:=C'^⋆×_C'C.

    ℋ^1(C,𝔾) [r] [d]_∏_L^+_   ℋ_e^1(U,𝔾) [d]_∏_L_
    ∏_∈π^-1()ℋ^1(_q, L^+𝔾_) [r]    ∏_∈π^-1()ℋ^1(_q, L𝐆_)
     .

Here ℋ_e^1 (U, 𝔾)(S) is the full subcategory of ℋ^1(U, 𝔾)(S) consisting of those 𝔾-torsors over U_S that can be extended to a 𝔾-torsor over C_S. Now the categories ℋ^1(C,𝔾) and ℋ^1_e(U, 𝔾) are by lemma <ref> equivalent to ℋ^1(C',π_⋆ G) and ℋ^1_e(C'^⋆,π_⋆). Furthermore the categories 

    ∏_∈π^-1()ℋ^1(_q, L^+𝔾_)   =∏_∈ℋ^1(_q, ∏_∈π^-1() L^+𝔾_)
    ∏_∈π^-1()ℋ^1(_q, L𝐆_)   =∏_∈ℋ^1(_q, ∏_∈π^-1() L𝐆_)

are 
by corollary <ref> equivalent to ∏_∈ℋ^1(_q, L^+π_⋆𝔾_) and ∏_∈ℋ^1(_q, Lπ_⋆𝐆_). Therefore the whole diagram (<ref>) is equivalent to the diagram

    ℋ^1(C',π_⋆𝔾) [r] [d]_∏_L^+_   ℋ_e^1(C'^⋆,π_⋆𝔾) [d]_∏_L_
    ∏_∈ℋ^1(_q, L^+π_⋆𝔾_) [r]    ∏_∈ℋ^1(_q, Lπ_⋆𝐆_) .


Now we choose some covering S' over S that trivializes L^+_, σ^⋆ L^+_ for all ∈π^-1() and fix some trivializations α_:L_^+()_S'→ L^+𝔾__S',  α'_:σ^⋆ L_^+()_S'→ L^+𝔾__S'. 

Then (, τ) defines a tuple ∏_∈(∏_|L^+𝔾_,S', ∏_|α_∘ L_(τ|_U)∘α'_^-1) and the equivalence of the diagrams shows that it corresponds to the tuple ∏_∈(L^+π_⋆𝔾__,S', α_∘ L_(π_⋆τ|_C'^⋆)∘α'_^-1) defined in the same way by the shtuka (π_⋆, π_⋆τ). Here α_,α_' are the trivializations corresponding to ∏_|α_ and ∏_|α'_.

Now choose some finite extension R⊃_qζ such that there are representatives Z_,R for all ∈. Using the 2-cartesian diagram 

    ∏_∈π^-1()ℱ l_𝔾_
    [r] [d]    _q
    [d] 
    ∏_∈π^-1()ℋ^1(_q, L^+𝔾_) 
    [r]    ∏_∈π^-1()ℋ^1(_q, L𝐆_)

the tuple (∏_|L^+𝔾_,S',  ∏_|α_∘ L_(τ|_U)∘α'_^-1) defines an S'×_R_ẐSpf R-valued point in ∏_|ℱ l_𝔾_×__qSpf R=∏_|∏_l∈/deg ℱ l_𝔾_×__Spf R. 

By lemma <ref> the boundedness of  at all the points v∈ (v∩π^-1(w)) by Z_v 

is equivalent to the boundedness of L^+_v() by 

∏_l∈/deg Z_,l with Z_v,0=Z_v for all v∈v and Z_,l=𝒮(1)_R for all v∉v and l≠ 0, which means by definition, that the above S'×_R_ẐSpf R valued point factors through ∏_|∏_l∈/deg Z_,l. 
The tuple (L^+π_⋆𝔾_, S', S',α_∘ L_(π_⋆τ|_C^⋆)∘α_'^-1) defines in the same way a morphism S'×_R_ẐSpf R→ℱ l_(π_⋆𝔾)_×__qSpf R=∏_l∈/deg ℱ l_π_⋆𝔾_×__Spf R. 
Composing with the isomorphism ∏_|ℱ l_𝔾_≃ℱ l_π_⋆𝔾_, the above morphism factors also through ∏_|∏_l∈/deg Z_,l. With lemma <ref> and the definition of π_⋆ Z_ on page defpiz it follows, that π_⋆𝒢 is bounded by π_⋆ Z_.
 

Next we have to define the π_⋆ H-level structure γ'. We fix a geometric base point s∈ S and we choose for all closed points ∈ C\ a trivialization L^+_𝒢_s≃ (L^+𝔾__,κ(s), τ=1), which exists by <cit.>. This provides also trivializations 

    L^+_(π_⋆𝒢_s)=∏_| L^+_𝒢_s≃∏_|L^+𝔾__,κ(s)=L^+π_⋆𝔾__,κ(s).

We denote by ℒ_ and ℒ_ the shtukas L^+_𝒢_s and L^+_(π_⋆𝒢_s).
Now these trivializations induce isomorphisms 

    β:ω^∘_𝕆^= ∏_∈ C\𝒯_L^+𝔾_≃   ∏_∈ C\𝒯_ℒ_ =𝒯_𝒢
    π_⋆β:ω^∘_𝕆^= ∏_∈ C'\𝒯_L^+π_⋆𝔾_≃   ∏_∈ C'\𝒯_ℒ_  =𝒯_π_⋆𝒢.

We write ω^∘_𝔸^:=ω^∘_𝕆^⊗_𝕆^𝔸^ and ω'_𝔸^^∘:=^∘_𝕆^⊗_𝕆^𝔸^. Now β^-1∘γ∈ Aut^⊗(ω^∘_𝔸^) is given by an element g∈𝔾(𝔸^) and the H-orbit of γ is β∘ gH. Now we can use the projection 𝔾(𝔸^)↠𝔾(𝔸^π^-1())=π_⋆𝔾(𝔸^) to define π_⋆ g∈π_⋆𝔾(𝔸^) as the image of g. This corresponds to an element in Aut(ω'_𝔸^^∘). Therefore π_⋆β∘π_⋆ g
defines an element 

γ' and consequently an π_⋆ H-orbit in Isom^⊗(ω'_𝔸^^∘, 𝒱̌_π_⋆𝒢). This orbit is independent of the representative γ since π_⋆ H was defined as the image of H under the above projection. 
Let ρ∈π_1(S,s) since γ H⊂ Isom^⊗(ω^∘_𝔸^, 𝒱̌_𝒢) is π_1(S,s) invariant, we know that there is h∈ H such that ργ=γ h. This defines a group homomorphism φ: π_1(S,s)→ H and we set
φ:π_1(S,s)→ H→π_⋆ H. Let ρ∈π_⋆(S,s) and  γ'∈ Isom^⊗(ω'^∘_𝔸^, 𝒱̌_π_⋆𝒢) be as above, then ρ operates by ργ =γ'φ(ρ) and in particular π_1(S,s)γ'⊂γ'π_⋆ H. This means that the orbit γ' π_⋆ H is π_1(S,s) invariant and defines a level structure γ' of '.
 

After constructing this morphism, we now prove that it is representable by a scheme and finite. 
By proposition <ref> it is clear that the morphism ∇_n^̂̂Z_vℋ^1(C,𝔾)→∇_n^Ẑ_wℋ^1(C',π_⋆𝔾) is finite.
















Now we find some finite subscheme D⊂ C such that H_D:=ker(𝔾(^)→𝔾(_D)) is a subgroup of finite index in H. 
Then we have by <ref> the following diagram: 

    ∇_n^Ẑ_v, Hℋ^1(C,𝔾) [d] 
       ∇_n^Ẑ_v,H_Dℋ^1(C,𝔾) [d] [r]^∼[l]_finite étale   ∇_n^Ẑ_vℋ^1_D(C,𝔾) [d] [r]^finite étale   ∇_n^Ẑ_vℋ^1(C,𝔾) [d]^finite
    
    
    
    
    
    
    ∇_n^Ẑ_w,π_⋆ Hℋ^1(C',π_⋆𝔾)
       ∇_n^Ẑ_w,H_π_⋆ Dℋ^1(C',π_⋆𝔾) [r]^∼[l]_finite étale   ∇_n^Ẑ_wℋ^1_π_⋆ D(C',π_⋆𝔾) [r]^finite étale   ∇_n^Ẑ_wℋ^1(C',π_⋆𝔾)


where the horizontal arrows are finite (and even étale) by <cit.>. This implies firstly that the morphism ∇_n^Ẑ_v,H_Dℋ^1(C,𝔾) →∇_n^Ẑ_w,H_π_⋆ Dℋ^1(C',π_⋆𝔾) is finite and consequently that the morphism ∇_n^Ẑ_v, Hℋ^1(C,𝔾)  →∇_n^Ẑ_w,π_⋆ Hℋ^1(C',π_⋆𝔾) is finite.





 §.§ Changing the Group 𝔾



Now let f:𝔾→𝔾' be a morphism of smooth, affine group schemes over C. In this subsection we explain how this induces a morphism between the moduli stacks of 𝔾-shtukas and 𝔾'-shtukas. Further we prove some of its properties, depending on f. First of all we recall, that given a sheaf M on C_Ét with an action of 𝔾, we can define the sheaf M×^𝔾𝔾' whose R-valued points are given by the set {(a,b) | a∈ M(R), b∈𝔾'(R)}/∼, where (a,b)∼ (c,d) if and only if (a,b)=(cg, f(g^-1)d) for some g∈𝔾(R). Actually this construction works for any sheaf of groups on any site. Now this construction is functorial for 𝔾-equivariant morphisms φ:M_1→ M_2 and commutes obviously with base change. We also write f_⋆ M=M×^𝔾𝔾' and note that if M is a 𝔾-torsor then f_⋆ M is a 𝔾'-torsor.
With these facts we see that for a given 𝔾-shtuka (𝒢, s_1,…, s_n, τ_𝒢) over S, the tuple (f_⋆𝒢, s_1, …, s_n, f_⋆τ_𝒢) defines a 𝔾'-shtuka over S. Therefore we get a morphism

    ∇_nℋ^1(C,𝔾) ⟶∇_nℋ^1(C,𝔾')     (𝒢, s_1,…, s_n, τ_𝒢)↦ (f_⋆𝒢, s_1, …, s_n, f_⋆τ_𝒢).



Now we want to show that this morphism also induces a morphism of these moduli stacks with additional H-level structure. So we fix n closed points v=(v_1,…,v_n) in C and let H⊂𝔾(^) be an open and compact subgroup. Let further S be a connected _q-scheme with a geometric base point s∈ S and (,γ)=(, s_1,…,s_n,τ_, γ) be a 𝔾-shtuka over S with an H-level structure γ H. We already mentioned that by <cit.> f:𝔾→𝔾' induces a continuous homomorphism f_^:𝔾(^)→𝔾'(^) (see also above definition <ref>). For an open, compact subgroup H'⊂𝔾'(^) satisfying f_^(H)⊂ H' we now construct an H'-level structure on the shtuka f_⋆=(f_⋆,s_1,…,s_n,f_⋆τ_). 

We choose for every v∈C=C\ a trivialization α_v:L^+_v(_s) (L^+𝔾_v_,s,1·σ^⋆) which exists by <cit.>. 
Since f_⋆ commutes with base change this induces trivializations
f_⋆α_v:L^+_v((f_⋆)_s) (L^+𝔾'_v_,s,1·σ^⋆). We denote by ω^∘_𝕆^:Rep_^𝔾→𝔐𝔬𝔡_^[π_1(S,s̅)] and ω'^∘_𝕆^:Rep_^𝔾'→𝔐𝔬𝔡_^[π_1(S,s̅)] the forgetful functors and by ℒ_v and ℒ'_v the local shtukas 
L^+_v(_s) and L^+_v((f_⋆)_s). Then the previous trivializations provide isomorphisms of tensor functors

    β:ω^∘_𝕆^= ∏_∈ C\𝒯_L^+𝔾_   ∏_∈ C\𝒯_ℒ_ =𝒯_𝒢
    
    f_⋆β:ω'^∘_𝕆^= ∏_∈ C\𝒯_L^+𝔾'_   ∏_∈ C\𝒯_ℒ'_  =𝒯_f_⋆𝒢.




It follows that β^-1∘γ∈ Aut^⊗(ω_^^∘) is given by an element g∈𝔾(^) and the H-orbit of γ is given by β∘ gH. Now we view the image f(g) of g under the map f_^:𝔾(^)→𝔾'(^) as an automorphism in 
Aut^⊗(ω'^∘_𝕆^)
and define γ':=f_⋆γ := f_⋆β∘ f(g). Since f_^(H)⊂ H' the H' orbit of f(g) is independent of the chosen representative γ in the orbit γ H. 
Since π_1(S,s̅) leaves γ H invariant there is for all ρ∈π_1(S,s̅) an h∈ H such that ρ·γ=γ· h. This defines a group homomorphism φ:π_1(S,s̅)→ H and we set φ ':π_1(S,s̅)→ HH'. Now ρ∈π_1(S,s̅) operates on γ'∈ Isom(ω',𝒱̌_) by ρ·γ'=γ'·φ̅'(ρ). In particular π_1(S,s̅)γ'⊂γ'H' so that γ'H' is π_1(S,s̅) invariant and defines a H'-level structure on f_⋆. A morphism (,γ) to (ℱ, η) induces naturally a morphism (f_⋆, γ')→ (f_⋆ℱ, η') so that we get a morphism of moduli stacks 

    ∇_n^Hℋ^1(C,𝔾)→∇_n^H'ℋ^1(C,𝔾')   (, γ)↦ (f_⋆, γ').









Next we show that this morphism behaves well with respect to boundedness conditions.
We note that for all ∈ the morphism f:𝔾→𝔾' induces a morphism L^+𝔾_v_i→ L^+𝔾'_v as well as a morphism L𝔾_v→ L𝔾'_v and consequently also a morphism ℱ𝑙_𝔾_v→ℱ𝑙_𝔾'_v.


Let Ẑ_v be a bound in ∏_i-1^nℱ𝑙_𝔾_v_i and 𝒢 a 𝔾-shtuka over S bounded by Ẑ_v.
Let further Ẑ'_v be a bound in  ∏_i=1^nℱ𝑙_𝔾'_v_i such that after choosing representatives over some DVR R the morphism Ẑ_v, R→∏_i=1^nℱ𝑙_𝔾'_v_i factors through Ẑ'_v,R.
Then f_⋆𝒢 is bounded by  Ẑ'_v.


We have to prove that for v∈ the local shtuka Γ_v(f_⋆𝒢) is bounded by Ẑ'_v. We choose some covering S'→ S with S'/Spec R that trivializes L^+_vσ^⋆𝒢 and L^+_v𝒢 at the same time and fix such trivializations, which we denote by
α:L^+_vσ^⋆𝒢_S'→ L^+𝔾_v,S' and α':L^+_v𝒢_S'→ L^+𝔾_v,S'. Then f_⋆α and f_⋆α' are trivializations of L_v^+(f_⋆σ^⋆𝒢)_S' and L_v^+(f_⋆𝒢)_S'. Now we have the automorphism α'∘τ^deg v∘α^-1:L𝐆_v,S' L𝐆_v,S' and we let 1_S':S'→ L𝐆_,S' be the unit morphism. The composition defines an S' valued point 1_S'∘α'∘τ^deg v∘α^-1 in L𝐆_v,S'. 
The composition of this point with the morphism L𝔾_v,S'→ L𝔾'_v,S' induced by f defines an S'-valued point in L𝔾'_v,S'. Since the diagram 

    S' [r]^1_S'  [rd]     L𝐆_v,S'[d] [rrr]^α'∘τ^deg v∘α^-1          L𝐆_v,S'[d] 
        L𝐆'_v,S'[rrr]^f_⋆(α'∘τ^deg v∘α^-1)          L𝐆'_v,S'

commutes, this is exactly the S' valued point defined by 

    f_⋆(α')f_⋆(τ^deg v)f_⋆(α^-1)=f_⋆(α'τ^deg vα^-1).

By assumption 𝒢 is bounded by Ẑ_ and consequently the morphism 1_S'∘α'∘τ^deg v∘α^-1 factors after projection to ℱ𝑙_𝔾_v,S' through Ẑ_v,R and maps then into Ẑ'_v,R. This means exactly that Γ_v(f_⋆𝒢) is bounded by Ẑ'_v, so that f_⋆𝒢 is bounded by Ẑ'_.



This lemma and the previous explanations show.

 
The morphism (id,f):(C,𝔾,,Ẑ_,H)→ (C,𝔾',,Ẑ'_,H') of shtuka data 

    f_⋆ :  ∇_n^Ẑ_,Hℋ^1(C,𝔾)    ⟶∇_n^Ẑ'_, H'ℋ^1(C,𝔾')
     
     (𝒢, s_1,…, s_n, τ_𝒢,γ)   ⟼ (f_⋆𝒢, s_1, …, s_n, f_⋆τ_𝒢,f_⋆γ).

Follows directly from lemma <ref> and the morphism (<ref>) on page changethegroup2.





Now we are interested in some special classes of morphisms f:𝔾→𝔾'.



  §.§.§ Generic Isomorphisms of 𝔾


First of all we want to consider morphisms f:𝔾→𝔾' which are generically an isomorphism, that means f× id_Q:𝐆𝐆'. In this case f:𝐆→𝐆' is already an isomorphism over some open subscheme in C. So we fix such an f:𝔾→𝔾' and denote by U the maximal open subscheme in C such that f× id_U:𝔾_U→𝔾'_U is an isomorphism and denote by w=(w_1,…,w_m) the finite set of closed points in the complement C\ U.

Before we come to the moduli stacks of the global 𝔾-shtukas, we prove a proposition that describes the morphism ℋ^1(C,𝔾)→ℋ^1(C,𝔾'). For this proposition we need the following lemma.


Let ℒ'_+ be an L^+𝔾'_w torsor over an _q-scheme S. Then the quotient stack [ℒ'_+/L^+𝔾_w] is represented by a scheme ℒ'_+/L^+𝔾_w over S that is étale locally on S isomorphic to 

L^+𝔾'_w/L^+𝔾_w. In the case that 𝔾_w is parahoric ℒ'_+/L^+𝔾_w is projective.



Let ℒ':=ℒ'_+×^L^+𝔾'_wL𝐆'_w be the associated L𝐆'_w-torsor of ℒ'_+. By <cit.> the quotient stack [ℒ'/L^+𝔾_w] is represented by an ind-quasi-projective ind-scheme ℒ'/L^+𝔾_w over S. The closed morphism ℒ'_+→ℒ' realizes [ℒ'_+/L^+𝔾_w]
 as a closed sub-sheaf of ℒ'/L^+𝔾_w. Since ℒ'_+ is affine over S the quotient [ℒ'_+/L^+𝔾_w] is given by a closed subscheme in ℒ'/L^+𝔾_w. It is clear that after passing to a covering S'→ S that trivializes ℒ'_+, the scheme ℒ'_+/L^+𝔾_w becomes isomorphic to L^+𝔾'_w/L^+𝔾_w×__qS'. Since ℒ'/L^+𝔾_w is by <cit.> ind-projective if 𝔾_w is parahoric, we see that the last statement about the projectivity of ℒ'_+/L^+𝔾_w follows. 
 




Now we can prove:


Let f:𝔾→𝔾' be a morphism of smooth, affine group schemes over C, which is an isomorphism over C\w. Then the morphism 

    f_⋆: ℋ^1(C,𝔾)→ℋ^1(C,𝔾'),   ↦ f_⋆

is schematic and quasi-projective. Étale locally it
is relatively representable by the morphism 

    (L^+𝔾'_w_1/ L^+𝔾_w_1)×__q…×__q(L^+𝔾'_w_m/ L^+𝔾_w_m)⟶_q.


That means that for any _q-morphism S→ℋ^1(C,𝔾') there is an étale covering S'→ S such that the fiber product S'×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) is given by S'×__q(∏_w∈wL^+𝔾'_w/ L^+𝔾_w), where the product is taken over _q.
In the case that the fibers 𝔾_ for all ∈ are parahoric group schemes this morphism is projective.


Let S→ℋ^1(C,𝔾') be given by a 𝔾-torsor ' over C_S. Let g:T→ S be an S-scheme. Then a T-valued point of the fiber product S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) is given by a tuple (g,,α) where ∈ℋ^1(C,𝔾) and α:f_⋆ g^⋆ '. Using the theorem of Beauville-Laszlo from <ref> we write =(|_U_T,∏_w∈wℒ_w, φ) with U_T:=(C\w)×__q T, ℒ_w∈ℋ^1(_q,L^+𝔾_w)(T) and φ=(φ_w)_w∈w:∏_w∈wL_w()∏_w∈wL(ℒ_w). In the same way we write '=('|_U_S,∏_w∈wℒ'_w, ψ). In particular f_⋆ is given by (f_⋆(|_U_T),f_w,⋆ℒ_w, f_⋆φ) and the isomorphism α is determined by  α_U:f_⋆→ g^⋆ ' and α_w:f_w,⋆ℒ_w→  L_w^+(g^⋆ ') satisfying

    L_w(f_⋆(|_U_T)) [r]^f_⋆φ[d]^L_w(α_U)    L(f_w,⋆(ℒ_w)) [d]^L(α_w)
    
    L_w(g^⋆('|_U_T)) [r]^g^⋆ψ    L(g_w^⋆ℒ'_w)
         .

Since f|_U=id we have f_⋆(|_U_T)=|_U_T and the point (|_U_T,∏_w∈wℒ_w, φ) is equivalent to the point (g^⋆'|_U_T,∏_w∈wℒ_w, (φ_w∘ L_w(α_U^-1))) by the isomorphism (α_U^-1,∏ id_ℒ_w).
This shows that the category of tuples (,α) as above is equivalent to the category of tuples (ℒ_w,α_w)_w∈w where ℒ_w∈ℋ^1(C,L^+𝔾_w)(T) and α_w:f_⋆ℒ_w g^⋆ℒ'_w. Namely we associate with some arbitrary tuple (ℒ_w,α_w)_w∈ D the tuple ((g^⋆'|_U_T,ℒ_w, φ),β) where β|_U=id and β_w=α_w and φ is uniquely determined by the condition f_⋆φ=ψ∘ L(α_w^-1). This is unique because f× id_Q_w:𝐆_w→𝐆'_w is an isomorphism. 

Now we note that the isomorphisms α_w:f_⋆ℒ_w g^⋆ℒ'_w are in bijection with the L^+𝔾_w equivariant morphisms ℒ_w→ g^⋆ℒ'_w. This shows that the tuples (ℒ_w,α_w) parametrize exactly the T-valued points of the quotient stack [ g^⋆ℒ_w/L^+𝔾_w] over _q. It follows with the lemma <ref> that the fiber product S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) is given by the scheme g^⋆ℒ_w_1/L^+𝔾_w_1×…× g^⋆ℒ_w_m/L^+𝔾_w_m. In particular the
morphism f_⋆ is representable and the remaining statements follow directly from the previous lemma.





Now let us turn to the moduli stacks of global 𝔾-shtukas. 
Let us firstly assume that w⊂ and that all the closed points  are _q-rational. In particular the group homomorphism f:𝔾→𝔾' is an isomorphism outside the fixed characteristic places _1,…,_n. Then we have the following theorem.


Let f:𝔾→𝔾' be a morphism of smooth, affine group schemes over C, which is an isomorphism over C\w with ⊂ and w_i∈ C(_q) for all w_i∈w. Let H⊂𝔾(^)=𝔾'(^) be an open, compact subgroup, let Ẑ'__i be a bound in _𝔾'__i for all i and let Ẑ__i be the base change of Ẑ'__i under the map _𝔾__i→_𝔾'__i. Then the morphism 

    f_⋆: ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ'_, Hℋ^1(C,𝔾'),    (,γ H) ↦ (f_⋆,γ H)

 is schematic and quasi-projective. Étale locally it is relatively representable by the morphism


    (L^+𝔾'_w_1/ L^+𝔾_w_1)×__q…×__q(L^+𝔾'_w_m/ L^+𝔾_w_m)⟶_q.

That means that for any _q-scheme S there is an étale covering S'→ S such that the fiber product S'×_∇_n^Ẑ'_, Hℋ^1(C,𝔾)∇_n^Ẑ_, Hℋ^1(C',𝔾') is given by S'×__q(∏_w∈wL^+𝔾'_w/ L^+𝔾_w), where the product is taken over _q. In particular f_⋆ is a surjective morphism.
In the case that 𝔾 is a parahoric Bruhat-Tits group scheme this morphism is projective.



Since f is an isomorphism outside , for two open subgroups H̃⊂ H⊂𝔾(^)=𝔾'(^) the diagram 

    ∇_n^Ẑ_, Hℋ^1(C, 𝔾) 
    [d]^f_⋆[r]
       ∇_n^Ẑ_, Hℋ^1(C, 𝔾)
    [d]^f_⋆
    ∇_n^Ẑ'_, Hℋ^1(C, 𝔾') 
    [r]
       ∇_n^Ẑ'_,  Hℋ^1(C, 𝔾')
 is cartesian. In particular we can assume H⊂𝔾(^)=𝔾'(^), because otherwise we can prove the theorem for the compact open subgroup H:=H∩𝔾(^). This implies the assertions of the theorem for the group H since the vertical arrows on the left and the right in the previous diagram are relatively represented by the same morphism.
Now for each S-valued point (,γ H) in ∇_n^Ẑ_, Hℋ^1(C,𝔾) we find an isomorphic point (',γ'H) with γ'∈ Isom^⊗(ω_^^∘, 𝒯̌_').

This is due to the fact, that we can pull back global 𝔾-shtukas along quasi-isogenies of local 𝔾_-shtukas <cit.> and is explained in the proof of <cit.>. We get a morphism ∇_n^Ẑ_,Hℋ^1(C,𝔾)→ℋ^1(C,𝔾) sending (,γ H)=(',γ'H) to '. This is the morphism ∇_n^Ẑ_,Hℋ^1(C,𝔾)→∇_n^Ẑ_,𝔾(^)ℋ^1(C,𝔾)
from (<ref>) in <ref> composed with the morphism (<ref>) with D=∅ in <ref> and the natural morphism ∇_nℋ^1(C,𝔾)→ℋ^1(C,𝔾).
  Now using proposition <ref> it suffices to prove that ∇_n^Ẑ_,Hℋ^1(C,𝔾) is given by the fiber product 
  
    ℳ:=∇_n^Ẑ'_,Hℋ^1(C,𝔾')×_ℋ^1(C,𝔾')ℋ^1(C,𝔾).

   There is a natural morphism p:∇_n^Ẑ_, Hℋ^1(C,𝔾)→ℳ which sends an S-valued point (, γ H), where we can assume as before γ∈ Isom^⊗(ω_^^∘, 𝒯̌_), to ((f_⋆,γ H), , id_f_⋆), which is well defined by <ref>. We need to prove that this morphism induces an equivalence of the fibered categories. First we see that it is fully faithful. Let (_1,γ_1 H) and (_2,γ_2 H) be two S-valued points in ∇_n^Ẑ_, Hℋ^1(C,𝔾), where we assume again γ_1,γ_2∈ Isom^⊗(ω_^^∘, 𝒯̌_). Let g∈ Hom((_1,γ_1 H),(_2,γ_2 H)). Since 𝒱̌_g∘γ_1=γ_2 mod H we see that 𝒱̌_g= γ_2∘ h∘γ_1^-1 for some h∈ H, which implies that 𝒱̌_g already comes from a tensor isomorphism in Isom^⊗(𝒯̌__1,𝒯̌__2). By <cit.> it follows that g is not only a quasi-isogeny but also a morphism of the global 𝔾-shtukas _1→_2. Therefore Hom((_1,γ_1 H),(_2,γ_2 H)) equals the morphisms of 𝔾-torsors such that g is a morphism of the global 𝔾-shtukas _1 and _2 compatible with the level structure. Since f_⋆ is an isomorphism outside of  the latter condition is equivalent to the statement that f_⋆ g is a morphism of 𝔾'-shtukas compatible with the level structure. But this says exactly that 
  
    Hom((_1,γ_1 H),(_2,γ_2 H))=Hom_ℳ(((f_⋆_1,γ_1 H), _1, id_f_⋆_1),((f_⋆_2,γ_2 H), _2, id_f_⋆_2)).

  For the essential surjectivity let ((E,s_1,…,s_n,τ_E,γ_EH),,ψ) be an S-valued point in ℳ, with γ_E∈ Isom^⊗(ω_^^∘, 𝒯̌_E) as before. This is isomorphic to 
  
    ((f_⋆, s_1,…,s_n,σ^⋆ψ∘τ_E∘ψ^-1, 𝒯̌_ψ∘γ_EH),,id_f_⋆)

   by (ψ^-1,id_). We need to show that it comes from an element 
  
    (, γ_ H)=(, s_1,…,s_n,τ_,γ_ H)∈∇_n^Ẑ_, Hℋ^1(C,𝔾).

   Here s_1,…,s_n and  are already uniquely defined. 
  Therefore we need to define the isomorphism τ_:σ^⋆|_C_S\i∪Γ_s_i→|_C_S\i∪Γ_s_i. Since all the closed points  are _q-rational (C\w)_S is contained in C_S\i∪Γ_s_i. Note that this is not the case if w_i splits, because in this case w_i×__qS has deg w_i components isomorphic to S and Γ_s_i surjects only to one of these. By assumption f× id_S:𝔾_S→𝔾'_S is an isomorphism over (C\w)_S. This together with the fact that τ_ has to satisfy f_⋆τ_=σ^⋆ψ∘τ_E∘ψ^-1 an the inclusion (C\w)_S⊂ C_S\i∪Γ_s_i defines therefore a unique τ_:σ^⋆|_C_S\i∪Γ_s_i→|_C_S\i∪Γ_s_i. 

  Now  is a global 𝔾-shtuka with H-level structure γ_:=𝒯̌_ψ∘γ_E that is mapped to ((E,γ_E H), , ψ)∈ℳ(S). It just remains to prove that  is bounded by Ẑ_ to see that (, γ_) lies indeed in ∇_n^Ẑ_, Hℋ^1(C,𝔾). Let R be an extension of A__i with representatives Ẑ__i,R and Ẑ'__i,R of the bounds Ẑ__i and Ẑ'__i. We choose an étale covering S' of S and trivializations α:L^+𝔾__i,S'→Γ__i(_S') and α':L^+𝔾__i,S'→Γ__i(σ^⋆_S'). Then α^-1∘τ_∘α'=(f_)_⋆(α^-1∘τ_∘α'):L𝐆__i,S'→ L𝐆__i,S'
defines an S'-valued point of L𝐆__i and hence an induced morphism S'→_𝔾__i,R→_𝔾'__i,R. By assumption E and hence f_⋆ is bounded by Ẑ'_. This means that this morphism factors through Ẑ'__i,R and since Ẑ__i arises from base change it factors by the universal property of the fiber product also through Ẑ__i,R. This shows that  is bounded by Ẑ__i for all _i∈.




If (id_C,f):(C,𝔾,,Ẑ_,H)→ (C,𝔾',,Ẑ'_,H) is a morphism of shtuka data, where Ẑ_ does not arise as a base change of Ẑ'_ or if f:𝔾→𝔾' is an isomorphism outside  w without any conditions relating w to the characteristic points  or their residue field, the morphism 

    f_⋆: ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ'_, Hℋ^1(C,𝔾'),    (,γ H) ↦ (f_⋆,γ H)

 is still representable, but in general not surjective anymore. More precisely, we have the following theorem.



Let w=(w_1,…,w_m) be a finite set of closed points in C and let 

    (id_C,f):(C,𝔾,,Ẑ_,H)→ (C,𝔾',,Ẑ'_,H)
 be a morphism of shtuka data, where f:𝔾→𝔾' is an isomorphism over C\w. Then the morphism 

    f_⋆: ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ'_, Hℋ^1(C,𝔾'),    (,γ H) ↦ (f_⋆,γ H)

is schematic and quasi-projective. In the case that 𝔾 is a parahoric Bruhat-Tits group scheme this morphism is projective.
For any morphism (', γ' H):S→∇_n^Ẑ'_, Hℋ^1(C,𝔾') 

    S×_∇_n^Ẑ'_, Hℋ^1(C,𝔾')∇_n^Ẑ'_, Hℋ^1(C,𝔾)


    S×__q((L^+_w_1(')/ L^+𝔾_w_1)×__q…×__q(L^+_w_m(')/ L^+𝔾_w_m)).

If Ẑ_ arises as a base change of Ẑ'_ for all ∈, the morphism f_⋆ is surjective.





In the case that Ẑ__i does not arise by base change from Ẑ__i the immersion Ẑ__i→_𝔾__i factors through the base change Ẑ”__i:=Ẑ'__i×__𝔾'__i_𝔾__i. Since ∇_n^Ẑ_, Hℋ^1(C,𝔾)→∇_n^Ẑ”_,Hℋ^1(C,𝔾) is a closed substack we may therefore assume from the beginning that Ẑ__i arises by base change from Ẑ'__i for all _i∈. Furthermore we can as in the previous theorem assume that H⊂𝔾(^).
Let S→∇_n^Ẑ'_, Hℋ^1(C,𝔾') be an S-valued point given by (',γ'H)=(',s_1',…,s_n',τ_', γ'H), where we can assume as before that γ'∈ Isom^⊗(ω_^^∘, 𝒯̌_'). There is a 
natural morphism 

    S×_∇_n^Ẑ'_, Hℋ^1(C,𝔾')∇_n^Ẑ_, Hℋ^1(C,𝔾)→ S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾)

 sending an T-valued point (g,, γ H, ψ) to (g,, ψ), where g:T→ S is a morphism of schemes, (, γ H) is a T-valued point in ∇_n^Ẑ_, Hℋ^1(C,𝔾) and ψ:f_⋆(, γ H) g^⋆(',γ'H) is an isomorphism of global 𝔾'-shtukas. By proposition <ref> it is now enough to show that this is a closed immersion.

Given a T-valued point (g,, ψ) in S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾), there can be at most one T-valued point (g,(, γ  H),ψ) in S×_∇_n^Ẑ'_, Hℋ^1(C,𝔾')∇_n^Ẑ_, Hℋ^1(C,𝔾) with =(, s_1,…, s_n, τ_) and γ∈ Isom^⊗(ω_^^∘, 𝒯̌_) mapping to (g,(, γ  H),ψ). This is because ψ:f_⋆(,γ H) g^⋆(',γ ' H) is an isomorphism of global 𝔾-shtukas. That means namely that s_1,…, s_n are determined by s_1'∘ g,…, s_n'∘ g, that γ H equals 𝒯̌_ψ^-1∘ g^⋆γ' H and that there is at most one τ_ since over the open subset X:=(C\w)_S⋂(C_S\i⋃Γ_s_i)⊂ C_S the isomorphism τ_ is determined by f_⋆τ_=σ^⋆ψ∘ g^⋆τ_'∘ψ. 

Therefore we have to answer the question if the morphism 
τ_|_X: σ^⋆|_X→|_X
can be extended to C_S\i⋃Γ_s_i. Note that if this is possible, then the global 𝔾-shtuka  is automatically bounded by Ẑ_ as we have seen at the end of the proof of the previous proposition <ref>.

Let  be the compositum of all _v_i with v_i∈v and let v_i^(0)∈ C_ be the closed point lying over v_i that equals the image of the characteristic morphism s_i. Then the definition C_:=C_\(i⋃ v_i^(0)) satisfies C_×_ S=C_S\( i⋃ Γ_s_i) 
Let further 

    I={w∈ C_ | w|w_j for some w_j∈w, w≠v_i^(0) for all v_i∈ }.

 In other words that means that I is determined by i⋃ Γ_s_i⊂(C_\ I)×_ S=:C_S^I
and ((C_\ I)×_ S)\( v_i∈w∩⋃ Γ_s_i)=(C\w)×_F_qS. The definition satisfies also the equation (C_\ I)×_ S=U.

Then by the theorem of Beauville-Laszlo from <ref> we have 
the following cartesian diagram

    ℋ^1(C_,𝔾_) [r] [d]_w∈ I∏L^+_w   ℋ_e^1(C_\ I,𝔾_) [d]^w∈ I∏ L_w
    w∈ I∏ℋ^1(, L^+𝔾_w) [r]    w∈ I∏ℋ^1(, L𝐆_w)

which means that σ^⋆|_C_S\i∪Γ_s_i and |_C_S\i∪Γ_s_i are given by tuples (σ^⋆|_U, w∈ I∏L^+_w(σ^⋆), w∈ I∏id_L_w(σ^⋆)) and (|_U, v∈ I∏L^+_(), v∈ I∏ id_L_()). The morphism τ_|_U:σ^⋆|_U→|_U determines for all w∈ I an isomorphism L_(τ_):L_(σ^⋆)→ L_w() . The question if τ_ can be extended to C_S\i⋃ Γ_s_i 
is then equivalent to the question if all the isomorphisms L_w(τ_) in ℋ^1(,L𝔾_) already come from an isomorphism L^+_w(σ^⋆)→ L_w^+() in ℋ^1(, L^+𝔾_w). Since L^+𝔾_⊂L𝔾_ is a quasi-compact closed subscheme, this is a closed condition on T which shows that 

    S×_∇_n^Ẑ'_, Hℋ^1(C,𝔾')∇_n^Ẑ_, Hℋ^1(C,𝔾)→ S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾)
 is a closed immersion.
It rests to show that under our assumption on Ẑ_ the morphism f_⋆ is surjective. This is not clear yet, since the closed subscheme

    S×_∇_n^Ẑ'_, Hℋ^1(C,𝔾')∇_n^Ẑ_, Hℋ^1(C,𝔾) ↪ S×__q((L^+_w_1(')/ L^+𝔾_w_1)×__q…×__q(L^+_w_m(')/ L^+𝔾_w_m))

does not necessarily surject to S. For the proof of the surjectivity we show that for any algebraically closed field K and every global 𝔾'-shtuka '=(',s_1,…, s_n, τ_') in ∇_n^Ẑ'_ℋ^1(C,𝔾')(K), there is a global 𝔾-shtuka =(, s_1,…, s_n, τ_) in ∇_n^Ẑ_ℋ^1(C,𝔾)(K) with f_⋆='. By proposition <ref> and the fact that K is algebraically closed the choice of a 𝔾-torsor  over C_K with f_⋆=' corresponds to an element in ∏_w∈w(L^+𝔾'_w/L^+𝔾_w)(K). Now let ' be the compositum of the fields _w for all w∈w. For a closed point w∈w⊂ C there are exactly deg w different closed points in C_ lying above w. We denote them by w^(0), …, w^(deg w-1), where w^(0) is a randomly chosen one and the others arise by applying successively σ on the residue field. If w∈ we choose w^(0) as before to be the image of the characteristic morphism s_i. Now once again Beauville and Laszlo help us with the diagram

    ℋ^1(C_',𝔾'_') [r] [d]^v∈ J∏L^+_v   ℋ_e^1(V,𝔾'_') [d]^v∈ J∏L_v
    v∈ J∏ℋ^1(', L^+𝔾'_v) [r]    v∈ J∏ℋ^1(', 𝐆'_v)
         ,

where J={v∈ C_' | v|w w∈w} and V:=C_'\ J. It allows us to identify ' with the tuple ('|_V_K, w∈w∏i=1deg w∏L^+𝔾'_w^(i), (ϵ_w^(i))_w^(i)∈ J) where ϵ_w^(i):L_w^(i)(') L𝐆'_w^(i) already comes from an isomorphism of L^+𝔾'_w^(i)-torsors. Consequently σ^⋆' is identified with 
(σ^⋆'|_V_K, w∈w∏i=1deg w∏L^+𝔾'_w^(i), (σ^⋆ϵ_w^(i-1))_w^(i)∈ J) where σ^⋆ϵ_w^(i-1):L_w^(i)(σ^⋆') L𝐆'_w^(i) is coming again from an isomorphism of L^+𝔾'_w^(i)-torsors.

 Note that the index i is computed in /deg w so that -1=deg w-1. We use again the intuitive notation τ'_w^(i):=L_w^(i)(τ_'):L_w^(i)(σ^⋆')→ L_w^(i)(') and define for all w^(i)∈ J the element c_w^(i):=ϵ_w^(i)∘τ'_w^(i)∘σ^⋆(ϵ_w^(i-1))^-1 in L𝐆_w^(i)(K).
The fact that τ_' is an isomorphism over C_K\k=1n⋃Γ_s_k

 implies that c_w^(i) is an element in L^+𝔾'_w^(i)(K) for all w^(i)∈ J_0:=J\(w∈w∩⋃w^(0)). Equivalently we have c_w^(i)∈ L^+𝔾'_w^(i)(K) for all w∈w∩ and i=1,…,deg w-1 as well as for all w∈w\(w∩) and all i=0,…, deg w-1. 
 We will now define the tuple (b_w^(i))_w^(i)∈ J∈w^(i)∈ J∏L^+𝔾'_w^(i)/L^+𝔾_w^(i)(K) that will determine by proposition <ref> the 𝔾-torsor  over C_K mapped under f_⋆:
 ℋ^1(C,𝔾)→ℋ^1(C,𝔾') to '.
If w∈w∩ we define

    b_w^(0):=1  b_w^(i):=σ^⋆ b_w^(i-1)· (c_w^(i))^-1∈ L^+𝔾'_w^(i)(K)i=1,…, deg w-1

Now if w∉ we write c_w^:=c_w^(0)·σ^⋆ (c_w^(deg w-1)) ·σ^2⋆(c_w^(deg w-2))·…·σ^(deg w-1)⋆(c_w^(1))∈ L^+𝔾'_w^(0)(K) and we can choose by <cit.>
an element b_w^(0)∈ L^+𝔾'_w^(0)(K) satisfying b_w^(0)·c_w^·σ^deg w⋆(b_w^(0))^-1=1. Additionally we define 
b_w^(i):=σ^⋆ b_w^(i-1)· (c_w^(i))^-1∈ L^+𝔾'_w^(i)(K)i=1,…, deg w-1,


so that we have b_w^(i)· c_w^(i)·σ^⋆ (b_w^(i-1))^-1=1. Further more we see


    b_w^(0)· c_w^(0)·σ^⋆ (b_w^(deg w-1))^-1
    
    =     b_w^(0)· c_w^(0)·
    σ( σ^deg w-1b_w^(0)·σ^deg w-2(c_w^(1))^-1·σ^deg w-3(c_w^(2))^-1·…· c_w^deg w-1)^-1
    
    =     b_w^(0)·c_w^·σ ^deg w⋆(b_w^(0))^-1=1 .

Now the 𝔾-torsor over C_K, determined by the choice of (b_w^(i))_w^(i)∈ J∈w∈w∏i=0deg w-1∏L^+𝔾'_w^(i)(K) and lying in the pre-image of ' under f_⋆:ℋ^1(C,𝔾)→ℋ^1(C,𝔾'), is given, as described in proposition <ref>, by 

    =('|_U, w∈w∏i=0deg w-1∏L^+𝔾_w^(i), (b_w^(i)∘ϵ_w^(i))_w^(i)∈ J).

It lies indeed in the pre-image of ' since f_⋆=('|_U, w∈w∏i=0deg w-1∏L^+𝔾'_w^(i), (b_w^(i)∘ϵ_w^(i))_w^(i)∈ J) is isomorphic to ' by (id_'|_U, ((b_w^(i))^-1)_w^(i)∈ J). Now we show that there is τ_:σ^⋆|_C_K\k∪Γ_s_k→|_C_K\k∪Γ_s_k with f_⋆τ_=τ_'. We set τ_|_U:=τ_' and need to convince ourself that it extends to C_K\k∪Γ_s_k. This is the case if and only if for all w^(i)∈ J_0 the vertical right hand side morphism b_w^(i)∘ϵ_w^(i)∘ L_w^(i)(τ_)∘σ^⋆(ϵ_w^(i-1))^-1∘σ^⋆(b_w^(i-1))^-1∈ L𝐆_w^(i)(K)
in the diagram

    L_w^(i)(σ^⋆) [d]_τ_w^(i):=L_w^(i)(τ_)[rr]^σ^⋆ϵ_w^(i-1)       L𝐆_w^(i)[rr]^σ^⋆ b_w^(i-1)[d]^c_w^(i)        L𝐆_w^(i)[d] 
    
    L_w^(i)() [rr]_ϵ_w^(i)       L𝐆_w^(i)[rr]_b_w^(i)       L𝐆_w^(i)

is given by an element in L^+𝔾_w^(i)(K).
By construction we have b_w^(i)∘ c_w^(i)∘σ^⋆ b_w^(i-1)=1 for all w^(i)∈ J_0.
This proves f_⋆=' and finally the theorem.





















  §.§.§ Closed Subgroups of 𝔾'

Secondly we take a closer look to the case that f:𝔾→𝔾' is a closed immersion of group schemes over C. 
We start with the following lemma that we mainly need for theorem <ref>.


Let f:𝔾→𝔾' be a closed immersion of smooth, affine group schemes over the curve C. Then the diagonal morphism Δ:ℋ^1(C,𝔾)→ℋ^1(C,𝔾)×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) of the induced morphism f_⋆:ℋ^1(C,𝔾)→ℋ^1(C,𝔾') is a monomorphism. 
The same is true for the diagonal morphism Δ:∇_nℋ^1(C,𝔾)→∇_nℋ^1(C,𝔾)×_∇_nℋ^1(C,𝔾')∇_nℋ^1(C,𝔾) of the induced morphism f_⋆:∇_n ℋ^1(C,𝔾)→∇_n ℋ^1(C,𝔾').


For the first diagonal morphism we have to prove that for any _q-scheme S the functor Δ_S:ℋ^1(C,𝔾)(S)→ℋ^1(C,𝔾)×_ℋ^1(C,𝔾')ℋ^1(C,𝔾)(S) is fully faithful. Let ∈ℋ^1(C,𝔾)(S), then this functor is cleary always faithful since φ∈Aut() is send to (φ,φ)∈Aut(Δ()), where Δ()=(,,id_f_⋆). Note that it suffices to consider φ∈Aut() since all morphisms in ℋ^1(C,𝔾) are isomorphisms. To show that Δ_S is full, let (φ,ψ)∈Aut(Δ()) which means by definition that 

    f_⋆[d]_id_f_⋆[r]^f_⋆φ    f_⋆[d]^id_f_⋆
     
    f_⋆[r]_f_⋆ψ    f_⋆

commutes. Therefore we have f_⋆φ=f_⋆ψ and since f:𝔾→𝔾' is a closed immersion this implies φ=ψ and hence that Δ_S is full.

More precisely, to see this, one chooses a covering U→ C_S that trivializes  so that φ and ψ correspond to morphisms φ, ψ:U→𝔾 satisfying the corresponding cocycle condition. The morphisms f_⋆φ and f_⋆ψ correspond to the compositions Uφ,ψ⟶𝔾f⟶𝔾' and the equality f_⋆φ=f_⋆ψ means f∘φ=f∘ψ. Since f is a closed immersion this implies φ=ψ, which proves that the first diagonal morphism is a monomorphism. The proof for the second diagonal morphism Δ:∇_nℋ^1(C,𝔾)→∇_nℋ^1(C,𝔾)×_∇_nℋ^1(C,𝔾')∇_nℋ^1(C,𝔾) works literally in the same way.



The morphism f_⋆:ℋ^1(C,𝔾)→ℋ^1(C,𝔾') is representable by an algebraic space. 
In particular for every _q-morphism ':S→ℋ^1(C,𝔾') and the natural projection p_S:C_S→ S, the Weil restriction p_S⋆('/𝔾_S)  is an algebraic space, that equals the fiber product S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾).


Since the diagonal morphism in lemma <ref> is a monomorphism it follows by <cit.> that f_⋆:ℋ^1(C,𝔾)→ℋ^1(C,𝔾') is representable by an algebraic space. 

By definition this means that the fiber product S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) is an algebraic space and in particular given by a functor (𝐒𝐜𝐡/S)^op→𝐒𝐞𝐭. We show that this functor coincides with the Weil restriction functor p_S⋆('/𝔾_S).

By definition a T-valued point of this fiber product S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) is given by a tuple (g, , α) where g:T→ S is a morphism of schemes,  is a 𝔾-torsor over C_T and α is an isomorphism of 𝔾'-torsors f_⋆ g^⋆ '. Since isomorphisms f_⋆ g^⋆ ' are in bijection with 𝔾-equivariant morphisms → ' the category of the tuples above is equivalent to the set of morphisms from C_T to the quotient '/𝔾_S.

Since Hom_C_S(C_T,[ '/𝔾_S])=Hom_S(T,p_S,⋆(['/𝔾_S])) by definition of the Weil restriction, the fiber product S×_ℋ^1(C,𝔾')ℋ^1(C,𝔾) is given by p_S⋆('/𝔾_S). 




Let f:𝔾→𝔾' be a closed immersion of smooth, affine group schemes over C. Then the induced morphism f_⋆: ∇_nℋ^1(C, 𝔾)→∇_nℋ^1(C,𝔾') is unramified and schematic.


We first show that f_⋆ is unramified and then conclude that it is representable by a scheme. Let B be any ring and I⊂ B an ideal with I^2=0 and p:Spec B:=Spec B/I→ Spec B the natural projection arising in a diagram of the form

    Spec B/I [d]_p[rr]^g       ∇_nℋ^1(C,𝔾) [d]^f_⋆
    
    Spec B [rr]_g'@–>@<0.6ex>[urr]^g_1@<-0.6ex>@–>[urr]_g_2      ∇_nℋ^1(C,𝔾')   .


To prove that f_⋆:∇_nℋ^1(C,𝔾)→∇_nℋ^1(C,𝔾') is unramified, we need to show that for any diagram of this kind there exists at most one dashed arrow making the diagram commutative, that means g_1=g_2.
This suffices since ∇_nℋ^1(C, 𝔾) and ∇_nℋ^1(C,𝔾') are locally of ind-finite type over the notherian scheme C^n.
The morphism g:Spec B→∇_nℋ^1(C,𝔾) corresponds to a global 𝔾-shtuka :=(, s_1,…, s_n,τ_) over Spec B, where g_1 and g_2 correspond to global 𝔾-shtukas _1=(_1,s_1',…,s_n',τ__1) and _2=(_2,s_1”,…,s_n”,τ__2) over Spec B. The commutativity of the upper triangle means
that there are isomorphisms β_1:p^⋆_1 and β_2:p^⋆_2 of global 𝔾-shtukas over Spec B. Therefore we have to prove, that the isomorphism β_2^-1∘β_1 arises already from an isomorphism _1→_2 of global 𝔾-shtukas over Spec B.
Furthermore we denote by ':=(',s_1,…, s_n,τ_') the global 𝔾'-shtuka over Spec B corresponding to g':Spec B→∇_nℋ^1(C,𝔾'). The commutativity of the lower triangle gives us isomorphisms α_1:f_⋆_1' and α_2:f_⋆_2' of global 𝔾'-shtukas over Spec B
satisfying γ=p^⋆α_2∘ f_⋆β^-1_2=p^⋆α_1∘ f_⋆β _1^-1 where γ:f_⋆ p_⋆' is the isomorphism of global 𝔾'-shtukas over Spec B coming from the commutativity of the square.


Now these isomorphisms imply directly that the paws s_i, s_i' and s_i” coincide for all i with 1 i n. Although f:𝔾→𝔾' is a closed immersion it is by the following remark <ref> a priori not so clear that the torsors _1 and _2 are isomorphic, but we now prove this as follows.

The 𝔾-torsors _1 and _2 over C_B come with 𝔾-equivariant maps to ' which are induced by α_1:f_⋆_1→' and α_2:f_⋆_2→'. Therefore they define two C_B-valued points h_1,h_2:C_B→'/𝔾_B. In other words one can describe them as follows. Since 

    Spec B [r]^_1[rd]_'   ℋ^1(C,𝔾) [d]^f_⋆
       ℋ^1(C,𝔾')
      and  
    Spec B [r]^_2[rd]_'   ℋ^1(C,𝔾) [d]^f_⋆
       ℋ^1(C,𝔾')

commute, that means f_⋆_1≃'≃ f_⋆_2, the 𝔾-torsors _1 and _2 induce morphisms h_1,h_2 from Spec B to the fiber product Spec B
×_ℋ^1(C,𝔾')ℋ^1(C,𝔾). In corollary <ref> this fiber product was seen to be p_B⋆('/𝔾_B). Therefore we have Hom_C_B(C_B, ('/𝔾_B))=Hom_B(Spec B, p_B⋆('/𝔾_B)) by definition of the Weil restriction, so that h_1 and h_2 correspond consequently to morphisms C_B→'/𝔾_B. First we show that they coincide on C_B:=C_B\⋃_iΓ_s_i.

The _q-Frobenius induces a morphism j:B/I→ B, b↦ b^q which is well defined, because I^2=0 and in particular I^q=0. We get the following commutative diagram:

    Spec B [rrd]_σ_B[rr]^j        Spec B/I [d]_p[rr]^      ℋ^1(C,𝔾)
           Spec B 
    @<0.7ex>[urr]^_1@<-0.7ex>[urr]__2

which implies 

    σ^⋆_1 = j^⋆ p^⋆_1j^⋆β_1 j^⋆j^⋆β_2^-1 j^⋆ p^⋆_2 = σ^⋆_2.

By restricting this isomorphism to C_B and composing with τ__1 and τ__2 we get 

    δ_0:_1|_C_Bτ^-1__1σ^⋆_1|_C_Bj^⋆β_1 j^⋆|_C_Bj^⋆β_2^-1σ^⋆_2|_C_Bτ__2_2|_C_B ,
 an isomorphism δ_0:=τ__2∘ j^⋆β_2^-1∘ j^⋆β_1∘τ^-1__1 of 𝔾-torsors over C_B. It satisfies 

    α_2|_C_B∘ f_⋆δ_0∘α_1^-1|_C_B = α_2∘ f_⋆τ__2_=τ_'∘σ^⋆α_2∘ f_⋆ j^⋆β_2^-1∘f_⋆ j^⋆β_1∘σ^⋆α_1^-1_=j^⋆(f_⋆β_1∘ p^⋆α_1^-1)∘τ_'^-1
    
    =   τ_'∘σ^⋆α_2∘ j^⋆ f_⋆β_2^-1∘ j^⋆(f_⋆β_2∘ p^⋆α_2^-1)∘τ_'^-1= id_'|_C_B.

In other words δ_0 is an isomorphism from (_1|_C_B, α_1|_C_B) to (_2|_C_B, α_2|_C_B)

of C_B-valued points  in '/𝔾_B. Therefore the restriction of (h_1,h_2):C_B→'/𝔾_B×_C_B'/𝔾_B to the open subscheme C_B in C_B

 factors through the diagonal in the following diagram

    C_B  [rr]^(h_1,h_2)@–>[rrd]        '/𝔾_B×_C_B'/𝔾_B  
    C_B [rr] [u]       '\𝔾_B [u]^Δ      .

To see that _1≃_2 over C_B we have to show that the morphism (h_1,h_2) factors through the diagonal Δ as well. Now since f is a closed immersion, the quotient '/𝔾_B exists as a scheme by <cit.> and it is smooth and separated by <cit.>. In particular the diagonal Δ is a closed immersion. Therefore C_B factors through the diagonal if 
the scheme theoretic image of C_B in C_B equals C_B. This was proven in lemma <ref>.
As a result of this, we conclude that δ_0 extends to an isomorphism δ:_1_2 of 𝔾-torsor over C_B.
The computation 

    δ_0^-1∘τ__2∘σ^⋆δ_0    = τ__1∘ j^⋆β_1^-1∘ j^⋆β_2 ∘τ__2^-1∘τ__2∘σ^⋆τ__2∘σ^⋆ j^⋆β_2^-1_=j^⋆τ_∘σ^⋆ j^⋆β_1∘σ^⋆τ__1^-1
       = τ__1∘ j^⋆β_1^-1∘ j^⋆τ_∘σ^⋆ j^⋆β_1_= σ^⋆τ__1∘σ^⋆τ__1^-1 = τ__1

shows that δ:_1_2 is an isomorphism of 𝔾-shtukas over B, which finishes the proof that the morphism f_⋆:∇_nℋ^1(C, 𝔾)→∇_nℋ^1(C,𝔾') is unramified. 


It rests to show that this morphism is schematic.
We have proven in lemma <ref> that the diagonal Δ_f_⋆ of f_⋆ is a monomorphism, which implies together with <cit.> that the morphism f_⋆ is representable by an algebraic space. It is clear that f_⋆ is a separated morphism, since the moduli spaces of global 𝔾-shtukas are separated. Furthermore we have proven that f_⋆ is unramified and in particular locally quasi-finite <cit.>. All together this allows us to apply <cit.> which states that a separated, locally quasi-finite morphism of algebraic stacks that is representable by an algebraic space is already schematic. This finishes the proof of the theorem.







Note that this is a particular property of the morphism of shtukas. Even if f:𝔾→𝔾' is a closed immersion, it is not true that ℋ^1(C,𝔾)→ℋ^1(C,𝔾') is an unramified morphism.



Let (id_C,f):(C,𝔾,, Ẑ_, H)→ (C,𝔾',, Ẑ'_, H') be a morphism of shtuka data, where f:𝔾→𝔾' is a closed immersion of smooth, affine group schemes over C. Then the induced morphism 
    f_⋆: ∇_n^Ẑ_, Hℋ^1(C, 𝔾)→∇_n^Ẑ'_, H'ℋ^1(C,𝔾')

is unramified and schematic. 


We first consider the induced morphism f_⋆: ∇_n^Ẑ_ℋ^1(C, 𝔾)→∇_n^Ẑ'_ℋ^1(C,𝔾') of the moduli spaces of global 𝔾-shtukas without level structures. We have the following commutative diagram

    ∇_n^Ẑ_ℋ^1(C, 𝔾)[rr] [d]_f_⋆      ∇_nℋ^1(C, 𝔾)^[d]^f_⋆
    ∇_n^Ẑ'_ℋ^1(C,𝔾') [rr]
          ∇_nℋ^1(C,𝔾')^.

The vertical arrow on the right is an unramified morphism by theorem <ref>, where the horizontal arrows are closed immersions and in particular also unramified. As a consequence the vertical arrow on the left is unramified as well. 
To prove the statement for the morphism f_⋆ of moduli spaces of global  𝔾-shtukas with level H-structure, we choose similar to  
<ref>
some finite subscheme D⊂ C such that 
H_D:=ker(𝔾(^)→𝔾(_D)) and H'_D:=ker(𝔾'(^)→𝔾'(_D)) are subgroups of finite index in H (resp. H'). Then we have by <ref> the following commutative diagram 


    ∇_n^Ẑ_v, Hℋ^1(C,𝔾) [d] 
          ∇_n^Ẑ_v,H_Dℋ^1(C,𝔾) @<6.6ex>[d] 
    [ll]_(0,6)finite^(0,6)étale∇_n^Ẑ_vℋ^1_D(C,𝔾) @<-6.6ex>[d] [rr]^(0,6)finite_(0,6)étale      ∇_n^Ẑ_vℋ^1(C,𝔾) [d]_unramified
    
    
    
    
    
    
    ∇_n^Ẑ_,H'ℋ^1(C,𝔾')
          ∇_n^Ẑ_,H'_ Dℋ^1(C,𝔾') 
     [ll]_(0,6)finite^(0,6)étale∇_n^Ẑ_ℋ^1_D(C,𝔾') [rr]^(0,6)finite_(0,6)étale      ∇_n^Ẑ_ℋ^1(C,𝔾')

All the horizontal arrows are étale and in particular unramified. Furthermore we have seen that the vertical arrow on the right is unramified. It follows

that ∇_n^Ẑ_vℋ^1_D(C,𝔾) →∇_n^Ẑ_ℋ^1_D(C,𝔾') is unramified <cit.> and finally that f_⋆: ∇_n^Ẑ_, Hℋ^1(C, 𝔾)→∇_n^Ẑ'_, H'ℋ^1(C,𝔾') is unramified <cit.>. It is clear that it is also schematic, which proves the corollary.













Let 𝔾 be a parahoric Bruhat-Tits group scheme and f:𝔾→𝔾' be a closed immersion of smooth, affine group schemes and =(_1,…,_n) be a set of closed points in C. Then the induced morphism

    f_⋆: ∇_nℋ^1(C, 𝔾)^→∇_nℋ^1(C,𝔾')^




We know by theorem <ref> that this morphism is unramified and schematic and in particular locally quasi-finite. Moreover the morphism is quasi-compact. Since ∇_nℋ^1(C,𝔾)→ℋ^1(C,𝔾) is of ind-finite type, this follows from <cit.> after choosing a representation ρ:𝔾'→GL(𝒱_0) for some vector bundle 𝒱_0 such that the quotient GL(𝒱_0)/𝔾 is quasi-affine (see <cit.>).
Therefore it suffices to prove that f_⋆ satisfies the valuative criterion for properness to see that this morphism is proper and consequently also finite, due to the quasi-finiteness. Thus let R be a complete discrete valuation ring with uniformizer π such that its residue field κ_R=R/π is algebraically closed and let K=Frac(R) be the fraction field of R. Let us further denote by K^alg an algebraic closure of K and by R^alg the integral closure of R in K^alg. We need to prove that in every diagram of the form 

    Spec K^alg[rr]^i_K[d]_j       Spec K [rr]^g_1[d]_(0,3)j      ∇_nℋ^1(C,𝔾)^[d]^f_⋆
    
    Spec R^alg[rr]_i_R@–>[rrrru]        Spec R [rr]_g_2      ∇ℋ^1(C,𝔾')^

there exists a unique dashed arrow making the diagram commutative. 

Here g_1,g_2,i_K,i_R,j and j are defined by the diagram. Choosing the closed embedding ρ:𝔾'↪GL(𝒱_0) it suffices, due to the separateness of the moduli spaces, to prove the valuative criterion for the composition ρ_⋆∘ f_⋆:∇_nℋ^1(C, 𝔾)^→∇_nℋ^1(C,GL(𝒱_0))^.  Therefore we may assume that 𝔾' equals GL(𝒱_0).
We denote by =(,s_1,…,s_n, τ_) the global 𝔾-shtuka over K corresponding to g_1 and by '=(',s_1',…,s_n',τ_') the global 𝔾'-shtuka over R corresponding to g_2. Furthermore the commutativity of the square gives an isomorphism α:f_⋆→ j^⋆' of global 𝔾'-shtukas over K. 
 

Let S:={v∈ C | 𝔾×_C_v}⋃. Then 𝔾'/𝔾×_C(C\ S) is by <cit.> an affine scheme over C\ S and in particular '/ 𝔾_R×_C_R(C\ S)_R is  an affine scheme over (C\ S)_R. Now 
the 𝔾-torsor |_(C\ S)_R with its 𝔾 equivariant morphism →' induced by α defines an (C\ S)_K valued point of the quotient '/𝔾_R.

    (C\ S)_K [rr]^s[rrd]       '/ 𝔾×_C_R (C\ S)_R [d] 
           (C\ S)_R


Now the proof consists of several steps.
In a first step we want to show that s factors through (C\ S)_R which means that it gives a section s_R:(C\ S)_R→'/𝔾_R×_C_R(C\ S)_R of the vertical morphism in diagram (<ref>). This morphism s_R corresponds to a unique 𝔾-torsor ℰ over (C\ S)_R together with an isomorphism α_R:f_⋆ℰ'|_(C\ S)_R satisfying j^⋆ℰ=|_(C\ S)_K.

In the second step of the proof we then show that the base change of ℰ to R^alg extends uniquely to a 𝔾-torsor over the whole relative curve C_R^alg. More precisely, we show that there is a 𝔾-torsor ℰ over C_R^alg such that firstly the restriction ℰ|_(C\ S)_R^alg is isomorphic to the 𝔾-torsor i_R^⋆ℰ over (C\ S)_R^alg and secondly f_⋆ℰ≃ i_R^⋆' and j^⋆ℰ≃ i_K^⋆.
Then we show in the third step that this 𝔾-torsor ℰ over C_R^alg gives rise to a unique 𝔾-shtuka (ℰ, r_1,…, r_n, τ_ℰ) in ∇_nℋ^1(C,𝔾)^(R^alg) making the diagram (<ref>) commutative. This will then finish the proof.


(Step 1)  We can assume that Spec A= C\ S is affine by enlarging S if necessary. Since we have seen that '/𝔾_R×_C_R(C\ S)_R is affine over (C\ S)_R=Spec A_R:=Spec (A×__qR) we can set '/𝔾_R×_C_RSpec A_R=:Spec B for some ring B. Therefore, to prove the assertion of the first step, namely that s in diagram (<ref>) factors through (C\ S)_R it is enough to show that the ring morphism s^⋆:B→ A⊗__qK=:A_K factors through A_R. We write L:=Frac(A_R) for the function field of C_R and 𝒪:=(A_R)_(π)⊂ L for the localisation of A_R at the prime ideal (π):=ker(A_R→ A_κ_R). The fact that A_R is normal due to the smoothness of C_R over R and the fact that the prime ideal (π)⊂ A_R corresponding to the generic point of Spec A_κ_R is of height 1 in A_R, implies that 𝒪 is a discrete valuation ring with uniformizer π. The normality of A_R allows us also by <cit.> to write A_R=⋂_𝔭⊂ A_R 𝔭 of height 1A_R,𝔭. For all prime ideals 𝔭⊂ A_R of height 1 we have either 𝔭=(π) or π∉𝔭. In the second case 𝔭 comes from a closed point in Spec A_K which means A_R,𝔭=A_K,𝔭. Since A_K=𝔮⊂ A_K max. ideal⋂A_K,𝔮 we conclude 

    A_R=𝒪∩ A_K⊂ L.

Due to this equation it is enough to show that the composition
s_L:Spec L Spec A_K Spec B of s|_A_K with η:Spec L→ Spec A_K factors through Spec 𝒪. 

The Frobenius pullback (σ^⋆_L,σ^⋆α_L) with σ^⋆α_L:(f_⋆σ^⋆)→σ^⋆'_L gives an L-valued point of the quotient σ^⋆'/𝔾_R. As before this quotient is affine over A_R and given exactly by σ^⋆'/𝔾_R×_C_RSpec A_R=Spec (B⊗_A_R,σA_R)→ Spec A_R, where the A_R-algebra structure of B⊗_A_R,σA_R is given by multiplication in the second component. This means that the L-valued point (σ^⋆_L, σ^⋆α_L) is given by an A_R-morphism Spec L→ Spec(B⊗_A_R,σA_R). In other words we can describe this morphism as follows. The Frobenius σ:=id_A⊗σ_R:A_R→ A_R induces of course a morphism of the fraction field L which we denote again by σ:L→ L, a/b↦σ(a)/σ(b) for a,b∈ A_R. It is not the absolut _q-Frobenius. Now the composition σ∘ s_L^⋆:B→ L is not an A_R-linear morphism, but it induces a unique A_R-linear morphism σ^⋆ s_L^⋆:B⊗_A_R,σA_R→ L making the following diagram commutative.

    b @|->[d]     B [rr]^s_L^⋆[d]        L [d]^σ
    
    b⊗ 1      B⊗_A_R,σ A_R [rr]^σ^⋆ s_L^⋆       L

This morphism σ^⋆ s_L^⋆ is the one coming from the tuple (σ^⋆_L, σ^⋆α_L).

The 𝔾'-shtuka ' is defined over R. In particular the restriction of τ_' to Spec A_R is an isomorphism σ^⋆'|_A_R→'|_A_R that induces an isomorphism τ_' of A_R-algebras 

    τ_':Spec (B⊗_A_R,σA_R)→ Spec B.

 It sends a T-valued point (ℰ_0, δ) with δ:f_⋆ℰ_0→σ^⋆' to (ℰ_0, τ_'∘δ). We then would like to know, that the following diagram 

    B [rd]_s_L^⋆[rr]^τ^⋆_'       B⊗_A_R,σA_R [ld]_σ^⋆ s_L^⋆
        L

of A_R-morphisms commutes, which can be seen as follows.
By assumption (see diagram (<ref>)) the diagram 

    f_⋆σ^⋆_L [rr]^σ^⋆α_L[d]_f_⋆τ_
          σ^⋆'_L [d]_τ_'
    
    
    f_⋆_L [rr]^α_L      '_L

is a commutative diagram of isomorphisms of 𝔾'-torsors over L. (Actually the whole diagram is already defined over A_K and the vertical arrow on the right is even defined over A_R.) 
Now σ^⋆ s_L was corresponding to (σ^⋆_L, σ^⋆α_L), so that by the description of the morphism (<ref>) the composition τ_'∘σ^⋆ s_L corresponds to the L-valued point (σ^⋆_L, τ_'∘σ^⋆α_L) of Spec B. This point in the fiber category (Spec B)(L) is by τ_^-1 isomorphic to (_L, τ_'∘σ^⋆α_L∘  f_⋆τ_^-1), which is by diagram (<ref>) equal to (_L, α_L). Since (_L, α_L) is exactly the L-valued point s_L the commutativity of diagram (<ref>) follows.


Now we choose a closed point ∈ C\ S. Then we can consider the associated étale local 𝔾'_-shtuka L_(')=(L_^+('), τ'_:=L_(τ_')) over R, which arises from the formal 𝔾̂'_-torsor '×_C_RSpf A_,R as described in <ref>. Since R is strictly henselian the L^+𝔾'_-torsor L^+_(') is trivial so that we choose a trivialization β:L^+_(') L^+𝔾'_. In particular the composition β∘τ'_∘σ^⋆β^-1:L^+𝔾'_ L^+𝔾'_ is given by an element b∈ L^+𝔾'_(R) so that β:L_(') (L^+𝔾'_,b).

We define R_i:=R/π^q^i and b_i∈ L^+𝔾'_(R_i) as the image of the projection of b under the map L^+𝔾'_(R)→ L^+𝔾'_(R_i), b↦ b_i.
Since R_0=κ_R is algebraically closed, there exists by <cit.> a c_0∈ L^+𝔾'_ with c_0=b_0·σ^⋆ c_0. Note that σ^⋆ c_0∈ L_^+(𝔾'_)(R_1). We set inductively c_i:=b_i·σ^⋆ c_i-1 for i 1 and c:=i→∞lim c_i=k→∞lim b·σ^⋆ b·…·σ^(k-1)⋆bσ^k⋆c_0∈ L^+_(𝔾'_)(R) which satisfies c=b·σ^⋆ c. Replacing the trivialization β by c^-1∘β gives therefore an isomorphism of local 𝔾'_-shtuka c^-1·β:L_(') (L^+_𝔾'_, id) as becomes clear from the diagram

    σ^⋆ L^+_(') [r]^σ^⋆β[d]^τ_    L^+_𝔾'_[r]^σ^⋆ c[d]^b    L^+_𝔾'_[d]^id
    
     L^+_(') [r]^β    L^+_𝔾'_[r]^c^-1    L^+_𝔾'_

Let A_,R:=A_⊗̂__R and Γ(A,𝔾'/𝔾) the ring of sections of 𝔾'/𝔾 over Spec A. The trivializations c^-1∘β: L^+_(') → L^+_𝔾'_ and σ^⋆ (c^-1∘β) : σ^⋆ L^+_(') → L^+_𝔾'_ and the isomorphism τ_ induces after passing to the v-adic completion morphisms c^-1β, σ^⋆ (c^-1β) and τ_ as in the following diagram

    Γ(A,𝔾'/𝔾)⊗_A A_,R[dd]^id[rr]^c^-1β       B ⊗_A_R A_,R[dd]^τ_^⋆[rrd]^s_L^⋆      
                  L^:=L⊗_A_R A_,R
    Γ(A,𝔾'/𝔾)⊗_A A_,R[rr]^σ^⋆(c^-1β)       B⊗_A_R,σ A_,R[rru]_σ^⋆ s_L^⋆


The right hand side of the diagram arises as the v-adic completion of the diagram (<ref>), where s_L^⋆ and σ^⋆ s_L^⋆ denote the induced morphism of the completion.
Since ord_π(σ(x))=q·ord_π(x) for all x∈ L^ the diagram (<ref>) implies 

    ord_π(s_L^⋆∘c^-1β(y))=ord_π(σ^⋆(s_L^⋆∘c^-1β)(y))=q· ord_π(s_L^⋆∘c^-1β(y))

for y∈Γ(A,𝔾'/𝔾). This means that ord_π(s_L^⋆∘c^-1β(y)) equals 0 or ∞. In particular we have

    s_L^⋆∘c^-1β:Γ(A,𝔾'/𝔾)⊗_A A_,R→{x∈ L^ | ord_π(x) 0 }

which implies 

    B⊗_A_RA_,R[rr]^s_L^⋆          {x∈ L^ | ord_π(x) 0 }        
    
    B [rr]^s_L^⋆    [u]       {x∈ L^ | ord_π(x) 0 }∩ L= 𝒪

This finishes the first step.



(Step 2) 
As we have described above, the proof of the first step gives us a 𝔾-torsor ℰ over (C\ S)_R with j^⋆ℰ=|_(C\ S)_K and an isomorphism α_R: f_⋆ℰ'|_(C\ S)_R. We now show that ℰ×_(C\ S)_R(C\ S)_R^alg extends to a 𝔾-torsor ℰ over C_R^alg with ℰ×_C_R^algC_K^alg=𝒢_K^alg and α_R:f_⋆ℰ'_R^alg. 

Now the field L:=Quot(A_R^alg) has transcendence degree one over K^alg so that its cohomological dimension equals one by <cit.>. Since 𝔾_L is reductive this implies by <cit.> that ℰ is trivial over L. Therefore we can choose a finite extension K'/K and a trivialization γ_L':ℰ→𝔾_L', where L':=Quot(A_R') and where R' is the integral closure in K'. We recall that we denoted by z_ a uniformizer of C at v, so that A_⊗̂R'=R'z_ (do not confuse A_ with A) and L' is contained in Quot(R'z_). In particular the trivialization γ_L implies that the 𝔾-torsor ℰ is trivial over Quot(R'z_). This fact allows us to apply <cit.>
to see that ℰ_Quot(R'z_pv) extends to a 𝔾-torsor ℰ_ over R'z_. (Note that <cit.> use the notation 𝒪_E for our ring κ_Rz_ and z for a uniformizer π' in our ring R'.) This corresponds by <cit.> to a L^+𝔾_-torsor L^+(ℰ_) over R' which becomes trivial after base change to the strictly henselian ring R^alg. We fix such a trivialization β_v:L^+_(ℰ__R^alg) L^+𝔾__R^alg for all ∈ S. 
They induce trivializations
L(β_v):L_(ℰ_R^alg) L𝐆__R^alg
and therefore isomorphisms 

    L_ℰ_R^alg/L^+𝔾__,R^alg(_𝔾_)×__qR^alg=:__R^alg.

An T-valued point (ℒ^+,δ) with δ:ℒ→ L_ℰ_R^alg is send to (ℒ^+,β_∘δ). By the theorem of Beauville-Laszlo in <ref> we have the following cartesian diagram

    ℋ^1(C,𝔾) [rr] [d]^∈ S∏L^+_      ℋ^1_e(C\ S,𝔾) [d]^∈ S∏L_
    ∈ S∏ℋ^1(_q,L^+𝔾_) 
    [rr]_L
          ∈ S∏ℋ^1(_q, L𝐆_)
    .

Due to this diagram the torsor _K^alg corresponds to the tuple 
(|_(C\ S)_K^alg, ∈ S∏L^+_(_K^alg) , (ϵ_)_∈ S)  with ϵ_=id:L(L_^+())→ L_(|_(C\ S)_K^alg). Now for all ∈ S the tuple (L_^+(_K^alg), (β_× id_K^alg)∘ϵ_) gives an K^alg-valued point of the affine flag variety __R^alg. By assumption 𝔾_ is parahoric so that __R^alg is ind-projective over R^alg by <cit.>. As a consequence we can lift (L_^+(_K^alg),(β_× id_K^alg)∘ϵ_) to a unique R^alg-valued point (ℰ_, β_∘δ_)∈_(R^alg) with (ℰ_×_R^algK^alg, (β_∘δ_)× id_K^alg)≃(L_^+(_K^alg),(β_× id_K^alg)∘ϵ_). 

In particular the tuple (ℰ_(C\ S)_R^alg, ∏_∈ Sℰ_,δ_) defines a unique R^alg-valued point in ℋ^1(C,𝔾) given by a 𝔾-torsor ℰ over C_R^alg with ℰ×_C_R^algC_K^alg=_K^alg. By diagram (<ref>) with 𝔾 replaced by 𝔾' we get an isomorphism α_R^alg:f_⋆ℰ'_R^alg. This finishes the second step.


(Step 3) We need to show that the 𝔾-torsor ℰ over C_R^alg is part of a global 𝔾-shtuka ℰ=(ℰ,r_1,…,r_n,τ_ℰ) defining the dashed arrow in diagram (<ref>). The condition that α_R^alg:f_⋆ℰ i_R^⋆' needs to be an isomorphism of global 𝔾'-shtukas defines r_i by r_i=s_i'∘ i_R for all 1 i n. So we have to construct τ_ℰ. 

From the proof of the first step we get the commutative diagram

    (C\ S)_R [dl]_σ^⋆ s_R[dr]^s_R   
    σ^⋆'/𝔾_R×_C_R(C\ S)_R [rr]^τ_'      '/𝔾_R×_C_R(C\ S)_R

We defined (ℰ, α_R) with α_R:f_⋆ℰ→' to be the (C\ S)_R-valued point in '/𝔾_R corresponding to s_R. Hence (σ^⋆ℰ, σ^⋆α_R) corresponds to σ^⋆ s_R and the composition τ_'∘σ^⋆ s_R corresponds to the tuple (σ^⋆ℰ, τ_'∘σ^⋆α_R). The commutativity of (<ref>) means that (σ^⋆ℰ, τ_'∘σ^⋆α_R) and (ℰ, α_R) are isomorphic as (C\ S)_R-valued points in '/𝔾_R. This gives us therefore an isomorphism τ_ℰ:σ^⋆ℰ→ℰ of 𝔾-torsors over (C\ S)_R satisfying τ_'∘σ^⋆α_R=α_R∘ f_⋆τ_ℰ. This defines the isomorphism τ_ℰ restricted to (C\ S)_R^alg by τ_ℰ|_(C\ S)_R^alg= τ_ℰ×_Rid_R^alg and we have to 
extend it to C_R^alg\i⋃Γ_r_i. We know additionally by ℰ|_C^K^alg= and α:f_⋆ j^⋆'_K^alg that τ_ℰ extends to C_K^alg\i⋃Γ_r_i. Therefore we only have to extend τ_ℰ at finitely many closed points of C_R^alg\i⋃Γ_r_i. This works similar as at the end of the proof of proposition <ref>. 
So for p∈ C_R^alg\ (i⋃Γ_r_i⋃ C_K^alg) we choose an open neighboorhood  V⊂ C_κ_R with (V×_κ_R R^alg)⋂((i⋃Γ_r_i)⋃ (C\ S)_R^alg)=p. We write V:=V\ p so that 










τ_ℰ is defined on V_R^alg and need to be extended to V_R^alg. Moreover the 𝔾-torsors σ^⋆ℰ|_V_R^alg and ℰ|_V_R^alg are two R^alg-valued points in ℋ^1(V,𝔾)(R^alg) so that τ_ℰ|_V_R^alg is an isomorphism in ℋ_e^1(V,𝔾)(R^alg). Thanks again to Beauville and Laszlo (<ref>) the cartesian diagram

    ℋ^1(V,𝔾) [rr] [d]^L^+_p      ℋ^1(V, 𝔾) [d]^L_p
    ℋ^1(κ_R, L^+𝔾_p) [rr]       ℋ^1(κ_R,L𝐆_p)

makes it sufficient to show that the isomorphism L_p(τ_ℰ):L_p(σ^⋆ℰ)→ L_p(ℰ) in ℋ^1(κ_R, L𝐆_p) comes from an isomorphism in ℋ^1(κ_R, L^+𝔾_p). After trivializing L_p(ℰ) the morphism L_p(τ_ℰ) is given by an element h∈ L𝐆_p(R^alg). Since τ_ℰ is already defined on V_K^alg the pullback of h to h_K∈ L𝐆_p(K^alg) is already given by an element in L^+𝔾_p(K^alg). Since L^+𝔾_p⊂ L𝐆_p is a closed subgroup we conclude that h is already an element in L^+𝔾_p(R^alg). This shows that τ_ℰ extends uniquely to V_R^alg and hence to C_R^alg\i⋃Γ_r_i. Hereby we found the 𝔾-shtuka ℰ over R^alg defining a unique dashed arrow in the diagram (<ref>), which ends the proof of the theorem.




Let 𝔾 be a parahoric Bruhat-Tits group scheme and (id_C,f):(C,𝔾,, Ẑ_, H)→ (C,𝔾',, Ẑ'_, H') be a morphism of shtuka data, where f:𝔾→𝔾' is a closed immersion of smooth, affine group schemes over C. Then the induced morphism 
    f_⋆: ∇_n^Ẑ_, Hℋ^1(C, 𝔾)→∇_n^Ẑ'_, H'ℋ^1(C,𝔾')

is finite.


The proof of this corollary works literally in the same way as the proof of corollary <ref> with replacing unramified by finite.




The results of this section can potentially be used 

to formulate and prove an analog of the André-Oort conjecture for global 𝔾-shtukas. To formulate such a conjecture one needs the notion of special points and special subvarieties. In the case of Drinfeld modular curves an analog of the André-Oort conjecture has been formulated and proved in <cit.>. Later the notion of special subvarieties and the formulation of the André Oort conjecture was generalized in <cit.> to the higher dimensional Drinfeld modular varieties. In the same paper this André-Oort conjecture was proven in some special cases. These results were extended in <cit.>. To define Drinfeld modular varieties, one fixes a point ∞∈ C so that C∖∞=:Spec A is affine and ℳ_A^r is the moduli space for Drinfeld A-modules of rank r. Now for certain finite extensions A'⊂ A coming from a morphism C→ C' of curves, Breuer shows that there is a proper morphism ℳ_A^r→ℳ_A'^r·[A:A'] of moduli spaces which is also finite by <cit.>. 
Then Breuer uses the image of this morphism to define special subvarieties.

Drinfelds modular variety ℳ_A^r can be embedded into ∇_2^Ẑ_ℋ^1(C,GL_r) for n=2 and some specific choosen bound Ẑ_. The morphism ℳ_A^r→ℳ_A'^r·[A:A'] then corresponds to a morphism ∇_2^Ẑ_ℋ^1(C,GL_r)→∇_2^Ẑ'_wℋ^1(C',GL_r·[C:C']) coming from a morphism of 
shtuka data (C,GL_r, , Ẑ_)→ (C',GL_r·[C:C'],w,Ẑ_w). So extending the coefficients for Drinfeld modules generalizes to changing the curve for global 𝔾-shtukas as in subsection <ref>, since we are not restricted to choose n=2, 𝔾=GL_r or some specific bound. Moreover we have seen that additionally to changing the curve, we can also change the group scheme 𝔾 as in subsection <ref>. 
Although we do not know if this is precisely the correct definition it is conceivable to define a special subvariety of ∇_n^Ẑ'_w, H'ℋ^1(C',𝔾') to be the image of the morphism 

    f_⋆∘π_⋆:∇_n^Ẑ_,Hℋ^1(C,𝔾)→∇_n^Ẑ'_w, H'ℋ^1(C',𝔾')

arising from a morphism (π,f) of shtuka data, where f:π_⋆𝔾↪𝔾'  is a closed immersion of (Bruhat-Tits) group schemes.  Special points in ∇_n^Ẑ'_w, H'ℋ^1(C',𝔾') would then be defined to be those points which arise in the image of a morphism f_⋆: ∇_n^Ẑ_w, Hℋ^1(C',𝕋)→∇_n^Ẑ'_w, H'ℋ^1(C',𝔾')^, where f:𝕋→𝔾' is a closed (Bruhat-Tits) group scheme that is generically a torus in 𝔾'.

Following this, an André-Oort conjecture for global 𝔾-shtukas would then say that given a set S of special points, the Zariski closure of these points is a finite union of special subvarieties.
Again, this is not a precise formulation but should give an impression of the flavor of a possible statement.











tocsectionReferences


Paul Breutmann, CNRS,

Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG)

4 place Jussieu, 75252 Paris Cedex 5, France


