# Integral models of moduli spaces of shtukas with deep Bruhat-Tits level
structures

Patrick Bieker  Bielefeld University, Faculty of Mathematics, Postfach 100131,
33501 Bielefeld, Germany<EMAIL_ADDRESS>
###### Abstract.

We construct integral models for moduli spaces of shtukas with deeper Bruhat-
Tits level structures. We embed the moduli space of global shtukas for a
deeper Bruhat-Tits group scheme into the limit of the moduli spaces of shtukas
for all associated parahoric group schemes. Its schematic image defines an
integral model of the moduli space of shtukas with deeper Bruhat-Tits level
with favourable properties: They admit proper, surjective and generically
étale level maps as well as a natural Newton stratification. In the Drinfeld
case, this general construction of integral models recovers the moduli space
of Drinfeld shtukas with Drinfeld level structures.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) via the CRC/TRR 326 Geometry and Arithmetic of
Uniformized Structures, project number 444845124, and the CRC/TRR 358 Integral
Structures in Geometry and Representation Theory, project number 491392403.

##  1\. Introduction

Following Langlands’ philosophy, the Hasse-Weil $\zeta$-function of a Shimura
variety should be computable in terms of automorphic $L$-functions. The
Langlands-Kottwitz method for computing the Hasse-Weil $\zeta$-function (or
more generally to study the cohomology of Shimura varieties) relies on two
geometric inputs. On the one hand one needs a description of the mod-$p$
points of the Shimura variety as conjectured by Langlands and Rapoport [LR87]
and on the other hand one needs to identify a certain test function that
serves as an input to the Arthur-Selberg trace formula. For both steps the
existence of good integral models of the Shimura variety is necessary. For
Shimura varieties with parahoric level structure there has been substantial
progress towards both steps. The Langlands-Rapoport conjecture is proved under
mild hypotheses in [Hof22] (and [AH16] in the function field setting), while
the test function conjecture is shown in the series of papers [HR21, HR20,
Ans+22]. The goal of this work is to provide a first step towards generalising
the results in the function field setting to more general (Bruhat-Tits-)level
structures. Namely, we construct analogues of the integral models of Shimura
varieties of [KP18] for these deeper level structures in the function field
setting.

The analogues of Shimura varieties over function fields are moduli spaces of
(global) shtukas. They were first introduced in [Dri87a] for
$\operatorname{GL}_{n}$ and later generalised to arbitrary split reductive
groups in [Var04] and further to flat affine group schemes of finite type in
[AH21]. They are used to great success in establishing a Langlands
correspondence over function fields in [Dri87] for $\operatorname{GL}_{2}$,
[Laf02] for $\operatorname{GL}_{n}$ and [Laf18] for arbitrary reductive
groups. Recently, a lot of progress has been made in studying the geometry of
moduli spaces of shtukas with parahoric level, compare for example [AH14],
[AH19], [Bre18], [YZ19] and [Zhu14]. However, little is recorded for deeper
level structures. A first result beyond the parahoric case is obtained in
[Bie23], where Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structures for
Drinfeld shtukas are defined (in the style of [Dri76, KM85]) and it is shown
that their moduli spaces admit well-behaved (that is finite flat and
generically étale) level maps as in the case of modular curves in [KM85]. The
goal of this work is to construct integral models of moduli spaces of shtukas
with deep Bruhat-Tits level structures for general reductive groups that
generalise both the parahoric case and in the Drinfeld case the moduli space
of shtukas with Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structures of
[Bie23], and to study properties of these integral models.

Let $X$ be a smooth, projective and geometrically connected curve over a
finite field ${\mathbb{F}_{q}}$. Let $G$ be a (connected) reductive group over
the function field $F$ of $X$ and let us fix a parahoric model $\mathcal{G}\to
X$ of $G$. In other words, $\mathcal{G}$ is a smooth affine group scheme with
connected fibres over $X$ and generic fibre $G$ such that for all closed
points $x$ of $X$ the pullback
$\mathcal{G}_{\operatorname{Spec}(\mathcal{O}_{x})}$ to the spectrum of the
completed local ring $\mathcal{O}_{x}$ at $x$ is a parahoric group scheme in
the sense of [BT84].

Let us fix a closed point $x_{0}$ of $X$. Let $\Omega$ be a bounded subset of
an apartment in the Bruhat-Tits building of $G_{F_{x_{0}}}$, where $F_{x_{0}}$
is the completion of $F$ at $x_{0}$. By Bruhat-Tits theory, we get a smooth
affine $\mathcal{O}_{x_{0}}$-group scheme $\mathcal{G}_{\Omega}$ with
connected fibres that we glue with $\mathcal{G}$ outside of $x_{0}$ to obtain
a (global) Bruhat-Tits group scheme $\mathcal{G}_{\Omega}\to X$ which is
smooth and affine with connected fibres by construction. Without changing
$\mathcal{G}_{\Omega}$, we may assume that $\Omega$ is convex, closed and a
union of facets.

Let $I$ be a finite set and let $\underline{\mu}=(\mu_{i})_{i\in I}$ be a
tuple of conjugacy classes of geometric cocharacters of $G$. For simplicity,
we assume in this introduction that $\underline{\mu}$ is defined over the
function field $F$ of $X$ (in general it will only be defined over a finite
separable extension of $F$). A global $\mathcal{G}_{\Omega}$-shtuka over a
scheme $S$ is a $\mathcal{G}_{\Omega}$-bundle $\mathcal{E}$ on $X_{S}$
together with an isomorphism
$\varphi\colon\mathcal{E}|_{X_{S}\setminus\Gamma_{\underline{x}}}\xrightarrow{\cong}\sigma^{\ast}\mathcal{E}|_{X_{S}\setminus\Gamma_{\underline{x}}}$
away from the graph $\Gamma_{\underline{x}}$ of an $I$-tuple $\underline{x}\in
X^{I}(S)$ of points of $X$. We denote by
$\operatorname{Sht}^{I,\leq\underline{\mu}}_{\mathcal{G}_{\Omega},X}$ the
moduli space of global $\mathcal{G}_{\Omega}$-shtukas whose zeroes and poles
are bounded by $\underline{\mu}$. It is a Deligne-Mumford stack locally of
finite type over ${\mathbb{F}_{q}}$.

While for a subset $\Omega^{\prime}$ of $\Omega$ there is still a natural map
$\operatorname{Sht}^{I,\leq\underline{\mu}}_{\mathcal{G}_{\Omega},X}\to\operatorname{Sht}^{I,\leq\underline{\mu}}_{\mathcal{G}_{\Omega^{\prime}},X}$
by [Bre18, Theorem 3.20] (and Theorem 3.5.5), already in the Drinfeld case
$G=\operatorname{GL}_{2}$, the level map
$\operatorname{Sht}_{\operatorname{GL}_{2,[0,n]},X}^{(1,2),\leq((0,-1),(1,0))}\to\operatorname{Sht}_{\operatorname{GL}_{2},X}^{(1,2),\leq((0,-1),(1,0))}$
is neither proper nor surjective for $n\geq 2$ by [Bie23, Remark 2.20]. In
order to remedy this issue, we propose the following construction to
relatively compactify
$\operatorname{Sht}^{I,\leq\underline{\mu}}_{\mathcal{G}_{\Omega},X}$.

######  Definition 1.0.1 (compare Definition 5.2.4).

In the situation above, that is, for a reductive group $G$ over $F$, and a
Bruhat-Tits group scheme $\mathcal{G}_{\Omega}\to X$ for a subset $\Omega$
(assumed to be convex, closed and a union of facets) of the Bruhat-Tits
building for $G_{F_{x_{0}}}$ at the fixed point $x_{0}$ of $X$ as above, the
_integral model of the moduli space of shtukas with
$\mathcal{G}_{\Omega}$-level structure_
$\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}$ is
defined to be the schematic image in the sense of [EG21] of the map

$\operatorname{Sht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{\mathcal{G}_{\mathfrak{f}},X}^{I,\leq\underline{\mu}},$

where the limit is taken over the set of all facets $\mathfrak{f}$ contained
in $\Omega$ partially ordered with respect to inclusion.

Clearly, in the parahoric case (that is, when $\Omega$ is a facet) we have

$\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X^{I}}^{\leq\underline{\mu}}=\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{\mathcal{G}_{\mathfrak{f}},X^{I}}^{\leq\underline{\mu}}=\operatorname{Sht}_{\mathcal{G}_{\Omega},X^{I}}^{\leq\underline{\mu}},$

as the index set has $\Omega$ as its final object. So our construction
generalises the parahoric case. Moreover, by [Bie23, Theorem 6.7], our notion
of integral models recovers in the Drinfeld case the moduli space of Drinfeld
shtukas with Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structure at
$x_{0}$.

The main result of this work is to show that our integral models have good
formal properties. Namely, that they admit proper, surjective and generically
finite étale level maps (similar to the integral models of Shimura varieties
with parahoric level structures):

######  Theorem 1.0.2 (compare Theorem 5.2.3 and Theorem 5.2.6).

In the situation of Definition 1.0.1, the map

$\operatorname{Sht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{\mathcal{G}_{\mathfrak{f}},X}^{I,\leq\underline{\mu}}$

is schematic and a quasi-compact locally closed immersion. It factors into an
open immersion
$\operatorname{Sht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}\to\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}$
followed by the closed immersion
$\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{\mathcal{G}_{\mathfrak{f}},X}^{I,\leq\underline{\mu}}$.
The restriction of the inclusion

$\operatorname{Sht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}|_{(X\setminus\\{x_{0}\\})^{I}}\xrightarrow{\cong}\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}|_{(X\setminus\\{x_{0}\\})^{I}}$

away from $x_{0}$ is an open and closed immersion. Moreover, for a subset
$\Omega^{\prime}\prec\Omega$, there is a natural level map

$\bar{\rho}_{\Omega^{\prime},\Omega}\colon\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}\to\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega^{\prime}},X}^{I,\leq\underline{\mu}}$

that is schematic, proper and surjective, and over
$(X\setminus\\{x_{0}\\})^{I}$ it is finite étale.

######  Remark 1.0.3.

In principle, we expect our methods to apply also to even more general level
structures, namely to stabilisers of bounded subsets $\Omega$ of the Bruhat-
Tits building that are not necessarily contained in a single apartment. This
includes in particular the case that $\Omega$ is a ball of radius $r$ around a
point $x$ in the Bruhat-Tits building, whose (connected) stabiliser should be
closely connected (up to the centre) to the Moy-Prasad subgroup
$G(F_{x_{0}})_{x,r}$.

However, the results from Bruhat-Tits theory we rely on in the proof of
Theorem 1.0.2 do not seem to be available in this case. Compare Section 5.3
for further discussion of this situation and the statements we can prove in
the more general setting.

The construction of integral models is a first step towards generalising
results used in the Langlands-Kottwitz method at parahoric level to the more
general level structures considered here. In particular, a fine analysis of
the (étale-local) structure of their special fibres together with the
calculation of the trace of Frobenius on the nearby cycles might yield
significant progress towards a geometric construction of test functions for
deeper Bruhat-Tits level structures generalising the results in the parahoric
case of [HR21, HR20, Ans+22] (and [HR12] in the Drinfeld case with
$\Gamma_{1}(p)$-level structures). Note that the test functions for
$\Gamma(p^{n})$-level (which are not Bruhat-Tits level structures in our
sense) constructed in [Sch13] have no apparent connection to geometry. A
geometric construction of test functions in our setting might shed some more
light onto this situation as well.

Let us briefly comment on the proof of Theorem 1.0.2. In the parahoric case,
level maps on moduli spaces of shtukas are also studied in [Bre18, Theorem
3.20]. However, the assumption on the bounds used there does not apply to the
bounds given by cocharacters. We provide the relevant background on bound for
global (and local) shtukas and generalise the result of [Bre18, Theorem 3.20]
to include bounds in this sense (compare Theorem 3.5.5). Using the assertion
in the parahoric case, we are then able to deduce the result also for level
maps at deeper level.

The first part of Theorem 1.0.2 is based on a corresponding result on the
moduli space of $\mathcal{G}_{\Omega}$-bundles.

######  Theorem 1.0.4 (compare Theorem 4.3.3).

In the situation of Definition 1.0.1, the natural map

$\operatorname{Bun}_{\mathcal{G}_{\Omega}}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Bun}_{\mathcal{G}_{\mathfrak{f}}}$

is schematic and representable by a quasi-compact open immersion.

As a first step in the proof of this theorem, we show in the local case (and
hence also for the corresponding global Bruhat-Tits group schemes), that the
not necessarily parahoric Bruhat-Tits group scheme $\mathcal{G}_{\Omega}$ is
the limit of all its associated parahoric group schemes
$\mathcal{G}_{\Omega}\xrightarrow{\cong}\varprojlim_{\mathfrak{f}\prec\Omega}\mathcal{G}_{\mathfrak{f}},$
compare Theorem 4.2.1.

Note that given a compatible system of $\mathcal{G}_{\mathfrak{f}}$-torsors
for all facets $\mathfrak{f}\prec\Omega$, it is in general not true that their
limit is a torsor for $\mathcal{G}_{\Omega}$, as it might be impossible to
construct a compatible system of sections. By controlling the deformation
theory of torsors for the $\mathcal{G}_{\mathfrak{f}}$, we are able to show
that the locus where the limit of a compatible system of
$\mathcal{G}_{\mathfrak{f}}$-bundles on $X$ is already a
$\mathcal{G}_{\Omega}$-bundle on $X$ is open.

As a first step in studying the structure of the special fibre of
$\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}$, we
show that the Newton stratification on the special fibre of the moduli space
of shtukas in the parahoric case induces a well-defined Newton stratification
on the special fibre in the case of deeper level. For a reductive group $H$
over a local field $E$ we denote by $B(H)$ the set of $\sigma$-conjugacy
classes in $H(\breve{E})$, where $\breve{E}$ is the completion of the maximal
unramified extension of $k$. Then $B(H)$ classifies quasi-isogeny classes of
local shtukas for (an integral model of) $H$.

We fix a tuple of pairwise distinct closed points $\underline{y}=(y_{i})_{i\in
I}$ in $X$ and denote by
$\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X,\mathbb{F}_{\underline{y}}}^{I,\leq\underline{\mu}}=\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X}^{I,\leq\underline{\mu}}\times_{X}\mathbb{F}_{\underline{y}}$
the special fibre over $\underline{y}$, where $\mathbb{F}_{\underline{y}}$ is
the compositum of the residue fields of the points $y_{i}$ of $X$.

Combining our compactification with the results of [HV11, Theorem 7.11] and
[Bre18, Section 5] in the parahoric case, we get the following result on the
Newton stratification for deep level.

######  Theorem 1.0.5 (compare Definition 5.4.3 and Corollary 5.4.4).

Let $\ell$ be an algebraically closed extension of
$\mathbb{F}_{\underline{y}}$. There is a well-defined map

$\bar{\delta}_{\mathcal{G}_{\Omega}}\colon\mathscr{S}\mathrm{ht}_{\mathcal{G}_{\Omega},X,\mathbb{F}_{\underline{y}}}^{I,\leq\underline{\mu}}(\ell)\to\prod_{i\in
I}B(G_{F_{y_{i}}})$

that is compatible with the level maps in the sense that for
$\Omega^{\prime}\prec\Omega$ we have

$\bar{\delta}_{\mathcal{G}_{\Omega}}=\bar{\delta}_{\mathcal{G}_{\Omega^{\prime}}}\circ\bar{\rho}_{\Omega^{\prime},\Omega}.$

Moreover, for $\underline{b}=(b_{i})_{i\in I}\in B(G_{F_{y_{i}}})$ the
preimage of $\underline{b}$ under $\bar{\delta}_{\mathcal{G}_{\Omega}}$ is the
set of $\ell$-valued points of a locally closed substack
$\operatorname{Sht}_{\mathcal{G}_{\Omega},X,\mathbb{F}_{\underline{y}}}^{I,\leq\underline{\mu},\underline{b}}$
of
$\operatorname{Sht}_{\mathcal{G}_{\Omega},X,\mathbb{F}_{\underline{y}}}^{I,\leq\underline{\mu}}$
called the _Newton stratum_ of
$\operatorname{Sht}_{\mathcal{G}_{\Omega},X,\mathbb{F}_{\underline{y}}}^{I,\leq\underline{\mu}}$
for $\underline{b}$.

In the parahoric case, the map $\bar{\delta}$ is given by associating to a
point in the special fibre over $\underline{y}$ the quasi-isogeny classes of
its local shtukas at the points $y_{i}$. We use the compatibility of the
Newton stratification with the level maps in the parahoric case to extend this
result to the case of deep level.

Moreover, we show that in the hyperspecial case the Newton stratification
satisfies the strong stratification property (as for Shimura varieties).
Recall that there is a natural order on $B(H)$ induced by the dominance order
on cocharacters. It is well-known in the parahoric case that the closure of a
Newton stratum

$\overline{\operatorname{Sht}^{I,\leq\underline{\mu},\underline{b}}_{\mathcal{G},X,\mathbb{F}_{\underline{y}}}}\subseteq\bigcup_{\underline{b^{\prime}}\leq\underline{b}}\operatorname{Sht}^{I,\leq\underline{\mu},\underline{b^{\prime}}}_{\mathcal{G},X,\mathbb{F}_{\underline{y}}}$

is contained in a union of Newton strata. Note that this also generalises to
deeper level. We say that the Newton stratification satisfies the strong
stratification property when we even have equality. However, the inclusion is
strict in general. For local shtukas for split reductive groups, the strong
stratification property is due to [Vie13]. We use the Serre-Tate theorem for
shtukas of [AH14] to deduce the corresponding result in the global case,
compare Theorem 5.4.5.

### Organisation

This paper is organised as follows. Sections 2 and 3 provide background on the
general theory of moduli spaces of shtukas. Our construction of integral
models for deeper level relies on functoriality results for moduli spaces of
shtukas. As the functoriality result of [Bre18, Theorem 3.20] is not
applicable in our setting, we first carefully set up (and slightly generalise)
the theory of bounds for (iterated, global and local shtukas) in Section 2. We
show basic properties including local-global compatibility. In Section 3, we
generalise various well-known results for moduli spaces of (bounded iterated)
shtukas to our setting. In particular, we show the relevant functoriality of
moduli spaces of shtukas in Theorem 3.5.5 which also proves the analogue of
Theorem 1.0.2 at parahoric level. In Section 4, we study (torsors under)
Bruhat-Tits group schemes as well as their deformation theory and show Theorem
1.0.4. In Section 5, we give the construction of integral models at deeper
Bruhat-Tits level and deduce our main results. We construct a Newton
stratification on the integral models with deeper level.

### Notation

We fix the following notation. For a Dedekind scheme $X$ over a field $k$
(usually $X$ will be a smooth (even projective) curve over $k$ or the spectrum
of $k\left\llbracket\varpi\right\rrbracket$ and $k$ will often assumed to be
finite) and a closed point $x$ of $X$ we denote by $\mathcal{O}_{X,x}$ the
local ring at $x$ and by $\mathcal{O}_{x}$ its completion, by
$\mathfrak{m}_{x}\subseteq\mathcal{O}_{x}$ the maximal ideal with uniformiser
$\varpi_{x}$ and by $\mathbb{F}_{x}$ the residue field. Moreover, we denote by
$F_{x}$ the completion of the function field $F=F(X)$ of $X$ at $x$. When
$k={\mathbb{F}_{q}}$ is a finite field, we denote by $\sigma$ the (absolute)
$q$-Frobenius endomorphism $\operatorname{Frob}_{S}$ of some
${\mathbb{F}_{q}}$-scheme $S$, and also the map
$\sigma=id_{X}\times\operatorname{Frob}_{S}\colon X_{S}\to X_{S}$.

### Acknowledgements

I thank my advisor Timo Richarz for introducing me to this topic, his steady
encouragement and his interest in my work. I thank the anonymous referee for
their careful reading of the paper and their valuable suggestions to clarify
the exposition. I thank Urs Hartl, Alexander Kutzim and Qihang Li for their
comments that lead to a significant improvement of the present work. I thank
Gebhard Böckle, Paul Hamacher, João Lourenço, Eva Viehmann and Torsten Wedhorn
for helpful conversations surrounding this work. I thank Catrin Mair and
Thibaud van den Hove for their comments on a preliminary version of this
paper. I thank Urs Hartl for sharing the revised version of [Bre18] and Tasho
Kaletha for sharing a preliminary version of [KP23].

##  2\. Bounds for (global and local) shtukas

Bounds for (global or local) shtukas usually are defined as certain
(equivalence classes of) subschemes of affine Beilinson-Drinfeld
Grassmannians, compare for example [Var04, Laf18, AH14, AH19]. In this section
we generalise the existing notion of bounds and provide a uniform treatment of
global and local bounds. An important class of bounds are generically defined
bounds, which arise as the closure of their generic fibre. In this case, the
theory simplifies. For example, we show that a bound always admits a
representative over its reflex field. Moreover, generically defined bounds are
the natural setting for our construction of integral models with deeper level
structure in the following.

This includes in particular the bounds given by Schubert varieties inside the
affine Grassmannian for parahoric group schemes that are often considered in
applications, compare for example [Laf18, YZ19, FYZ24]. We discuss the
relation of our notion of bounds to various other notions of bounds in the
literature.

The discussion of bounds is used to formulate and prove analogues of the
functoriality results for moduli spaces of shtukas in the next section. In
particular, generically defined bounds provide the natural setting to
formulate the functoriality of moduli spaces of shtukas under generic
isomorphisms of group schemes in Theorem 3.5.5.

###  2.1. Beilinson-Drinfeld affine Grassmannians

We recall the definition of Beilinson-Drinfeld affine Grassmannians. We work
in the following setting. Let $k$ be a field and let $X$ be either a smooth
geometrically connected curve over $k$ or
$X=\operatorname{Spec}(k\left\llbracket\varpi\right\rrbracket)$. Let $I$ be a
finite set and $I_{\bullet}=(I_{1},\ldots,I_{m})$ a partition of $I$. Let
$\mathcal{G}\to X$ be a flat affine $X$-group scheme of finite type.

We need the following iterated version of Beilinson-Drinfeld affine
Grassmannians first introduced by [BD96] in the case of constant split
reductive group schemes.

######  Definition 2.1.1.

We denote by $\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ the functor on
$k$-algebras whose $R$-valued points are given by tuples $((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\varepsilon),$
where

  * •

$x_{i}\in X(R)$ are points on $X$,

  * •

$\mathcal{E}_{j}$ are $\mathcal{G}$-bundles on $X_{R}$ for $0\leq j\leq m$,

  * •

$\varphi_{j}\colon\mathcal{E}_{j}|_{X_{R}\setminus\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}}\xrightarrow{\cong}\mathcal{E}_{j-1}|_{X_{R}\setminus\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}}$ are isomorphisms of $\mathcal{G}$-bundles, and

  * •

$\varepsilon\colon\mathcal{E}_{0}\xrightarrow{\cong}\mathcal{G}\times_{X}X_{R}$
is a trivialisation of $\mathcal{E}_{0}$.

Then $\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ is representable by a
separated (strict) ind-scheme of ind-finite type over $X^{I}$ by [Hei10,
Ric14] in the global setting generalising the previous works [BD96, Gai01],
and by [Ric16, Ric21] in the local setting.

As a shorthand notation we denote a point $((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\varepsilon)\in\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}(R)$
by

$\left(\mathcal{E}_{m}\overset{\varphi_{m}}{\underset{\Gamma_{\underline{x}_{m}}}{\dashrightarrow}}\mathcal{E}_{m-1}\overset{\varphi_{m-1}}{\underset{\Gamma_{\underline{x}_{m-1}}}{\dashrightarrow}}\ldots\overset{\varphi_{1}}{\underset{\Gamma_{\underline{x}_{1}}}{\dashrightarrow}}\mathcal{E}_{0}\overset{\varepsilon}{\rightarrow}\mathcal{G}\right),$

where for a point $\underline{x}=(x_{i})_{i\in I}\in X^{I}(R)$ we denote by
$\underline{x}_{j}=(x_{i})_{i\in I_{j}}$ the projection to the
$I_{j}$-components and by $\Gamma_{\underline{x}_{j}}=\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}$ the union of the graphs $\Gamma_{x_{i}}$ of
$x_{i}\colon\operatorname{Spec}(R)\to X$. When $I$ is a singleton set, we
denote the corresponding Beilinson-Drinfeld affine Grassmannian by
$\operatorname{Gr}_{\mathcal{G},X}$. In the local setting
$X=\operatorname{Spec}(\mathcal{O})$ we also write
$\operatorname{Gr}_{\mathcal{G},\mathcal{O}}=\operatorname{Gr}_{\mathcal{G},X}$.
Compare [Ric21] for an alternative description of
$\operatorname{Gr}_{\mathcal{G},\mathcal{O}}$. Note that when
$\mathcal{O}=\mathcal{O}_{x_{0}}$ is the complete local ring of a point
$x_{0}\in X$ of some smooth curve $X$, we have
$\operatorname{Gr}_{\mathcal{G}_{\mathcal{O}},\mathcal{O}}=\operatorname{Gr}_{\mathcal{G},X}\times_{X}\operatorname{Spec}(\mathcal{O})$,
where
$\mathcal{G}_{\mathcal{O}}=\mathcal{G}\times_{X}\operatorname{Spec}(\mathcal{O})$
is the pullback of $\mathcal{G}$ to $\operatorname{Spec}\mathcal{O}$ by
[Ric21].

Let $R$ be a $k$-algebra. For a relative effective Cartier divisor $D\subseteq
X_{R}$, the formal completion of $X_{R}$ along $D$ is a formal affine scheme.
We denote by $\hat{\mathcal{O}}_{D}$ the underlying $R$-algebra and by
$\hat{D}=\operatorname{Spec}(\hat{\mathcal{O}}_{D})$ the corresponding affine
scheme. Then $D$ is a closed subscheme of $\hat{D}$ and we set
$\hat{D}^{0}=\hat{D}\setminus D$. Moreover, we denote by
$D^{(r)}=\operatorname{Spec}(\hat{\mathcal{O}}_{D}/\mathfrak{m}^{r})$ the
$r$-the formal neighbourhood of $D$, where
$\mathfrak{m}\subseteq\hat{\mathcal{O}}_{D}$ is the ideal defining $D$. In
particular, when $D$ is the graph $\Gamma_{x}$ of a section $x\in X(R)$ (or a
union thereof), we use the corresponding notation $\hat{\Gamma}_{x}$,
$\hat{\Gamma}_{x}^{\circ}$ and $\Gamma_{x}^{(r)}$.

######  Remark 2.1.2.

Using Beauville-Laszlo descent, compare [BL95], a point of
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ is given by a tuple

$((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\varepsilon),$

where the $\mathcal{E}_{j}$ are now $\mathcal{G}$-torsors on
$\hat{\Gamma}_{\underline{x}}$, the
$\varphi_{j}\colon\mathcal{E}_{j}\to\mathcal{E}_{j-1}$ are isomorphisms over
$\hat{\Gamma}_{\underline{x}}\setminus\Gamma_{\underline{x}_{j}}$, and
$\varepsilon$ is a trivialisation of $\mathcal{E}_{0}$ defined over
$\hat{\Gamma}_{\underline{x}}$. This also shows that the natural map
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}|_{U}\xrightarrow{\cong}\left(\prod_{i\in
I}\operatorname{Gr}_{\mathcal{G},X}\right)|_{U}$ is an isomorphism, where
$U\subseteq X^{I}$ is the complement of all diagonals.

We use the following global version of the (positive) loop group.

######  Definition 2.1.3.

The _global loop group_ $\mathcal{L}_{X^{I}}\mathcal{G}$ is the functor on the
category of $k$-algebras

$R\mapsto\left\\{(\underline{x},g)\colon\underline{x}\in
X^{I}(R),g\in\mathcal{G}(\hat{\Gamma}_{\underline{x}}^{0})\right\\}.$

The _positive global loop group_ $\mathcal{L}_{X^{I}}^{+}\mathcal{G}$ is the
functor on the category of $k$-algebras

$R\mapsto\left\\{(\underline{x},g)\colon\underline{x}\in
X^{I}(R),g\in\mathcal{G}(\hat{\Gamma}_{\underline{x}})\right\\}.$

Moreover, for $r\in\mathbb{N}$ let $\mathcal{L}_{X^{I}}^{(r)}\mathcal{G}$ be
the functor on $k$-algebras

$R\mapsto\left\\{(\underline{x},g)\colon\underline{x}\in
X^{I}(R),g\in\mathcal{G}(\Gamma_{\underline{x}}^{(r)})\right\\}.$

By [Hei10, Proposition 2], $\mathcal{L}_{X^{I}}\mathcal{G}$ is representable
by an ind-affine group ind-scheme over $X^{I}$ and
$\mathcal{L}_{X^{I}}^{+}\mathcal{G}$ is representable by a closed affine group
subgroup scheme of $\mathcal{L}_{X^{I}}\mathcal{G}$ over $X^{I}$. Moreover by
[Ric16, Lemma 2.11], all $\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}\to X^{I}$ are
representable by flat affine group schemes of finite type and the transition
maps for varying $r$ are affine. The canonical map given by the projections

(1)
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}\xrightarrow{\cong}\varprojlim_{r}\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}$

is an isomorphism. Assume moreover that $\mathcal{G}$ is smooth. Then all
$\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}$ are smooth and by (1) we have that
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}\to X^{I}$ is flat.

There is a natural $\mathcal{L}^{+}_{X^{I}}\mathcal{G}$-action on
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ by changing the
trivialisation $\varepsilon$.

###  2.2. Definition of bounds

We define bounds inside the iterated affine Grassmannian by adapting the
definition in the non-iterated setting of [AH19, Definition 3.1.3]. We work in
the setup of [HX23, Section 2.6] and allow different types of bounds
(global/local/finite) at different legs. More precisely we define bounds in
the following cases.

######  Definition 2.2.1.

  1. (1)

For each $i\in I$ a ground field $F^{(i)}$ is fixed, which can be either

    1. (a)

_of global type_ $F^{(i)}=F$ (only in the _global setting_ when $X$ is a
smooth geometrically connected curve over $k$),

    2. (b)

_of local type_ $F^{(i)}=F_{y}$, the completion of $F$ at a closed point $y$
of $X$,

    3. (c)

or _of finite type_ $F^{(i)}=\mathbb{F}_{y}$, the residue field at $y$.

  2. (2)

All finite extension of $F^{(i)}$ are without further mention assumed to be
separable and contained in a fixed separable closure $\widebar{F}^{(i)}$ of
$F^{(i)}$. We set $\widetilde{X}_{F^{(i)}}$ to be either $X$,
$\operatorname{Spec}(\mathcal{O}_{y})$ or
$\operatorname{Spec}(\mathbb{F}_{y})$ according to the type of $F^{(i)}$.
Moreover, let $\underline{F}=(F^{(i)})_{i\in I}$ and
$\widetilde{X}_{\underline{F}}=\prod_{i\in I}\widetilde{X}_{F^{(i)}}$, where
the product is taken over $k$. For a finite extension $(F^{(i)})^{\prime}$ of
$F^{(i)}$ we set $\widetilde{X}_{(F^{(i)})^{\prime}}$ to be the normalisation
of $\widetilde{X}_{F^{(i)}}$ in $(F^{(i)})^{\prime}$. It is either a smooth
curve in the global type, the (spectrum of the) valuation ring of
$(F^{(i)})^{\prime}$ in the local type or
$\operatorname{Spec}(F^{(i)})^{\prime}$ in the finite type. In each case the
natural map $\widetilde{X}_{(F^{(i)})^{\prime}}\to\widetilde{X}_{F^{(i)}}$ is
finite and faithfully flat. For an $I$-tuple
$\underline{F}^{\prime}=((F^{(i)})^{\prime})_{i\in I}$ of componentwise finite
extensions of $\underline{F}$ we set
$\widetilde{X}^{I}_{\underline{F}^{\prime}}=\prod_{i\in
I}\widetilde{X}_{(F^{(i)})^{\prime}}$.

  3. (3)

Let $\underline{F}_{1}$ and $\underline{F}_{2}$ be two $I$-tuples of finite
extensions of $\underline{F}$. Two quasi-compact closed subschemes
$Z_{1}\subseteq\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}_{1}}^{I}$
and
$Z_{2}\subseteq\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}_{2}}^{I}$
are called _equivalent_ if there is an $I$-tuple of finite extensions
$\underline{F}^{\prime}$ of $F$ such that componentwise
$\underline{F}^{\prime}$ is a finite extension of both $\underline{F}_{1}$ and
$\underline{F}_{2}$ with
$Z_{1}\times_{\widetilde{X}_{\underline{F}_{1}}^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}=Z_{2}\times_{\widetilde{X}_{\underline{F}_{2}}^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$
in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$.

  4. (4)

Let $\mathcal{Z}$ be an equivalence class of quasi-compact closed subschemes
$Z_{\underline{F}^{\prime}}\subseteq\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$
and let us fix a representative $Z_{\underline{F}^{\prime}}$ defined over some
$\underline{F}^{\prime}$ such that all components $(F^{(i)})^{\prime}$ are
Galois extensions of $F^{(i)}$. We set
$\operatorname{Aut}_{\mathcal{Z}}(\underline{F}^{\prime})=\\{(g_{i})_{i\in
I}\in\operatorname{Gal}((F^{(i)})^{\prime}/F^{(i)}):(g_{i})_{i\in
I}^{\ast}(\mathcal{Z})=\mathcal{Z}\\}$. The _reflex scheme_
$\widetilde{X}^{I}_{\mathcal{Z}}$ of $\mathcal{Z}$ is defined to be the
quotient

$\widetilde{X}^{I}_{\mathcal{Z}}=\widetilde{X}^{I}_{\underline{F}^{\prime}}/\operatorname{Aut}_{\mathcal{Z}}(\underline{F}^{\prime}).$

  5. (5)

By Lemma 2.2.3 the reflex scheme is independent of the choice of
representative. We say that $\mathcal{Z}$ _admits a representative over
$\widetilde{X}^{I}_{\mathcal{Z}}$_ if $Z_{\underline{F}^{\prime}}$ (or by
Lemma 2.2.3 any other representative of $\mathcal{Z}$) descends to
$\widetilde{X}^{I}_{\mathcal{Z}}$.

  6. (6)

A _bound_ is an equivalence class $\mathcal{Z}$ of quasi-compact closed
subschemes
$Z_{\underline{F}^{\prime}}\subset\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$,
such that all its representatives $Z_{\underline{F}^{\prime}}$ are stable
under the left
$\mathcal{L}_{X^{I}}^{+}\mathcal{G}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$-action
on
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$.

######  Remark 2.2.2.

The definition of the reflex scheme is due to [HX23, Definition 2.6.3]. We
thank U.Hartl for suggesting to use it in the present work.

  1. (1)

Later only global bounds (i.e. when $X$ is a curve) of global type and local
bounds (i.e. when
$X=\operatorname{Spec}(k\left\llbracket\varpi\right\rrbracket))$) of local
type will be relevant and the reader who is only interested in these
situations can safely ignore the extra generality we allow in the definition.
However, restricting to this situation does not significantly simplify the
arguments in this section. Bounds (in the global setting) with legs of
different types are for example used in [HX23].

  2. (2)

In general, the reflex scheme can be complicated, but in all cases relevant
later it will be a product $\prod_{i\in I}\tilde{X}_{F_{i}^{\prime}}$ for
finite separable extensions $F_{i}^{\prime}/F$. In particular, when
$I=\\{\ast\\}$ contains only one element
$\widetilde{X}_{\mathcal{Z}}=\widetilde{X}_{F_{\mathcal{Z}}}$ where
$F_{\mathcal{Z}}$ is the field of definition of the bound $\mathcal{Z}$ in the
sense of [AH19, Definition 3.1.3]. In many cases the reflex scheme of a bound
can be determined as in [AH19, Remark 3.1.4] (and [AH14, Remark 4.7] in the
local setting). It is however not clear if there always exists a
representative over the reflex scheme in general.

  3. (3)

By construction, the reflex scheme comes equipped with maps
$\widetilde{X}^{I}_{\underline{F}^{\prime}}\to\widetilde{X}^{I}_{\mathcal{Z}}\to\widetilde{X}_{\underline{F}}^{I}$
for any $I$-tuple $\underline{F}^{\prime}$ of finite extensions of
$\underline{F}$ such that $\mathcal{Z}$ admits a representative over
$\underline{F}^{\prime}$.

The following lemma is an analogue of [AH14, Remark 4.6].

######  Lemma 2.2.3.

Let $\mathcal{Z}$ be an equivalence class of closed subschemes of
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ and let
$\underline{F}^{\prime}\subseteq\underline{F}^{\prime\prime}$ be two
$I$-tuples of finite extensions of $F$ such that $\mathcal{Z}$ admits
representatives $Z_{\underline{F}^{\prime}}$ and
$Z_{\underline{F}^{\prime\prime}}$ over $\underline{F}^{\prime}$ and
$\underline{F}^{\prime\prime}$, respectively. Then
$Z_{\underline{F}^{\prime}}$ is the scheme-theoretic image of the map
$Z_{\underline{F}^{\prime\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime}}$.
In particular, if $\mathcal{Z}$ has a representative defined over some
$\underline{F}^{\prime}$, it is unique.

###### Proof.

Let $\underline{F}^{\prime\prime\prime}$ be componentwise a common extension
of $\underline{F}^{\prime}$ and $\underline{F}^{\prime\prime}$ such that
$Z_{\underline{F}^{\prime}}\times_{\tilde{X}^{I}_{\underline{F}^{\prime}}}\tilde{X}^{I}_{\underline{F}^{\prime\prime\prime}}=Z_{\underline{F}^{\prime\prime}}\times_{\tilde{X}^{I}_{\underline{F}^{\prime\prime}}}\tilde{X}^{I}_{\underline{F}^{\prime\prime\prime}}$.
Then the projection
$Z_{\underline{F}^{\prime\prime\prime}}\hookrightarrow\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime\prime\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime}}$
factors through $Z_{\underline{F}^{\prime}}$. But as the question is Zariski-
local on $Z_{\underline{F}^{\prime}}$, we may assume that
$\tilde{X}^{I}_{\underline{F}^{\prime}}=\operatorname{Spec}(R)$ and
$Z_{\underline{F}^{\prime}}=\operatorname{Spec}(A)$ are affine. Then both
$\tilde{X}^{I}_{\underline{F}^{\prime\prime\prime}}=\operatorname{Spec}(R^{\prime})$
and
$Z_{\underline{F}^{\prime\prime\prime}}=\operatorname{Spec}(A\otimes_{R}R^{\prime})$
are affine as well. The map $R\to R^{\prime}$ is a finite and faithfully flat
and thus a universally injective map of $R$-modules. Hence, the map $A\to
A\otimes_{R}R^{\prime}$ is injective. This shows that
$Z_{\underline{F}^{\prime}}$ is the scheme-theoretic image of
$Z_{\underline{F}^{\prime\prime\prime}}\to Z_{\underline{F}^{\prime}}$. By the
functoriality properties of scheme-theoretic images the projection
$Z_{\underline{F}^{\prime\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime}}$
factors through $Z_{\underline{F}^{\prime}}$ and has
$Z_{\underline{F}^{\prime}}$ as its scheme-theoretic image. ∎

One way to construct bounds is to prescribe a bound for each leg as follows.

######  Construction 2.2.4.

Assume $\mathcal{L}^{+}_{X^{I}}\mathcal{G}\to X^{I}$ is flat. Recall that this
is satisfied when $\mathcal{G}\to X$ is smooth by [Ric16, Lemma 2.11]. Let
$(\mathcal{Z}^{(i)})_{i\in I}$ be an $I$-tuple of bounds in
$\operatorname{Gr}_{\mathcal{G},X}$ with reflex schemes
$\widetilde{X}_{\mathcal{Z}^{(i)}}$. From the $(\mathcal{Z}^{(i)})_{i\in I}$
we construct a bound $\mathcal{Z}=\prod_{i\in I}\mathcal{Z}^{(i)}$ in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ as follows. Let
$\underline{F}^{\prime}=({F^{(i)}}^{\prime})_{i\in I}$ be an $I$-tuple of
finite extensions of $F$ such that $\mathcal{Z}^{(i)}$ admits a representative
over ${F^{(i)}}^{\prime}$. Let $U\subseteq X^{I}$ be the complement of all
diagonals and set
$U_{\underline{F}^{\prime}}=U\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime}}$.
If no two different components $\tilde{X}_{F^{(i)}}$ are of the finite type
(1c) in Definition 2.2.1, then $U_{\underline{F}^{\prime}}$ is non-empty. In
this situation we define

$Z_{\underline{F}^{\prime}}=\mathrm{image}\left(\prod_{i\in
I}Z^{(i)}_{{F^{(i)}}^{\prime}}|_{U_{\underline{F}^{\prime}}}\hookrightarrow\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}_{\underline{F}^{\prime}}^{I}\right)$

as the scheme-theoretic closure of the product of the
$Z^{(i)}_{{F^{(i)}}^{\prime}}$ away from the diagonals via the map of Remark
2.1.2. We denote by $\mathcal{Z}$ the equivalence class of
$Z_{\underline{F}^{\prime}}$.

In a similar fashion, we can define a bound $\mathcal{Z}$ in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ for any refinement
$I_{\bullet}^{\prime}=(I^{\prime}_{j^{\prime}})_{1\leq j^{\prime}\leq
m^{\prime}}$ of $I_{\bullet}$ and bounds
$\mathcal{Z}^{(I^{\prime}_{j^{\prime}})}$ in
$\operatorname{Gr}_{\mathcal{G},X}^{(I^{\prime}_{j^{\prime}})}$.

######  Lemma 2.2.5.

In the situation of Construction 2.2.4, let $\underline{F}^{\prime}$ and
$\underline{F}^{\prime\prime}$ be two $I$-tuples of finite extensions of
$\underline{F}$ such that the $\mathcal{Z}^{(i)}$ admit representatives over
both $(F^{(i)})^{\prime}$ and $(F^{(i)})^{\prime\prime}$. Let
$Z_{\underline{F}^{\prime}}$ and $Z_{\underline{F}^{\prime\prime}}$ be the
corresponding subschemes of the affine Grassmannian as constructed in
Construction 2.2.4.

  1. (1)

Then $Z_{\underline{F}^{\prime}}$ and $Z_{\underline{F}^{\prime\prime}}$ are
equivalent and stable under the $\mathcal{L}_{X^{I}}^{+}\mathcal{G}$-action.
In particular, $\mathcal{Z}$ is a bound and independent of the choice of
$\underline{F}^{\prime}$. The reflex scheme of $\mathcal{Z}=\prod_{i\in
I}\mathcal{Z}^{(i)}$ is $\widetilde{X}^{I}_{\mathcal{Z}}=\prod_{i\in
I}\widetilde{X}_{\mathcal{Z}^{(i)}}$.

  2. (2)

Assume that $\mathcal{Z}$ admits a representative
$Z_{\underline{F}^{\prime\prime\prime}}$ over
$\underline{F}^{\prime\prime\prime}$. Then for every $i\in I$ the bound
$\mathcal{Z}^{(i)}$ admits a representative
$Z^{(i)}_{(F^{(i)})^{\prime\prime\prime}}$ over
$(F^{(i)})^{\prime\prime\prime}$ and $Z_{\underline{F}^{\prime\prime\prime}}$
arises via Construction 2.2.4 from the
$Z^{(i)}_{(F^{(i)})^{\prime\prime\prime}}$.

###### Proof.

  1. (1)

That $Z_{\underline{F}^{\prime}}$ and $Z_{\underline{F}^{\prime\prime}}$ are
equivalent follows from the fact that the formation of scheme-theoretic images
commutes with flat base change.

By assumption, the action map
$\mathcal{L}_{X^{I}}^{+}\mathcal{G}|_{U_{\underline{F}^{\prime}}}\times_{U_{\underline{F}^{\prime}}}\prod_{i\in
I}Z^{(i)}_{(F^{(i)})^{\prime}}|_{U_{\underline{F}^{\prime}}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}|_{U_{\underline{F}^{\prime}}}$
factors through $Z_{U_{\underline{F}^{\prime}}}$. By the functoriality of
scheme-theoretic images together with the fact that
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}\to X^{I}$ is flat, also
$Z_{\underline{F}^{\prime}}$ is stable under the action of
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$.

  2. (2)

We first note that the representative $Z_{{F^{(i)}}^{\prime}}^{(i)}$ of
$\mathcal{Z}^{(i)}$ can be recovered from $Z_{\underline{F}^{\prime}}$ as the
scheme-theoretic image

$Z_{(F^{(i)})^{\prime}}^{(i)}=\mathrm{image}\left(Z_{\underline{F}^{\prime}}|_{U_{\underline{F}^{\prime}}}\to\prod_{i\in
I}\operatorname{Gr}_{\mathcal{G},X}|_{U_{\underline{F}^{\prime}}}\xrightarrow{\mathrm{pr^{(i)}}}\operatorname{Gr}_{\mathcal{G},X}\times_{X}\tilde{X}_{(F^{(i)})^{\prime}}\right),$

where $\mathrm{pr}^{(i)}$ denotes the projection to the $i$-th component.

Let now $\underline{F}^{\prime\prime\prime}$ be an $I$-tuple of extensions of
$\underline{F}$ as in the assertion and let
$Z_{\underline{F}^{\prime\prime\prime}}$ be a representative of $\mathcal{Z}$
over $\underline{F}^{\prime\prime\prime}$. As above, we denote by
$Z_{(F^{(i)})^{\prime\prime\prime}}^{(i)}$ the scheme-theoretic image of the
map
$Z_{\underline{F}^{\prime\prime\prime}}|_{U_{\underline{F}^{\prime\prime\prime}}}\xrightarrow{\mathrm{pr}^{(i)}}\operatorname{Gr}_{\mathcal{G},X}\times_{X}\tilde{X}_{(F^{(i)})^{\prime\prime\prime}}$.
Without loss of generality, we may assume that $F^{\prime}$ is an extension of
$F^{\prime\prime\prime}$. As the formation of scheme-theoretic images commutes
with flat base change, $Z_{(F^{(i)})^{\prime\prime\prime}}^{(i)}$ and
$Z_{(F^{(i)})^{\prime}}^{(i)}$ are equivalent. In particular,
$\mathcal{Z}^{(i)}$ admits a representative over
$(F^{(i)})^{\prime\prime\prime}$. That
$Z_{\underline{F}^{\prime\prime\prime}}$ arises from Construction 2.2.4 then
follows from the fact that representatives of a bound over a given field
extension are unique by Lemma 2.2.3.

∎

######  Remark 2.2.6.

Note that we crucially used the the flatness of
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$ to show that the representatives of
$\mathcal{Z}$ are stable under the
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$-action. It does not seem to be clear
whether or not $\mathcal{L}^{+}_{X^{I}}\mathcal{G}\to X^{I}$ is flat in
general for flat affine group schemes $\mathcal{G}\to X$ of finite type.

On the other hand, from a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ we can for every subset
$I^{\prime}\subseteq I$ construct a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{(I^{\prime}_{\bullet})}$ essentially by
projection to the $I^{\prime}$-components, where
$I^{\prime}_{\bullet}=(I^{\prime}\cap I_{j})_{1\leq j\leq m}$. This
generalises the construction from [AH19, proof of Proposition 4.3.3].

######  Construction 2.2.7.

Let $\mathcal{Z}$ be a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ with a representative
$Z_{\underline{F}^{\prime}}$ defined over $\underline{F}^{\prime}$. Let
$I^{\prime}\subseteq I$. We denote by
$Z_{\underline{F}^{\prime}}^{(I^{\prime})}$ the scheme-theoretic image of the
projection
$Z_{\underline{F}^{\prime}}|_{U_{\underline{F}^{\prime}}}\hookrightarrow(\prod_{i\in
I}\operatorname{Gr}_{\mathcal{G},X})|_{U_{\underline{F}^{\prime}}}\to\operatorname{Gr}_{\mathcal{G},X}^{I^{\prime}_{\bullet}}\times_{X^{I^{\prime}}}\widetilde{X}_{\underline{F}^{\prime}}^{I^{\prime}}$
to the $I^{\prime}$-components. We denote by $\mathcal{Z}^{(I^{\prime})}$ the
equivalence class of $Z_{\underline{F}^{\prime}}^{(I^{\prime})}$. Clearly,
$\mathcal{Z}^{(I^{\prime})}$ does not depend on the choice of representative
$Z_{\underline{F}^{\prime}}$.

For the computation of the reflex scheme of $\mathcal{Z}^{(I^{\prime})}$ we
note that
$\operatorname{Aut}_{\mathcal{Z}^{(I^{\prime})}}(\underline{F}^{\prime})$ is
just the projection of
$\operatorname{Aut}_{\mathcal{Z}}(\underline{F}^{\prime})$ to the
$I^{\prime}$-components. In particular, when $\mathcal{Z}=\prod_{i\in
I}\mathcal{Z}^{(i)}$ as in 2.2.4, we find
$\mathcal{Z}^{(I^{\prime})}=\prod_{i\in I^{\prime}}\mathcal{Z}^{(i)}$ and the
map $\prod_{i\in
I}\widetilde{X}^{I}_{\mathcal{Z}^{(i)}}=\widetilde{X}^{I}_{\mathcal{Z}}\to\widetilde{X}^{I^{\prime}}_{\mathcal{Z}^{(I^{\prime})}}=\prod_{i\in
I^{\prime}}\widetilde{X}^{I}_{\mathcal{Z}^{(i)}}$ is projection to the
$I^{\prime}$-components.

Assume additionally that $\mathcal{L}^{+}_{X}\mathcal{G}\to X$ is flat. By a
similar argument as in Lemma 2.2.5, we see that
$Z^{(I^{\prime})}_{\underline{F}^{\prime}}$ is stable under
$\mathcal{L}^{+}_{X^{I^{\prime}}}\mathcal{G}\times_{X^{I^{\prime}}}\widetilde{X}_{\underline{F}^{\prime}}$.
Hence, $\mathcal{Z}^{(I^{\prime})}$ is a bound.

We give two conditions for points on the affine Grassmannian to factor through
(representatives of) a bound. These lemmas will be helpful to check
boundedness of (global or local) shtukas in the next section.

######  Lemma 2.2.8.

Let $\mathcal{Z}$ be a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. Let $S$ be a $k$-scheme and
let
$g\in\left(\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}^{I}_{\mathcal{Z}}\right)(S)$.
Then $g\times\mathrm{id}_{\widetilde{X}_{\underline{F}^{\prime}}}\colon
S\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}$
factors through $Z_{\underline{F}^{\prime}}$ for one $I$-tuple of finite
extensions $\underline{F}^{\prime}$ for which a representative
$Z_{\underline{F}^{\prime}}$ of $\mathcal{Z}$ exists if and only if
$g\times\mathrm{id}_{\tilde{X}_{\underline{F}^{\prime}}}$ factors through
$Z_{\underline{F}^{\prime}}$ for all such $I$-tuples of finite extensions
$\underline{F}^{\prime}$.

###### Proof.

Assume that $g\times\mathrm{id}_{\widetilde{X}_{\underline{F}^{\prime}}}$
factors through $Z_{\underline{F}^{\prime}}$ for some $\underline{F}^{\prime}$
and let $\underline{F}^{\prime\prime}$ be a second $I$-tuple of finite
extensions such that $\mathcal{Z}$ admits a representative over
$\underline{F}^{\prime\prime}$. By replacing $\underline{F}^{\prime}$ by the
componentwise compositum of $\underline{F}^{\prime}$ and
$\underline{F}^{\prime\prime}$, we may assume that $\underline{F}^{\prime}$ is
a componentwise extension of $\underline{F}^{\prime\prime}$. The claim then
follows from Lemma 2.2.3. ∎

######  Lemma 2.2.9.

Let $\mathcal{Z}$ be a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. Let $S$ be a $k$-scheme and
let $g\colon
S\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}$
be a point in the affine Grassmannian for some $I$-tuple of finite extensions
$\underline{F}^{\prime}$ such that $\mathcal{Z}$ admits a representative over
$\underline{F}^{\prime}$.

  1. (1)

Assume that
$g\times\mathrm{id}_{\widetilde{X}_{\underline{F}^{\prime}}^{I}}\colon
S\times_{Y}\tilde{X}^{I}_{\underline{F}^{\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime}}$
factors through $Z_{\underline{F}^{\prime}}$ for some scheme
$Y\to\widetilde{X}^{I}_{\mathcal{Z}}$ such that the map
$\widetilde{X}^{I}_{\underline{F}^{\prime\prime}}\to\widetilde{X}^{I}_{\mathcal{Z}}$
factors through $Y$. Then $g$ factors through $Z_{\underline{F}^{\prime}}$.

  2. (2)

Assume that $\mathcal{Z}$ admits a representative over
$\widetilde{X}^{I}_{\mathcal{Z}}$. If $g$ factors through
$Z_{\underline{F}^{\prime}}$, then the induced map

$g\times\mathrm{id}_{\tilde{X}_{\underline{F}^{\prime\prime}}^{I}}\colon
S\times_{\widetilde{X}^{I}_{\mathcal{Z}}}\tilde{X}^{I}_{\underline{F}^{\prime\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime\prime}}$

factors through $Z_{\underline{F}^{\prime\prime}}$ for all finite extensions
$\underline{F}^{\prime\prime}$ such that $\mathcal{Z}$ admits a representative
$Z_{\underline{F}^{\prime\prime}}$ over $\underline{F}^{\prime\prime}$.

###### Proof.

  1. (1)

By construction, $g$ factors through $S\to
S\times_{Y}\tilde{X}^{I}_{\underline{F}^{\prime}}\xrightarrow{g\times\mathrm{id}_{\tilde{X}_{\underline{F}^{\prime}}^{I}}}\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times\tilde{X}^{I}_{\underline{F}^{\prime}}$.

  2. (2)

Assume that $g\colon
S\to\operatorname{Gr}_{G,X^{I},I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$
factors through $Z_{\underline{F}^{\prime}}$. By Lemma 2.2.3, the induced map
$g\colon
S\to\operatorname{Gr}_{G,X^{I},I_{\bullet}}\times_{X^{I}}X_{\mathcal{Z}}^{I}$
factors through the representative $Z_{\widetilde{X}_{\mathcal{Z}}}$ over the
reflex scheme. Hence, $g\times\mathrm{id}_{\tilde{X}_{\tilde{F}}^{I}}\colon
S\times_{X^{I}_{\mathcal{Z}}}\widetilde{X}^{I}_{\underline{F}^{\prime\prime}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}^{I}_{\underline{F}^{\prime\prime}}$
factors through $Z_{\underline{F}^{\prime\prime}}$ for all finite extensions
$\underline{F}^{\prime\prime}$ by the argument of Lemma 2.2.8.

∎

######  Remark 2.2.10.

We do not know whether (2) holds without the assumption that $\mathcal{Z}$
admits a representative over the reflex field. However, this assumption is
satisfied for all examples of bounds considered in practice, in particular for
all bounds considered in the following two subsections.

###  2.3. Generically defined bounds

We now discuss a special class of bounds, namely those that are _generically
defined_ in the sense that they are given as the scheme-theoretic closure of
their generic fibre. In this case, the theory simplifies significantly, for
example we show that in this case representatives always exist over the reflex
field. In this subsection we always assume without further mention that
$\mathcal{L}_{X^{I}}^{+}\mathcal{G}$ is flat over $X^{I}$, which is the case
for example if $\mathcal{G}\to X$ is smooth by [Ric16, Lemma 2.11].

For this subsection we restrict ourselves to bounds of global type in the
global setting or bounds of local type in the local setting.

######  Definition 2.3.1.

Let $\mathcal{Z}$ be a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. We say that $\mathcal{Z}$
is _generically defined_ if there is an $I$-tuple of finite extensions
$\underline{F}^{\prime}$ of $F$ such that there exists a representative
$Z_{\underline{F}^{\prime}}$ over $\underline{F}^{\prime}$ which is the
scheme-theoretic closure of its generic fibre over every connected component
of $\widetilde{X}^{I}_{\underline{F}^{\prime}}$.

######  Remark 2.3.2.

A bound with a representative $Z_{\underline{F}^{\prime}}$ that is flat over
$\widetilde{X}_{\underline{F}^{\prime}}$ is clearly generically defined, and
all its representatives will then be flat. When $I$ has only one element, any
representative of a generically defined bound in
$\operatorname{Gr}_{\mathcal{G},X}$ is automatically flat over
$\widetilde{X}_{F^{\prime}}$ (as the base is a Dedekind scheme in this case).

######  Lemma 2.3.3.

Let $\mathcal{Z}$ be a generically defined bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. Then every representative
of $\mathcal{Z}$ arises as the closure of its generic fibre.

###### Proof.

Let $\underline{F}^{\prime}$ be such that the representative
$Z_{\underline{F}^{\prime}}$ of $\mathcal{Z}$ is the closure of its generic
fibre. As taking the scheme theoretic image commutes with flat base change,
this property remains true for all representatives defined over some finite
extension of $\underline{F}^{\prime}$. Let $\underline{F}^{\prime\prime}$ be a
second $I$-tuple of field extensions over which a representative of
$\mathcal{Z}$ exists. Without loss of generality we may assume that
$\underline{F}^{\prime}$ is a componentwise extension of
$\underline{F}^{\prime\prime}$. Then the induced map of the generic fibre
$Z_{\underline{F}^{\prime},\eta_{\underline{F}^{\prime}}^{I}}\to
Z_{\underline{F}^{\prime\prime}}$ factors through
$Z_{\underline{F}^{\prime\prime},\eta_{\underline{F}^{\prime\prime}}^{I}}$ and
both the maps $Z_{\underline{F}^{\prime},\eta_{\underline{F}^{\prime}}^{I}}\to
Z_{\underline{F}^{\prime}}\to Z_{\underline{F}^{\prime\prime}}$ are scheme-
theoretically surjective. This shows the claim. ∎

We show that Constructions 2.2.4 and 2.2.7 preserve the property of being
generically defined.

######  Lemma 2.3.4.

Let $\mathcal{Z}$ be a bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. Then $\mathcal{Z}$ is
generically defined if and only if there exist generically defined bounds
$(\mathcal{Z}^{(i)})_{i\in I}$ in $\operatorname{Gr}_{\mathcal{G},X}$ such
that $\mathcal{Z}$ arises from $(\mathcal{Z}^{(i)})_{i\in I}$ as in
Construction 2.2.4.

The $\mathcal{Z}^{(i)}$ are unique by the Lemma 2.2.5. We call them the
components of $\mathcal{Z}$.

###### Proof.

Assume first that $\mathcal{Z}$ arises from generically defined bounds
$\mathcal{Z}^{(i)}$ via Construction 2.2.4. Let $\underline{F}^{\prime}$ be an
$I$-tuple of finite extensions of $F$ such that $\mathcal{Z}^{(i)}$ admits a
representative over $(F^{(i)})^{\prime}$ for every $i\in I$. As the generic
point of $X^{I}$ lies in the complement $U$ of all diagonals and $\prod_{i\in
I}Z^{(i)}_{(F^{(i)})^{\prime}}$ is flat over $X^{I}$, $\left(\prod_{i\in
I}Z^{(i)}_{F^{\prime}}\right)|_{U_{\underline{F}^{\prime}}}$ is the closure of
its generic fibre.

Conversely, assume that $\mathcal{Z}$ is generically defined and let
$(\mathcal{Z}^{(i)})_{i\in I}$ be the components of $\mathcal{Z}$ from
Construction 2.2.7. By construction, all $\mathcal{Z}^{(i)}$ are generically
defined and the generic fibre of any representative of $\mathcal{Z}$ is the
product of the corresponding representatives of the $\mathcal{Z}^{(i)}$. This
shows that $\mathcal{Z}$ arises from the $\mathcal{Z}^{(i)}$ via Construction
2.2.4. ∎

######  Lemma 2.3.5.

Let $\mathcal{Z}$ be a generically defined bound. Its reflex scheme is given
by $\widetilde{X}^{I}_{\mathcal{Z}}=\prod_{i\in
I}\widetilde{X}_{\mathcal{Z}^{(i)}}$ where the $\mathcal{Z}^{(i)}$ are the
components of $\mathcal{Z}$ as in Lemma 2.3.4. Moreover, $\mathcal{Z}$ admits
a representative over its reflex scheme.

###### Proof.

The description of the reflex scheme follows from Lemma 2.2.5. Moreover, it
suffices to show that $\mathcal{Z}$ admits a representative over its reflex
scheme in the case $I=\\{\ast\\}$. In this case
$\widetilde{X}_{\mathcal{Z}}=\widetilde{X}_{F_{\mathcal{Z}}}$ for some finite
extension $F_{\mathcal{Z}}/F$. In particular there is a finite Galois
extension $F^{\prime}/F_{\mathcal{Z}}$ such that there exists a representative
$Z_{F^{\prime}}$ of $\mathcal{Z}$ over $F^{\prime}$. By assumption it suffices
to descend $Z_{F^{\prime},\eta_{F^{\prime}}}$ to some subscheme
$Z_{{F_{\mathcal{Z}}},\eta_{{F_{\mathcal{Z}}}}}$ of the affine Grassmannian
over $\eta_{F_{\mathcal{Z}}}$, the representative of $\mathcal{Z}$ over
$F_{\mathcal{Z}}$ will then be given by the scheme-theoretic closure of
$Z_{{F_{\mathcal{Z}}},\eta_{{F_{\mathcal{Z}}}}^{I}}$. But
$\eta_{F^{\prime}}\to\eta_{{F_{\mathcal{Z}}}}$ is a finite étale Galois cover
with Galois group $\operatorname{Gal}(F^{\prime}/{F_{\mathcal{Z}}})$ and
$Z_{F^{\prime},\eta_{F^{\prime}}}$ is fixed by the Galois action by
assumption. Thus, $Z_{F^{\prime},\eta_{F^{\prime}}}$ descends to
${F_{\mathcal{Z}}}$. ∎

######  Remark 2.3.6.

By Lemma 2.3.4, a generically defined bound is uniquely determined by an
$I$-tuple $(Z^{(i)})$ of (equivalence classes of)
$\mathcal{L}_{X}^{+}\mathcal{G}\times_{X}\eta$-stable closed subschemes of the
generic fibre $\operatorname{Gr}_{\mathcal{G},X}\times_{X}\eta$ of the
Beilinson-Drinfeld affine Grassmannian.

If the generic fibre $G$ of $\mathcal{G}$ is reductive, the generic fibre of
Beilinson-Drinfeld Grassmannian for $\mathcal{G}$ can be identified (non
canonically) with the (classical) affine Grassmannian $\operatorname{Gr}_{G}$
for
$G\times_{\operatorname{Spec}(F)}\operatorname{Spec}(F\left\llbracket\varpi\right\rrbracket)$
by [Ric21, Section 0.2]. Hence, $I$-tuples of equivalence classes of closed
subschemes (that are stable under the loop group action) in
$\operatorname{Gr}_{G}$ determine bounds in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ for all smooth models
$\mathcal{G}\to X$ of $G$ by taking closures. Note that the bounds do not
depend on the choice of isomorphism
$\operatorname{Gr}_{\mathcal{G},X}\times_{X}\eta\cong\operatorname{Gr}_{G}$.

By a slight abuse of notation, we call such an equivalence class
$\mathcal{Z}=(\mathcal{Z}^{(i)})_{i\in I}$ in $\operatorname{Gr}_{G}$ a
_generically defined bound for $G$_ and denote the resulting bound in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ for any smooth model
$\mathcal{G}$ of $G$ by $\mathcal{Z}$ as well.

###  2.4. Examples of bounds

####  2.4.1. Beilinson-Drinfeld Schubert varieties

The bounds commonly used in practice are bounds given by Beilinson-Drinfeld
Schubert varieties. We recall and generalise their construction following
[Laf18, Définition 1.12]. We assume now that the generic fibre $G$ of
$\mathcal{G}$ is reductive. Let $\mu$ be a conjugacy class of geometric
cocharacters of $G$ with reflex field $F_{\mu}$ and let $F^{\prime}/F$ be a
finite separable extension that splits $G$. By a slight abuse of notation, we
denote by $\operatorname{Gr}_{G}$ the (classical) affine Grassmannian
$\operatorname{Gr}_{G\times_{F}\operatorname{Spec}(F\left\llbracket
t\right\rrbracket)}$ over $F$. We denote by
$\operatorname{Gr}_{G_{F^{\prime}}}^{\leq\mu}\subseteq\operatorname{Gr}_{G_{F^{\prime}}}=\operatorname{Gr}_{G}\times_{F}F^{\prime}$
the Schubert variety corresponding to $\mu$, which descends to the reflex
field $F_{\mu}$ of $\mu$. For an $I$-tuple of conjugacy classes of geometric
cocharacters $\underline{\mu}=(\mu_{i})_{i\in I}$ the (iterated) BD-Schubert
variety $\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\leq\underline{\mu}}$
is the bound induced by the $\operatorname{Gr}_{G}^{\leq\mu_{i}}$ as in the
previous section, compare Remark 2.3.6. More explicitly, it is defined as
follows.

######  Definition 2.4.1 (compare [Zhu14, Definition 3.1], [Laf18, Définition
1.12]).

  1. (1)

Let $\mu$ be a conjugacy class of cocharacters of $G$. Then
$\operatorname{Gr}^{\leq\mu}_{\mathcal{G},X}$ is defined as the scheme-
theoretic image

$\operatorname{Gr}^{\leq\mu}_{\mathcal{G},X}=\mathrm{image}\left(\operatorname{Gr}_{G}^{\leq\mu}\hookrightarrow\operatorname{Gr}_{\mathcal{G},X}\times_{X}X_{\mu}\right)$

where $X_{\mu}=\widetilde{X}_{F_{\mu}}$ is the reflex curve of $\mu$.

  2. (2)

Let $\underline{\mu}=(\mu_{i})_{i\in I}$ be an $I$-tuple of conjugacy classes
of geometric cocharacters of $G$ and let
$F_{\underline{\mu}}=(F_{\mu_{i}})_{i\in I}$ be the $I$-tuple of reflex fields
$F_{\mu_{i}}$ of the $\mu_{i}$. Then
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\leq\underline{\mu}}$ is
defined from the $(\operatorname{Gr}_{\mathcal{G},X}^{\leq\mu_{i}})_{i\in I}$
as in Construction 2.2.4. More precisely, it admits a representative defined
over $F_{\underline{\mu}}$ given by

$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\leq\underline{\mu}}=\mathrm{image}\left(\prod_{i\in
I}\operatorname{Gr}^{\leq\mu_{i}}_{\mathcal{G},X}|_{U_{F_{\underline{\mu}}}}\hookrightarrow\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}X^{I}_{\underline{\mu}}\right),$

where $U$ denotes the complement of all diagonals in $X^{I}$,
$U_{F_{\underline{\mu}}}=U\times_{X^{I}}X_{\underline{\mu}}^{I}$ and the map
$\prod_{i\in
I}\operatorname{Gr}^{\leq\mu_{i}}_{\mathcal{G},X}|_{U_{F_{\underline{\mu}}}}\hookrightarrow\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}X^{I}_{\underline{\mu}}$
is induced by the isomorphism of Remark 2.1.2.

This construction generalises many of the boundedness conditions used in
applications. When $\mathcal{G}$ is constant split reductive, these bounds are
constructed and used in [Laf18]. In particular, when
$\mathcal{G}=\operatorname{GL}_{n}$, $I_{\bullet}=(\\{1\\},\\{2\\})$ and
$\underline{\mu}=((1,0,\ldots,0),(0,\ldots,0,-1))$ this bound gives rise to
the notion of Drinfeld shtuka as defined in [Dri87a]. Further examples include
the shtukas for unitary groups considered in [FYZ24] for $\mathcal{G}=U(n)$
and $\underline{\mu}=((1,0,\ldots,0),\ldots,(1,0,\ldots,0))$. Moreover, in the
local setting (in the case that $\mathcal{G}$ is constant split reductive
these bounds are considered in [GL18, Li-23].

Both $\operatorname{Gr}^{\leq\mu}_{\mathcal{G},X}$ and
$\operatorname{Gr}^{I_{\bullet},\leq\underline{\mu}}_{\mathcal{G},X}$ are
bounds in the sense of Definition 2.2.1 with reflex fields $F_{\mu}$ and
$F_{\underline{\mu}}$, respectively. In particular, they are generically
defined. For
$\operatorname{Gr}^{I_{\bullet},\leq\underline{\mu}}_{\mathcal{G},X}$ this
follows from Lemmas 2.2.5 and 2.3.4.

####  2.4.2. Further examples of global bounds

Let now $X$ be a smooth projective and geometrically connected curve over $k$.
We call bounds (in the sense of Definition 2.2.1) defined in this situation
also _global bounds_. We now discuss various other constructions of global
bounds used in the literature.

######  Example 2.4.2.

  1. (1)

Let $W$ be an $E$-linear representation of $(^{L}G)^{I}$, where ${}^{L}G$
denotes the $L$-group of $G$ and where $E$ is a finite extension of
$\mathbb{Q}_{\ell}$ such that all highest weights of the dual group of $G$ are
defined over $E$. Then
$\bigcup_{\underline{\mu}}\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\leq\underline{\mu}}$,
where $\underline{\mu}$ runs over all highest weights appearing in $W$, is
defined over the compositum of all reflex fields of all the highest weights
appearing and naturally descends to a closed subscheme
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},W}$ of
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ defined over $X^{I}$.

  2. (2)

In the case that $\mathcal{G}$ is constant split reductive, [Var04] defines
bounds as follows. For a $\mathcal{G}$-torsor $\mathcal{E}$ and a dominant
coweight $\lambda$ of $\mathcal{G}$ we denote by
$\mathcal{E}_{\lambda}=\mathcal{E}\times^{\mathcal{G}}V_{\lambda}$, where
$V_{\lambda}$ is the Weyl module for $\lambda$. Then
$\operatorname{Gr}_{G,X}^{I_{\bullet},\lesssim\underline{\mu}}$ is defined to
be the closed subscheme of $\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$
parametrising points $((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\varepsilon)$
such that
$\varphi_{j,\lambda}(\mathcal{E}_{j,\lambda})\subseteq\mathcal{E}_{j-1,\lambda}\left(\sum_{i\in
I_{j}}\langle\lambda,\mu_{i}\rangle\Gamma_{x_{i}}\right)$ for all $1\leq j\leq
m$ and all dominant weights $\lambda$ of $\mathcal{G}$.

In contrast to the Beilinson-Drinfeld Schubert varieties defined above, these
bounds admit a moduli description by definition. However,
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\lesssim\underline{\mu}}$
might fail to be flat or reduced. [Neu16, Definition 3.3.1] gives a variant of
this definition also for quasi-split reductive groups over ${\mathbb{F}_{q}}$.

  3. (3)

For general flat affine group schemes of finite type $\mathcal{G}\to X$,
[AH21] give a boundedness condition by choosing a closed immersion
$f\colon\mathcal{G}\hookrightarrow\operatorname{SL}(\mathcal{V})$ for some
vector bundle $\mathcal{V}$ on $X$ and an $I$-tuple of cocharacters
$\underline{\mu}=(\mu_{i})_{i\in I}$ of $\operatorname{SL}_{n}$, where $n$ is
the rank of $\mathcal{V}$. The condition given in corresponds to the bound
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\lesssim\underline{\mu}}:=\operatorname{Gr}_{\operatorname{SL}(\mathcal{V}),X}^{I_{\bullet},\lesssim\underline{\mu}}\times_{\operatorname{Gr}_{\operatorname{SL}(\mathcal{V}),X}^{I_{\bullet}}}\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$
defined via pullback of the corresponding bound for $\operatorname{SL}_{n}$
along the map
$f_{\ast}\colon\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\to\operatorname{Gr}_{\operatorname{SL}(\mathcal{V}),X}^{I_{\bullet}}$.

As noted in [AH21, Remark 3.20], when $\mathcal{G}$ is constant split
reductive, this definition of bounds agrees with the bound
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},\lesssim
f\circ\underline{\mu}}$ of [Var04] as in (2) for the $I$-tuple of cocharacters
$f\circ\underline{\mu}=(f\circ\mu_{i})_{i\in I}$ of $\mathcal{G}$.

All the bounds considered in this example clearly have reflex scheme $X^{I}$
and a representative defined over $X^{I}$. Moreover,
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet},W}$ is generically defined by
construction.

####  2.4.3. Local bounds

By a _local bound_ we mean a bound in the sense of Definition 2.2.1 when
$X=\operatorname{Spec}(k\left\llbracket\varpi\right\rrbracket)$ for a finite
field $k$. Below, these are used as bounds for local shtukas. [HV11,
Definition 3.5] give a definition of local bounds in the style of [Var04] for
constant split reductive groups when $I=\\{\ast\\}$, compare also [AH14,
Example 4.11].

A different definition of local bounds is given in [AH14, Definition 4.5],
where bounds are not defined as subschemes of
$\operatorname{Gr}_{\mathcal{G},X}$, but rather as formal subschemes of
$\operatorname{Gr}_{\mathcal{G}}\hat{\times}_{\operatorname{Spec}k}\operatorname{Spf}(k\left\llbracket\varpi\right\rrbracket)$.
By [Li-23, Lemma 1.4], we have

$\operatorname{Gr}_{\mathcal{G}}\hat{\times}_{\mathbb{F}}\operatorname{Spf}(\mathcal{O})\cong\operatorname{Gr}_{\mathcal{G},k\left\llbracket\varpi\right\rrbracket}\hat{\times}_{\operatorname{Spec}k\left\llbracket\varpi\right\rrbracket}\operatorname{Spf}(k\left\llbracket\varpi\right\rrbracket),$

so the local bounds in our setting (for $I=\\{\ast\\})$) define bounds in the
sense of [AH14] by pullback under this isomorphism.

###  2.5. Construction of local bounds from global bounds

We explain how to construct local bounds from global ones in our setting
following the construction in [AH19, Proposition 4.3.3]. We generalise their
construction to the iterated setting and moreover also construct local bounds
at points on the diagonal of $X^{I}$.

We consider the global setting. In other words, $X$ denotes a smooth
geometrically connected curve over $k$ for the remainder of this section. Let
$\mathcal{Z}$ be an equivalence class of quasi-compact closed subschemes in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. We fix a closed point
$y^{0}$ of $X$ and assume there is a closed point $\underline{y}$ of the
reflex scheme $\widetilde{X}_{\mathcal{Z}}^{I}$ of $\mathcal{Z}$ lying over
$\Delta_{I}(y^{0})$, where $\Delta_{I}\colon X\to X^{I}$ denotes the diagonal
embedding. This is the case if and only if all components of local or finite
type in $\mathcal{Z}$ are defined at $y^{0}$. This is satisfied in particular
in the case that $\mathcal{Z}$ only has components of global type that will be
relevant later.

Let $\underline{F}^{\prime}$ be an $I$-tuple of finite extensions of
$\underline{F}$ such that $\mathcal{Z}$ admits a representative
$Z_{\underline{F}^{\prime}}$ over $\underline{F}^{\prime}$. We fix a closed
point $\underline{y}^{\prime}$ of $\widetilde{X}_{\underline{F}^{\prime}}$
over $\underline{y}$ and let us denote by
$y_{i}^{\prime}\in\widetilde{X}_{(F^{(i)})^{\prime}}$ its $i$-th component. We
set
$Z_{\underline{F}^{\prime},\underline{y}^{\prime}}=Z_{\underline{F}^{\prime}}\times_{\tilde{X}_{\underline{F}^{\prime}}^{I}}\operatorname{Spec}(\bigotimes_{\mathbb{F}_{y^{0}}}\mathcal{O}_{y^{\prime}_{i}})$
where we denote by $\mathcal{O}_{y^{\prime}_{i}}$ the completed local ring at
$y_{i}^{\prime}$ in $\widetilde{X}_{(F^{(i)})^{\prime}}$. We denote by
$\mathcal{Z}_{\underline{y}}$ the equivalence class of
$Z_{\underline{F}^{\prime},\underline{y}^{\prime}}$ in
$\operatorname{Gr}_{\mathcal{G},\operatorname{Spec}(\mathcal{O}_{y^{0}})}^{I_{\bullet}}$
(which will be of local type in the sense of Definition 2.2.1). Note that
components of global type for $\mathcal{Z}$ become components of local type
for $\mathcal{Z}_{\underline{y}}$ while components of local (respectively
finite) type for $\mathcal{Z}$ remain of local (respectively finite) type for
$\mathcal{Z}_{\underline{y}}$.

######  Lemma 2.5.1 (compare [AH19, Proposition 4.3.3]).

The equivalence class $\mathcal{Z}_{\underline{y}}$ does neither depend on the
choice of representative $Z_{\underline{F}^{\prime}}$ nor the choice of the
point $\underline{y}^{\prime}$. Moreover, when $\mathcal{Z}$ is a (global)
bound, $\mathcal{Z}_{\underline{y}}$ is a (local) bound.

###### Proof.

We first assume that $\underline{F}^{\prime}$ is a componentwise Galois
extension of $\underline{F}$. Let $\underline{y}^{\prime\prime}$ be a second
place of $\tilde{X}_{\underline{F}^{\prime}}$ lying over $\underline{y}$. As
$Z_{\underline{F}^{\prime}}$ is Galois-invariant,
$Z_{\underline{F}^{\prime},\underline{y}^{\prime}}$ and
$Z_{\underline{F}^{\prime},\underline{y}^{\prime\prime}}$ are equivalent. Let
now $\underline{F}^{\prime\prime}$ be a componentwise finite extension of
$\underline{F}^{\prime}$ and let $\underline{y}^{\prime\prime}$ be a closed
point of $\underline{F}^{\prime\prime}$ lying over $\underline{y}^{\prime}$.
Then clearly
$Z_{\underline{F}^{\prime\prime},\underline{y}^{\prime\prime}}=Z_{\underline{F}^{\prime},\underline{y}^{\prime}}\times_{\operatorname{Spec}(\bigotimes_{\mathbb{F}_{y^{0}}}\mathcal{O}_{y^{\prime}_{i}})}\operatorname{Spec}(\bigotimes_{\mathbb{F}_{y^{0}}}\mathcal{O}_{y^{\prime\prime}_{i}})$.
This shows that $\mathcal{Z}_{\underline{y}}$ is independent of the auxiliary
choices. Moreover, when $\mathcal{Z}$ is stable under the loop group action so
is $\mathcal{Z}_{\underline{y}}$. ∎

We slightly change notation and fix an $I$-tuple
$\underline{y}^{0}=(y^{0}_{i})_{i\in I}$ of closed points of $X$ for the
remainder of this subsection. For a closed point $y^{0}$ of $X$ we write
$y^{0}\in\underline{y}^{0}$ if $y^{0}=y^{0}_{i}$ for some $i\in I$. We set
$I^{y^{0}}=\\{i\in I\colon y^{0}_{i}=y^{0}\\}$ and define
$I^{y^{0}}_{\bullet}=(I_{j}^{y^{0}})_{j\in J}$ analogously.

######  Definition 2.5.2.

  1. (1)

An _$I$ -tuple of local bounds at $\underline{y}^{0}$_ is a tuple of (local)
bounds $(\mathcal{Z}_{y^{0}})_{y^{0}\in\underline{y}^{0}}$, where
$\mathcal{Z}_{y^{0}}$ is defined in
$\operatorname{Gr}_{\mathcal{G},\operatorname{Spec}(\mathcal{O}_{y^{0}})}^{I^{y^{0}}_{\bullet}}$.

  2. (2)

Let $\mathcal{Z}$ be a global bound and assume that
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}\to X^{I}$ is flat. For each
${y^{0}}\in\underline{y}^{0}$ we fix a closed point $\underline{y}$ on
$\widetilde{X}^{I^{y^{0}}}_{\mathcal{Z}^{(I^{y^{0}})}}$ lying over
$\Delta_{I^{y^{0}}}(y^{0})$. Its associated $I$-tuple of local bounds at
$\underline{y}^{0}$ is
$(\mathcal{Z}_{\underline{y}})_{y^{0}\in\underline{y}^{0}}$, where
$\mathcal{Z}_{\underline{y}}$ is as constructed above (using Construction
2.2.7).

Note that when all $y^{0}_{i}$ are pairwise distinct, an $I$-tuple of bounds
in the sense of the previous definition is an actual $I$-tuple of bounds
$(\mathcal{Z}_{y^{0}_{i}})_{i\in I}$, where each $\mathcal{Z}_{y^{0}_{i}}$
lives inside $\operatorname{Gr}_{\mathcal{G},\mathcal{O}_{y^{0}_{i}}}$. Our
definition gives the natural generalisation to the case where
$\underline{y}^{0}$ is also allowed to lie on a diagonal of $X^{I}$.

######  Example 2.5.3.

By construction, the local bounds attached to a Schubert variety in the global
setting is again a Schubert variety (in the local setting). Compare also
[Neu16, Theorem 3.3.9] for a local-global compatibility for global bounds in
the style of [Var04] and local bounds in the style of [HV11].

##  3\. Moduli spaces of bounded, iterated, global shtukas

Global shtukas for $\operatorname{GL}_{n}$ were first introduced in [Dri87a]
and generalised to split reductive groups (respectively to flat affine group
schemes of finite type) by [Var04] and [AH21], respectively. In this section,
we recall the definition and basic properties of moduli spaces of (bounded,
iterated, global) shtukas. We rely of the definition of bounds given in the
previous section and generalise the Serre-Tate theorem, the local model
theorem and functoriality results to this setting. The functoriality results
will be crucial to the construction of integral models with deeper level
structure below.

###  3.1. Global shtukas

We consider the “global setting” of the previous section: Let $X$ be a smooth
projective and geometrically connected curve over $k={\mathbb{F}_{q}}$, let
$\mathcal{G}\to X$ be a smooth affine group scheme, let $I$ be a finite set,
and let $I_{\bullet}=(I_{1},\ldots,I_{m})$ be a partition of $I$.

We denote by $\operatorname{Bun}_{\mathcal{G}}$ the stack of
$\mathcal{G}$-bundles on $X$. It is representable by a smooth Artin stack
locally of finite type over ${\mathbb{F}_{q}}$ by [Hei10, Proposition 1].

######  Definition 3.1.1.

We denote by $\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet}}$ the stack
fibred in groupoids over ${\mathbb{F}_{q}}$ whose $S$ valued points are given
by tuples $((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m}),$ where

  * •

$x_{i}\in X(S)$ are points on $X$ called the _characteristic sections_ (or
_legs_) for $i\in I$,

  * •

$\mathcal{E}_{j}\in\operatorname{Bun}_{\mathcal{G}}(S)$ are
$\mathcal{G}$-bundles on $X_{S}$ for $0\leq j\leq m$, and

  * •

$\varphi_{j}\colon\mathcal{E}_{j}|_{X_{S}\setminus\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}}\xrightarrow{\cong}\mathcal{E}_{j-1}|_{X_{S}\setminus\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}}$ are isomorphisms of $\mathcal{G}$-bundles away from the
graphs $\Gamma_{x_{i}}\subseteq X_{S}$ of the sections $x_{i}$.

As above, we use the shorthand notation
$\left(\mathcal{E}_{m}\overset{\varphi_{m}}{\underset{\Gamma_{\underline{x}_{m}}}{\dashrightarrow}}\mathcal{E}_{m-1}\overset{\varphi_{m-1}}{\underset{\Gamma_{\underline{x}_{m-1}}}{\dashrightarrow}}\ldots\overset{\varphi_{1}}{\underset{\Gamma_{\underline{x}_{1}}}{\dashrightarrow}}\mathcal{E}_{0}\right)$
for points of $\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet}}$.

The projection to $\mathcal{E}_{j}$ defines a map
$p_{j}\colon\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet}}\to\operatorname{Bun}_{\mathcal{G}}$
for all $0\leq j\leq m$.

######  Definition 3.1.2 ([AH21, Definition 3.3]).

The moduli space of shtukas $\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}$
is defined by the Cartesian diagram

${\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}}$${\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet}}}$${\operatorname{Bun}_{\mathcal{G}}}$${\operatorname{Bun}_{\mathcal{G}}\times\operatorname{Bun}_{\mathcal{G}}.}$$\scriptstyle{(p_{m},p_{0})}$$\scriptstyle{(\operatorname{id},\sigma^{\ast})}$

More explicitly, $\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}$ is the
stack fibred in groupoids over ${\mathbb{F}_{q}}$ whose $R$-valued points for
an ${\mathbb{F}_{q}}$-algebra $R$ are given by tuples $((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\theta),$
where

  * •

$x_{i}\in X(R)$ are points on $X$ called the _characteristic sections_ (or
_legs_) for $i\in I$,

  * •

$\mathcal{E}_{j}\in\operatorname{Bun}_{\mathcal{G}}(R)$ are
$\mathcal{G}$-bundles on $X_{R}$ for $0\leq j\leq m$,

  * •

$\varphi_{j}\colon\mathcal{E}_{j}|_{X_{R}\setminus\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}}\xrightarrow{\cong}\mathcal{E}_{j-1}|_{X_{R}\setminus\bigcup_{i\in
I_{j}}\Gamma_{x_{i}}}$ are isomorphisms of $\mathcal{G}$-bundles away from the
graphs $\Gamma_{x_{i}}\subseteq X_{R}$ of the sections $x_{i}$, and

  * •

$\theta\colon\mathcal{E}_{0}\xrightarrow{\cong}\sigma^{\ast}\mathcal{E}_{m}$
is an isomorphism of $\mathcal{G}$-bundles on $X_{R}$.

We denote a point of $\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}$ by
$\left(\mathcal{E}_{m}\overset{\varphi_{m}}{\underset{\Gamma_{\underline{x}_{m}}}{\dashrightarrow}}\mathcal{E}_{m-1}\overset{\varphi_{m-1}}{\underset{\Gamma_{\underline{x}_{m-1}}}{\dashrightarrow}}\ldots\overset{\varphi_{1}}{\underset{\Gamma_{\underline{x}_{1}}}{\dashrightarrow}}\mathcal{E}_{0}\xrightarrow{\theta}\sigma^{\ast}\mathcal{E}_{m}\right).$

The projection to the characteristic sections defines a map
$\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}\to X^{I}$. By [AH21, Theorem
3.15], $\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}$ is an ind-Deligne
Mumford stack that is separated and locally of ind-finite type over $X^{I}$.

Let $I_{\bullet}^{\prime}$ be a second partition of $I$ that is finer than
$I_{\bullet}$. The forgetful map
$\operatorname{Sht}_{\mathcal{G},X}^{I^{\prime}_{\bullet}}\to\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}$
is an isomorphism over the complement $U$ of all diagonals in $X^{I}$ by the
argument in [Var04, Lemma A.8 a)]. Let us fix a closed point $\underline{y}\in
X^{I}$. We denote by
$\operatorname{Sht}^{I_{\bullet}}_{\mathcal{G},\underline{y}}=\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\operatorname{Spf}(\mathcal{O}_{\underline{y}})$
the restriction of the moduli space of shtukas to the formal neighbourhood
$\operatorname{Spf}\left(\mathcal{O}_{\underline{y}}\right)$ of
$\underline{y}$. By the previous observation, this stack does not depend on
the choice of the partition $I_{\bullet}$ of $I$ whenever the $y_{i}$ are
pairwise different. Usually, when the index set $I$, its partition
$I_{\bullet}$ and the curve $X$ are understood from the context, we drop them
from the notation and denote
$\operatorname{Sht}_{\mathcal{G}}=\operatorname{Sht}^{I_{\bullet}}_{\mathcal{G},X}.$

######  Definition 3.1.3.

Let $\mathcal{Z}$ be a bound in the sense of Definition 2.2.1. Let

$\underline{\mathcal{E}}=((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\theta)\in(\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\mathcal{Z}}^{I})(R).$

By [HR20, Lemma 3.4], there exists an étale cover
$\operatorname{Spec}(R^{\prime})\to\operatorname{Spec}(R)$ such that
$\hat{\Gamma}_{\underline{x}_{R^{\prime}}}\to\hat{\Gamma}_{\underline{x}}$
trivializes $\mathcal{E}_{0}|_{\hat{\Gamma}_{\underline{x}}}$. Fixing a
trivialisation
$\alpha\colon\mathcal{E}_{0}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\xrightarrow{\cong}\mathcal{G}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}$
defines a point

$g=\left(\mathcal{E}_{m}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\overset{\varphi_{m}}{\underset{\Gamma_{\underline{x}_{m}}}{\dashrightarrow}}\mathcal{E}_{m-1}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\overset{\varphi_{m-1}}{\underset{\Gamma_{\underline{x}_{m-1}}}{\dashrightarrow}}\ldots\overset{\varphi_{1}}{\underset{\Gamma_{\underline{x}_{1}}}{\dashrightarrow}}\mathcal{E}_{0}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\overset{\varepsilon}{\rightarrow}\mathcal{G}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\right)\in(\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\mathcal{Z}}^{I})(R^{\prime}),$

using the description of the Beilinson-Drinfeld affine Grassmannian of Remark
2.1.2. We say that $\underline{\mathcal{E}}$ is _bounded by_ $\mathcal{Z}$ if
for some (or by Lemma 2.2.8 equivalently for all) $I$-tuples of finite
extensions $\underline{F}^{\prime}$ such that $\mathcal{Z}$ admits a
representative over $\underline{F}^{\prime}$, the map

$g\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\operatorname{id}_{\tilde{X}^{I}_{F^{\prime}}}\colon\operatorname{Spec}(R^{\prime})\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\tilde{X}^{I}_{\underline{F}^{\prime}}\to(\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\tilde{X}_{\underline{F}^{\prime}}^{I})$

factors through $Z_{\underline{F}^{\prime}}$. As $\mathcal{Z}$ is invariant
under the left $\mathcal{L}_{X^{I}}^{+}\mathcal{G}$-action, the definition is
independent of the choices of the étale cover $R\to R^{\prime}$ and the
trivialisation $\alpha$.

We denote the moduli stack of shtukas bounded by $\mathcal{Z}$ by
$\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}=\operatorname{Sht}^{I_{\bullet},\mathcal{Z}}_{\mathcal{G},X}$.
It is a Deligne-Mumford stack defined over $\widetilde{X}_{\mathcal{Z}}^{I}$
and a closed substack of
$\operatorname{Sht}_{\mathcal{G}}\times_{X^{I}}\widetilde{X}_{\mathcal{Z}}^{I}$.
When the bound $\mathcal{Z}$ is given by Beilinson-Drinfeld Schubert varieties
corresponding to an $I$-tuple $\underline{\mu}$ of conjugacy classes of
geometric cocharacters of $G$, we write
$\operatorname{Sht}_{\mathcal{G}}^{\leq\underline{\mu}}=\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet},\leq\underline{\mu}}=\operatorname{Sht}_{\mathcal{G},X}^{I_{\bullet},\mathcal{Z}}$.

###  3.2. Local shtukas with multiple legs and local-global compatibility

We again consider the local setting. Let $\mathcal{O}$ be the ring of integers
of a local field in characteristic $p$ with residue field $k$ a finite
extension of ${\mathbb{F}_{q}}$. The choice of a uniformiser $\varpi$ of
$\mathcal{O}$ induces an isomorphism $\mathcal{O}\cong
k\left\llbracket\varpi\right\rrbracket$. Let
$\mathcal{G}\to\operatorname{Spec}\mathcal{O}$ be a smooth affine
$\mathcal{O}$-group scheme, let $G$ be its generic fibre. As before, let $I$
be a finite set and let $I_{\bullet}=(I_{1},\ldots,I_{m})$ be a partition of
$I$. We set
$\mathcal{O}_{I}=\mathcal{O}\hat{\otimes}_{k}\ldots\hat{\otimes}_{k}\mathcal{O}$,
i.e., $\mathcal{O}_{I}\cong k\left\llbracket\varpi_{i}\right\rrbracket_{i\in
I}$. We denote by $\mathcal{N}ilp_{\mathcal{O}_{I}}$ the category of
$\mathcal{O}_{I}$-algebras $R$ such that the images $\zeta_{i}$ of
$\varpi_{i}$ in $R$ under the structure map $\mathcal{O}_{I}\to R$ are
nilpotent.

Similarly to the global loop group defined in the previous section, the
(local) loop group $LG$ of $G$ is defined as the functor on $k$-algebras

$R\mapsto G(R\left(\\!\left(\varpi\right)\\!\right)).$

Analogously, the positive loop group $L^{+}\mathcal{G}$ is defined as the
functor on $k$-algebras

$R\mapsto\mathcal{G}(R\left\llbracket\varpi\right\rrbracket).$

The functor $LG$ is representable by an ind-affine group ind-scheme while
$L^{+}\mathcal{G}$ is representable by an affine group scheme over $k$.

By [AH14, Proposition 2.4], the category of $L^{+}\mathcal{G}$-torsors over a
$k$-algebra $R$ is equivalent to the category of formal $\mathcal{G}$-torsors
on $\operatorname{Spf}(R\left\llbracket\varpi\right\rrbracket)$. By a slight
abuse of notation, we denote by
$\mathcal{E}\left[\frac{1}{\varpi}\right]=\mathcal{E}\times^{L^{+}\mathcal{G}}LG$
the associated $LG$-torsor of an $L^{+}\mathcal{G}$-torsor $\mathcal{E}$. Note
that for $R\in\mathcal{N}ilp_{\mathcal{O}_{I}}$, we have
$R\left(\\!\left(\varpi\right)\\!\right)=R\left\llbracket\varpi\right\rrbracket\left[\frac{1}{\varpi-\zeta_{i}}\right]$
for all $i\in I$ as the $\zeta_{i}$ are nilpotent.

######  Definition 3.2.1.

A _local shtuka_ (with $I$-legs) over $R\in\mathcal{N}ilp_{\mathcal{O}_{I}}$
is given by a tuple
$((\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\theta),$
where

  * •

the $\mathcal{E}_{j}$ are $L^{+}\mathcal{G}$-torsors on $R$,

  * •

the
$\varphi_{j}\colon\mathcal{E}_{j}[\frac{1}{\varpi}]\xrightarrow{\cong}\mathcal{E}_{j-1}[\frac{1}{\varpi}]$
are isomorphisms of the associated $L\mathcal{G}$-torsors,

  * •

and
$\theta\colon\mathcal{E}_{0}\xrightarrow{\cong}\sigma^{\ast}\mathcal{E}_{m}$
is an isomorphism of $L^{+}\mathcal{G}$-torsors on $R$.

In a similar fashion as in the global setting, we define bounded local
shtukas.

######  Definition 3.2.2.

Let $\mathcal{Z}$ be a (local) bound in the sense of Definition 2.2.1 and let
$\underline{y}$ be a closed point of $\widetilde{X}^{I}_{\mathcal{Z}}$. Let
$\underline{\mathcal{E}}=(\mathcal{E},\varphi)$ be a local shtuka over
$R\in\mathcal{N}ilp_{\mathcal{O}_{\underline{y}}}$. Let $R\to R^{\prime}$ be
an fppf-cover that trivialises $\mathcal{E}_{0}$ and choose a trivialisation
$\alpha$ of $\mathcal{E}_{0}$ over $R^{\prime}$. Using the equivalence of
$L^{+}\mathcal{G}$-torsors and formal $\mathcal{G}$-torsors of [AH14,
Proposition 2.4],
$\left(\mathcal{E}_{m}\overset{\varphi_{m}}{\dashrightarrow}\mathcal{E}_{m-1}\overset{\varphi_{m-1}}{\dashrightarrow}\ldots\overset{\varphi_{1}}{\dashrightarrow}\mathcal{E}_{0}\overset{\alpha}{\rightarrow}\mathcal{G}\right)$
defines a point
$g\in\left(\operatorname{Gr}_{\mathcal{G},\mathcal{O}}^{I_{\bullet}}\hat{\times}_{\operatorname{Spec}(\mathcal{O}_{I})}\operatorname{Spf}(\mathcal{O}_{I})\right)(R^{\prime})$.
We say that $\underline{\mathcal{E}}$ _is bounded by $\mathcal{Z}$_ if for
some (or equivalently for all) representatives $Z_{\underline{F}^{\prime}}$ of
$\mathcal{Z}$ over an $I$-tuple of finite extensions $\underline{F}^{\prime}$
such that $\mathcal{Z}$ admits a representative over $\underline{F}^{\prime}$,
the map

$g\times\operatorname{id}_{\mathcal{O}_{I}^{\prime}}\colon\operatorname{Spec}(R^{\prime})\times_{\operatorname{Spf}(\mathcal{O}_{\underline{y}})}\operatorname{Spf}(\mathcal{O}_{\underline{y}^{\prime}})\to(\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\hat{\times}_{\operatorname{Spec}(\mathcal{O}_{I})}\operatorname{Spf}(\mathcal{O}_{\underline{y}^{\prime}}))$

factors through
$Z_{\underline{F}^{\prime}}\times_{\tilde{X}_{\underline{F}^{\prime}}^{I}}\operatorname{Spf}(\mathcal{O}_{\underline{y}^{\prime}})$
for all closed points $\underline{y}^{\prime}$ of
$\widetilde{X}^{I}_{\underline{F}^{\prime}}$ lying over $\underline{y}$. As
$Z_{\underline{F}^{\prime}}$ is invariant under the left
$\mathcal{L}_{\widetilde{X}^{I}_{\underline{F}^{\prime}}}^{+}\mathcal{G}$-action,
the definition is independent of the choice of the trivialisation $\alpha$.

######  Remark 3.2.3.

In the literature on local shtukas the Frobenius-linear map is often defined
to be an isomorphism
$\theta^{\prime}\colon\sigma^{*}\mathcal{E}_{0}\cong\mathcal{E}_{m}$, compare
for example [HV11, AH14]. In particular, for a local shtuka
$\underline{\mathcal{E}}$ in our sense inverting all maps yields a local
shtuka

$\underline{\mathcal{E}}^{-1}=\left(\mathcal{E}_{m}\overset{\varphi^{-1}_{m}}{\dashleftarrow}\mathcal{E}_{m-1}\overset{\varphi^{-1}_{m-1}}{\dashleftarrow}\ldots\overset{\varphi^{-1}_{1}}{\dashleftarrow}\mathcal{E}_{0}\overset{\theta^{-1}}{\leftarrow}\sigma^{*}\mathcal{E}_{m}\right)$

in the more classical sense (and vice versa). Moreover,
$\underline{\mathcal{E}}$ is bounded by some bound $\mathcal{Z}$ if and only
if $\underline{\mathcal{E}}^{-1}$ is bounded by the inverse bound
$\mathcal{Z}^{-1}$ as defined in [HV23, Lemma 2.12].

We explain how to associate to a global shtuka an $I$-tuple of local shtukas.
At points away from the diagonal, this is contained in [AH14]. We extend their
construction to also include points on the diagonal using local shtukas with
multiple legs.

We consider the global setting from above. Let $X$ be a smooth, projective,
and geometrically connected curve over ${\mathbb{F}_{q}}$, and let
$\mathcal{G}\to X$ be a smooth affine group scheme. We use the notation
following [AH14, Remark 5.2]. Let $y\in X$ be a closed point. We denote by
$\mathcal{O}_{y}$ the completed local ring at $y$, and by
$\mathfrak{m}_{y}\subseteq\mathcal{O}_{y}$ and
$\mathbb{F}_{y}=\mathcal{O}_{y}/\mathfrak{m}_{y}$ its maximal ideal with
uniformiser $\varpi_{y}$ and residue field, respectively. Moreover, let
$d=[\mathbb{F}_{y}\colon{\mathbb{F}_{q}}]$ be the degree of $y$. Then the
$\mathfrak{m}$-adic completion of $\mathcal{O}_{y}\otimes_{\mathbb{F}_{q}}R$
factors as

$\mathcal{O}_{y}{\widehat{\otimes}}_{\mathbb{F}_{q}}R=(\mathbb{F}_{y}\otimes_{\mathbb{F}_{q}}R)\left\llbracket\varpi_{y}\right\rrbracket=\prod_{\ell\in\mathbb{Z}/d\mathbb{Z}}\mathcal{O}_{y}{\widehat{\otimes}}_{\mathbb{F}_{y}}R=\prod_{\ell\in\mathbb{Z}/d\mathbb{Z}}R\left\llbracket\varpi_{y}\right\rrbracket,$

where the $\ell$-th factor is defined by the ideal
$\mathfrak{a}_{\ell}=\langle a\times 1-1\otimes x(a)^{q^{\ell}}\colon
a\in\kappa_{y}\rangle$ in
$\mathcal{O}_{y}{\widehat{\otimes}}_{\mathbb{F}_{q}}R$. The Frobenius $\sigma$
cyclically permutes the factors.

Let us fix an $I$-tuple of closed points $\underline{y}=(y_{i})_{i\in I}$ of
$X$. We use the notation of Section 2.5 and say $y\in\underline{y}$ if
$y=y_{i}$ for some $i\in I$. Moreover, let $I^{y}=\\{i\in I\colon y_{i}=y\\}$
and let $I_{\bullet}^{y}=(I_{1}^{y},\ldots,I_{m}^{y})$. Let
$\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(R)$.
By the observation above, the $y$-adic completion of $\mathcal{E}_{j}$
decomposes as

$\mathcal{E}_{j}{\widehat{\times}}_{X_{R}}\operatorname{Spf}(\mathcal{O}_{y}{\widehat{\otimes}}_{{\mathbb{F}_{q}}}R)=\coprod_{\ell\in\mathbb{Z}/d\mathbb{Z}}\mathcal{E}_{j}{\widehat{\times}}_{X_{R}}\operatorname{Spf}(R\left\llbracket\varpi_{y}\right\rrbracket)=:\coprod_{\ell\in\mathbb{Z}/d\mathbb{Z}}\mathcal{E}_{j,y}^{(\ell)}.$

Hence, each component
$\mathcal{E}_{j,y}^{(\ell)}=\mathcal{E}_{j}{\widehat{\times}}_{X_{R}}V(\mathfrak{a}_{\ell})$
is a formal
$\hat{\mathcal{G}}_{y}=\mathcal{G}\times_{X}\operatorname{Spf}(\mathcal{O}_{y})$-torsor
over $R$ and corresponds to a $\mathcal{L}^{+}\mathcal{G}_{y}$-torsor
$\widehat{\mathcal{E}_{j,y}}^{(\ell)}$ over $R$ by [AH14, Proposition 2.4].

Since $\sigma$ permutes the components cyclically, we find
$\widehat{\mathcal{E}_{j,y}}^{(\ell)}=(\sigma^{(\ell)})^{\ast}\widehat{\mathcal{E}_{j,y}}^{(0)}$.
By [AH14, Lemma 5.3], the component $V(\mathfrak{a}_{0})$ is also given by the
completion of $X_{R}$ along $\Gamma_{x_{i}}$ for any $i\in I_{j}$. Hence,

$\varphi_{j}|_{V(\mathfrak{a}_{\ell})}\colon(\sigma^{(\ell)})^{\ast}\widehat{\mathcal{E}_{j,y}}^{(0)}=\widehat{\mathcal{E}_{j,y}}^{(\ell)}\xrightarrow{\cong}\widehat{\mathcal{E}_{j-1,y}}^{(\ell)}=(\sigma^{(\ell)})^{\ast}\widehat{\mathcal{E}_{j-1,y}}^{(0)}$

is an isomorphism for all $\ell\neq 0$. We define the isomorphism
$\tilde{\theta}\colon\widehat{\mathcal{E}_{0,y}}^{(0)}\xrightarrow{\cong}(\sigma^{(d)})^{\ast}\widehat{\mathcal{E}_{m,y}}^{(0)}$
by

$\tilde{\theta}=\theta|_{V(\mathfrak{a}_{d-1})}\circ\varphi_{1}|_{V(\mathfrak{a}_{d-1})}\circ\ldots\circ\varphi_{m}|_{V(\mathfrak{a}_{2})}\circ\theta|_{V(\mathfrak{a}_{1})}\circ\varphi_{1}|_{V(\mathfrak{a}_{1})}\circ\ldots\circ\varphi_{m}|_{V(\mathfrak{a}_{1})}\circ\theta|_{V(\mathfrak{a}_{0})}.$

Then,
$\widehat{\underline{\mathcal{E}}_{y}}=\left(\left(\widehat{\mathcal{E}_{j,y}}^{(0)}\right)_{0\leq
j\leq m},\left(\varphi_{j}|_{V(\mathfrak{a}_{0})}\right)_{1\leq j\leq
m},\tilde{\theta}\right)$ is a local $\mathcal{G}_{\mathcal{O}_{y}}$-shtuka
over $R$.

######  Definition 3.2.4.

Let $\underline{y}$ be an $I$-tuple of closed points of $X$. The _global-to-
local functor_ associates to a global shtuka
$\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(R)$
a tuple of local $\mathcal{G}_{\mathcal{O}_{y}}$-shtukas for
$y\in\underline{y}$ given by
$\widehat{\underline{\mathcal{E}}_{\underline{y}}}=(\widehat{\underline{\mathcal{E}}_{y}})_{y\in\underline{y}}.$
Then, $\widehat{\underline{\mathcal{E}}_{y}}$ is called the _local shtuka_ of
$\underline{\mathcal{E}}$ at $y$.

######  Remark 3.2.5.

In a similar fashion, for a closed point $y$ of $X$ we can associate to a
global shtuka
$\underline{\mathcal{E}}=((x_{i}),(\mathcal{E}_{j}),(\varphi_{j}),\theta)\in\operatorname{Sht}_{\mathcal{G}}|_{(X\setminus\\{y\\})^{I}}(R)$
with characteristic sections away from $y$ an étale local shtuka at $y$ by
[AH14, Remark 5.6] as follows. We denote by
$\widetilde{\mathcal{G}}_{y}=\mathrm{Res}_{\mathcal{O}_{y}/\mathbb{F}_{q}\left\llbracket\varpi_{y}\right\rrbracket}\mathcal{G}_{\mathcal{O}_{y}}$.
Then $\widetilde{\mathcal{G}}_{y}$ is a smooth affine group scheme over
${\mathbb{F}_{q}}\left\llbracket\varpi_{y}\right\rrbracket$. The _étale local
$\widetilde{\mathcal{G}}_{y}$-shtuka_ associated to $\mathcal{E}$ is then
given by
$\underline{\widetilde{\mathcal{E}}}_{y}=(\widetilde{\mathcal{E}}_{y},\varphi)$
with
$\widetilde{\mathcal{E}}_{y}=\mathrm{Res}_{(\mathcal{O}_{y}\widehat{\otimes}_{\mathbb{F}_{q}}R)/R\left\llbracket\varpi_{y}\right\rrbracket}(\mathcal{E}_{0}\widehat{\times}_{X_{R}}\operatorname{Spf}(\mathcal{O}_{y}\widehat{\otimes}_{\mathbb{F}_{q}}R))$
and $\varphi=\theta\circ\varphi_{1}\circ\ldots\circ\varphi_{m}$. Note that
$\varphi$ is an isomorphism by assumption.

The global-to-local functor is compatible with our notion of bounds in the
following sense. Let us fix a global bound
$\mathcal{Z}=\prod_{y\in\underline{y}}\mathcal{Z}^{(I^{y})}$ in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$ and for each
$y\in\underline{y}$ a closed point $y^{\prime}$ of
$\widetilde{X}^{I^{y}}_{\mathcal{Z}^{(I^{y})}}$ over $\Delta(y)\in X^{I^{y}}$.
We denote by
$\operatorname{Sht}^{\mathcal{Z}}_{\mathcal{G},\underline{y}^{\prime}}=\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\prod_{y\in\underline{y}}\operatorname{Spf}(\mathcal{O}_{y^{\prime}})$.

######  Proposition 3.2.6.

Let $\mathcal{Z}=\prod_{y\in\underline{y}}\mathcal{Z}^{(I^{y})}$ be a global
bound and let $(\mathcal{Z}_{y^{\prime}})_{y\in\underline{y}}$ be the
associated $I$-tuple of local bounds at $\underline{y}$ as in Definition
2.5.2. A global shtuka
$\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},\underline{y}^{\prime}}(R)$
is bounded by $\mathcal{Z}$ if and only if for all $y\in\underline{y}$ its
associated local shtuka $\widehat{\underline{\mathcal{E}}_{y}}$ at $y$ is
bounded by $\mathcal{Z}_{y^{\prime}}$.

###### Proof.

Let $R^{\prime}\to R$ be an étale cover trivialising
$\mathcal{E}|_{\hat{\Gamma}_{\underline{x}}}$ and let us fix a trivialisation
$\alpha\colon\mathcal{E}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\xrightarrow{\cong}\mathcal{G}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}$.
The pair $(\underline{\mathcal{E}},\alpha)$ defines a point
$g\in\left(\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\mathcal{Z}}^{I}\right)(R^{\prime})$
that even lies in
$\left(\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\prod_{y\in\underline{y}}\operatorname{Spf}(\mathcal{O}_{y^{\prime}})\right)(R^{\prime})$
by assumption on $R$. Restricting $\alpha$ yields trivialisations of
$\widehat{\underline{\mathcal{E}}_{y}}$ for all $y\in\underline{y}$ and hence
points
$\widehat{g_{y}}=(\widehat{\underline{\mathcal{E}}_{y}},\widehat{\alpha_{y}})\in\left(\operatorname{Gr}_{\mathcal{G},\mathcal{O}_{y}}^{I^{y}_{\bullet}}\times_{\operatorname{Spec}(\mathcal{O}_{y})^{I}}\operatorname{Spf}(\mathcal{O}_{y^{\prime}})\right)(R^{\prime})$.
By Remark 2.1.2 we find $g=\prod_{y\in\underline{y}}\widehat{g_{y}}$.

Let us fix an $I$-tuple of finite extensions $\underline{F}^{\prime}$ of $F$
such that $\mathcal{Z}$ admits a representative over $\underline{F}^{\prime}$.
Let us fix for every $y\in\underline{y}$ a closed point $y_{F^{\prime}}$ of
$\widetilde{X}^{I^{y}}_{\underline{F}^{\prime}}$ lying over $y^{\prime}$. By
construction, the representative $Z_{\underline{F}^{\prime}}$ of $\mathcal{Z}$
factors over
$\operatorname{Spf}(\mathcal{O}_{\underline{y}_{\underline{F^{\prime}}}})$ as
a product of the corresponding representatives of the local bounds. The
assertion follows. ∎

###  3.3. A Serre-Tate theorem

The classical Serre-Tate theorem relates the deformation theory of an abelian
variety in characteristic $p$ to the deformation theory of its $p$-divisible
group. [AH14, Theorem 5.10] proves an analogue of the Serre-Tate theorem for
$\mathcal{G}$-shtukas. We generalise the Serre-Tate theorem for shtukas to the
iterated case using local shtukas with multiple legs. The local-global
compatibility then also gives a bounded version of the Serre-Tate theorem.

We follow [AH14, Section 5] and start by analysing quasi-isogenies of local
and global shtukas.

######  Definition 3.3.1.

  1. (1)

Let $\underline{\mathcal{E}}$ and $\underline{\mathcal{E}}^{\prime}$ be two
global shtukas over an ${\mathbb{F}_{q}}$-algebra $R$ with the same
characteristic sections. A _quasi-isogeny_
$f\colon\underline{\mathcal{E}}\dashrightarrow\underline{\mathcal{E}}^{\prime}$
is a tuple of isomorphisms $f_{j}\colon\mathcal{E}_{j}|_{X_{R}\setminus
D_{R}}\xrightarrow{\cong}\mathcal{E}^{\prime}_{j}|_{X_{R}\setminus D_{R}}$ for
all $0\leq j\leq m$ away from some effective divisor $D$ of $X$, such that
$f_{j-1}\circ\varphi_{j}=\varphi^{\prime}_{j}\circ f_{j}$ for all $1\leq j\leq
m$ and $f_{m}\circ\theta=\theta^{\prime}\circ\sigma^{\ast}f_{0}$.

  2. (2)

Let $\underline{\mathcal{E}}$ and $\underline{\mathcal{E}}^{\prime}$ be two
local shtukas over an ${\mathbb{F}_{q}}$-algebra $R$. A _quasi-isogeny_
$f\colon\underline{\mathcal{E}}\dashrightarrow\underline{\mathcal{E}}^{\prime}$
is a tuple of isomorphisms
$f_{j}\colon\mathcal{E}_{j}\left[\frac{1}{\varpi}\right]\xrightarrow{\cong}\mathcal{E}^{\prime}_{j}\left[\frac{1}{\varpi}\right]$
of the associated $LG$-torsors for all $0\leq j\leq m$, such that
$f_{j-1}\circ\varphi_{j}=\varphi^{\prime}_{j}\circ f_{j}$ for all $1\leq j\leq
m$ and $\sigma^{*}f_{m}\circ\theta=\theta^{\prime}\circ f_{0}$.

In either case, we denote by
$\mathrm{QIsog}(\underline{\mathcal{E}},\underline{\mathcal{E}}^{\prime})$ the
set of quasi-isogenies between $\underline{\mathcal{E}}$ and
$\underline{\mathcal{E}}^{\prime}$.

######  Example 3.3.2.

Let $\underline{y}$ be an $I$-tuple of closed points of $X$ and let
$\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(R)$
for some $R\in\mathcal{N}ilp_{\mathcal{O}_{\underline{y}}}$. Let
$D=\sum_{y\in\underline{y}}[y]$ be the corresponding Cartier divisor on $X$.
The maps
$(\sigma^{*}\varphi_{j}\ldots\circ\sigma^{*}\varphi_{m}\circ\theta\circ\varphi_{1}\circ\ldots\varphi_{j-1})|_{X_{R}\setminus
D_{R}}\colon\mathcal{E}_{j}|_{X_{R}\setminus
D_{R}}\xrightarrow{\cong}\sigma^{\ast}\mathcal{E}_{j}|_{X_{R}\setminus D_{R}}$
for $0\leq j\leq m$ define a quasi-isogeny
$\varphi\colon\underline{\mathcal{E}}\dashrightarrow\sigma^{\ast}\underline{\mathcal{E}}$.

We also have the following analogues of [AH14, Proposition 5.7, Remark 5.8,
Proposition 5.9] in our setting. Their proofs carry over essentially verbatim.
For the convenience of the reader we include sketches of the proofs in our
setting.

######  Proposition 3.3.3.

Let $\underline{y}$ be an $I$-tuple of closed points of $X$, and let
$\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(R)$
for some $R\in\mathcal{N}ilp_{\mathcal{O}_{\underline{y}}}$. Let
$f_{y}\colon\underline{\mathcal{E}^{\prime}}_{y}\dashrightarrow\widehat{\underline{\mathcal{E}}_{y}}$
be a quasi-isogeny of local shtukas at $y$. There exists a global shtuka
$\underline{\mathcal{E}}^{\prime}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(R)$
and a quasi-isogeny
$f\colon\underline{\mathcal{E}}^{\prime}\dashrightarrow\underline{\mathcal{E}}$
of global shtukas such that
$\widehat{\underline{\mathcal{E}}_{y}^{\prime}}=\underline{\mathcal{E}}^{\prime}_{y}$
and $\widehat{f_{y}}=f_{y}$.

###### Proof.

For $0\leq j\leq m$ we construct a $\mathcal{G}$-bundle
$\mathcal{E}^{\prime}_{j}$ on $X_{R}$ by glueing
$\mathcal{E}_{j}|_{(X\setminus\\{y\\})_{R}}$ and $\mathcal{E}^{\prime}_{y,j}$
along $f_{y,j}$ using [AH14, Lemma 5.1] together with the argument of [AH14,
Remark 5.5]. The map $f_{y,j}$ glues with the identity on
$\mathcal{E}_{j}|_{(X\setminus\\{y\\})_{R}}$ to an isomorphism
$f_{j}\colon\mathcal{E}^{\prime}_{j}|_{(X\setminus\\{y\\})_{R}}\xrightarrow{\cong}\mathcal{E}_{j}|_{(X\setminus\\{y\\})_{R}}$.
Moreover, the maps $\varphi_{j}$ glue with $\varphi^{\prime}_{y,j}$ to
isomorphisms
$\varphi^{\prime}_{j}\colon\mathcal{E}_{j}^{\prime}|_{X_{R}\setminus\Gamma_{\underline{x}_{j}}}\xrightarrow{\cong}\mathcal{E}_{j-1}^{\prime}|_{X_{R}\setminus\Gamma_{\underline{x}_{j}}}$
(again using [AH14, Remark 5.5]). Similarly, we obtain $\theta^{\prime}$. Then
$\underline{\mathcal{E}}^{\prime}=((x_{i})_{i\in
I},(\mathcal{E}^{\prime}_{j}),(\varphi^{\prime}_{j}),\theta^{\prime})$ and
$f=(f_{j})\colon\underline{\mathcal{E}}^{\prime}\dashrightarrow\underline{\mathcal{E}}$
clearly satisfies all required properties. ∎

######  Proposition 3.3.4 (Rigidity of quasi-isogenies).

Let $R\in\mathcal{N}ilp_{\mathcal{O}_{\underline{y}}}$ and let $I\subseteq R$
be a nilpotent ideal with quotient $\bar{R}=R/I$. Let
$s\colon\operatorname{Spec}(\bar{R})\hookrightarrow\operatorname{Spec}(R)$ the
corresponding closed immersion on spectra. Moreover, let
$\underline{\mathcal{E}},\underline{\mathcal{E}}^{\prime}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(R)$
two global shtukas over $R$ with the same characteristic sections. Then the
pullback along $s$ induces a bijection

$\mathrm{QIsog}(\underline{\mathcal{E}},\underline{\mathcal{E}}^{\prime})\xrightarrow{\cong}\mathrm{QIsog}(s^{\ast}\underline{\mathcal{E}},s^{\ast}\underline{\mathcal{E}}^{\prime}).$

Note that rigidity of quasi-isogenies for local shtukas is shown in [AH14,
Proposition 2.11] and [Li-23, Proposition 2.3] in the iterated setting.

###### Proof.

It suffices to treat the case $I^{q}=0$, in which case the Frobenius
$\sigma\colon\operatorname{Spec}(R)\to\operatorname{Spec}(R)$ factors as
$\operatorname{Spec}(R)\xrightarrow{\sigma^{\prime}}\operatorname{Spec}(\bar{R})\xrightarrow{s}\operatorname{Spec}(R)$.
We can then recover
$f_{j}\colon\mathcal{E}_{j}\dashrightarrow\mathcal{E}_{j}^{\prime}$ from
$f_{j-1}$ (respectively
$\sigma^{\ast}f_{m}=(\sigma^{\prime})^{\ast}s^{\ast}f_{m}$) and
$\varphi_{j},\varphi_{j}^{\prime}$ (respectively $\theta$ and
$\theta^{\prime}$) as in the proof of [AH14, Proposition 5.9]. ∎

We keep the notation from the previous proposition. Let
$\underline{\bar{\mathcal{E}}}$ be a global shtuka over $\bar{R}$. We denote
by $\mathrm{Defo}_{R}(\underline{\bar{\mathcal{E}}})$ the category of pairs
$(\underline{\mathcal{E}},\alpha)$ of a global shtuka
$\underline{\mathcal{E}}$ over $R$ together with an isomorphism $\alpha\colon
s^{\ast}\underline{\mathcal{E}}\xrightarrow{\cong}\underline{\bar{\mathcal{E}}}$.
Similarly, we define $\mathrm{Defo}_{R}(\underline{\bar{\mathcal{E}}})$ for a
local shtuka $\underline{\bar{\mathcal{E}}}$ over $\bar{R}$. Note that by
rigidity of quasi-isogenies the Hom-sets in
$\mathrm{Defo}(\underline{\bar{\mathcal{E}}})$ are trivial (both in the local
and in the global setting). Let $\mathcal{Z}$ be a global (respectively local)
bound such that $\underline{\bar{\mathcal{E}}}$ is bounded by $\mathcal{Z}$.
Let $\mathrm{Defo}^{\mathcal{Z}}_{R}(\underline{\bar{\mathcal{E}}})$ be the
full subcategory of $\mathrm{Defo}_{R}(\underline{\bar{\mathcal{E}}})$ where
$\underline{\mathcal{E}}$ is bounded by $\mathcal{Z}$ as well.

######  Proposition 3.3.5 (Serre-Tate Theorem for iterated (bounded) shtukas,
compare [AH14, Theorem 5.10]).

Let $\underline{y}$ be an $I$-tuple of closed points of $X$, let
$R\in\mathcal{N}ilp_{\mathcal{O}_{\underline{y}}}$ together with a nilpotent
ideal $I\subseteq R$ and quotient $\bar{R}=R/I$. Let
$\underline{\bar{\mathcal{E}}}\in\operatorname{Sht}_{\mathcal{G},\underline{y}}(\bar{R})$.
The global-to-local functor induces an equivalence of categories

$\mathrm{Defo}_{R}(\underline{\overline{\mathcal{E}}})\to\prod_{y\in\underline{y}}\mathrm{Defo}_{R}(\widehat{\underline{\overline{\mathcal{E}}}_{y}})$

Moreover, let $\mathcal{Z}=\prod_{y\in\underline{y}}\mathcal{Z}^{(I^{y})}$ be
a global bound and for each $y\in\underline{y}$ let $y^{\prime}$ be a closed
point of $\widetilde{X}^{I^{y}}_{\mathcal{Z}}$ lying over $\Delta(y)$. Let
$\underline{\overline{\mathcal{E}}}\in\operatorname{Sht}^{\mathcal{Z}}_{\mathcal{G},\underline{y}^{\prime}}(\bar{R})$.
Then the global-to-local functor induces an equivalence of categories

$\mathrm{Defo}^{\mathcal{Z}}_{R}(\underline{\overline{\mathcal{E}}})\to\prod_{y\in\underline{y}}\mathrm{Defo}^{\mathcal{Z}_{y^{\prime}}}_{R}(\widehat{\underline{\overline{\mathcal{E}}}_{y}}).$

###### Proof.

The second statement clearly follows from the first and Proposition 3.2.6. We
prove the first following the proof of [AH14, Theorem 5.10] by constructing an
inverse functor. It suffices to treat the case $I^{q}=0$. Then $\sigma$
factors as $\sigma=s\circ\sigma^{\prime}$. We set
$\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}=(\sigma^{\prime})^{\ast}\underline{\overline{\mathcal{E}}}$,
hence
$s^{\ast}\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}=\sigma^{\ast}\underline{\overline{\mathcal{E}}}$
and the quasi-isogeny constructed in Example 3.3.2 gives a quasi-isogeny
$\varphi\colon\underline{\overline{\mathcal{E}}}\dashrightarrow
s^{\ast}\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}$. Let
$(\underline{\mathcal{E}}^{\prime}_{y},\alpha^{\prime}_{y})_{y\in\underline{y}}$
be an object of
$\prod_{y\in\underline{y}}\mathrm{Defo}_{R}(\widehat{\underline{\overline{\mathcal{E}}}_{y}})$.
Then $\widehat{\varphi_{y}}\circ\alpha^{\prime}_{y}\colon
s^{\ast}\underline{\mathcal{E}}_{y}^{\prime}\dashrightarrow
s^{\ast}\widehat{\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}_{y}}$ is
a quasi-isogeny for all $y\in\underline{y}$ and lifts by rigidity of quasi-
isogenies for local shtukas [Li-23, Proposition 2.3] to a quasi-isogeny
$\tilde{f}_{y}\colon\underline{\mathcal{E}}_{y}^{\prime}\dashrightarrow\widehat{\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}_{y}}$.
We define the global shtuka $\underline{\mathcal{E}}$ over $R$ as the pullback
of $\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}$ along the
$\tilde{f}_{y}$ as constructed in Proposition 3.3.3, it comes equipped with a
quasi-isogeny
$f\colon\underline{\mathcal{E}}\to\underline{\overline{\mathcal{E}}}^{\sigma^{\prime}}$
which is an isomorphism outside the characteristic sections. Moreover, we
consider the quasi-isogeny $\alpha=\varphi^{-1}\circ s^{*}f\colon
s^{*}\underline{\mathcal{E}}\to\underline{\overline{\mathcal{E}}}$. It is an
isomorphism outside the characteristic sections by construction and at the
characteristic sections we find
$\widehat{\alpha_{y}}=\widehat{\varphi_{y}}^{-1}\circ\widehat{s^{*}f_{y}}=\alpha_{y}^{\prime}$.
Hence, $\alpha$ is an isomorphism and
$(\underline{\mathcal{E}},\alpha)\in\mathrm{Defo}_{R}(\underline{\overline{\mathcal{E}}})$.
As in [AH14, Theorem 5.10] one checks that the constructions are indeed
inverse functors. ∎

###  3.4. Local model theorems

Local model theorems for moduli spaces of shtukas were shown under varying
hypotheses in [Var04, Theorem 2.20] (for $\mathcal{G}$ constant split
reductive), [Laf18, Proposition 2.8], and [AH19, Theorem 3.2.1] (for smooth
affine $\mathcal{G}$), compare also [Fen20, Theorem 4.19] for a different
argument. We slightly generalise their results. While the results are
certainly known to the experts, they do not seem to be contained in the
literature in this generality.

Let $\operatorname{Bun}_{\mathcal{G},\hat{\Gamma}^{I}}$ be the stack such that
for a ${\mathbb{F}_{q}}$-algebra $R$ an $R$ point is given by the data
$(\underline{x},\mathcal{E},\alpha)$, where $\underline{x}\in X^{I}(R)$ is an
$I$-tuple of sections of $X$,
$\mathcal{E}\in\operatorname{Bun}_{\mathcal{G}}(R)$ a $\mathcal{G}$-bundle on
$X_{R}$, and
$\alpha\colon\mathcal{E}|_{\hat{\Gamma}_{\underline{x}}}\xrightarrow{\cong}\mathcal{G}$
is a trivialisation of $\mathcal{E}$ on $\hat{\Gamma}_{\underline{x}}$. For a
positive integer $r$ the stack $\operatorname{Bun}_{\mathcal{G},r\Gamma^{I}}$
is defined to parametrise the same data as
$\operatorname{Bun}_{\mathcal{G},\hat{\Gamma}^{I}}$, but where $\alpha$ is a
trivialisation of $\mathcal{E}$ on $\Gamma^{(r)}_{\underline{x}}$ where
$\Gamma^{(r)}_{\underline{x}}$ is the $r$-th infinitesimal neighbourhood of
the union of the graphs $\Gamma_{x_{i}}$ as introduced in Section 2.1.

Then, $\operatorname{Bun}_{\mathcal{G},\hat{\Gamma}^{I}}$ admits a
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$-action (given by the operation on
$\alpha$), and the forgetful map
$\operatorname{Bun}_{\mathcal{G},\hat{\Gamma}^{I}}\to
X^{I}\times\operatorname{Bun}_{\mathcal{G}}$ is a
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$-torsor. Similarly,
$\operatorname{Bun}_{\mathcal{G},r\Gamma^{I}}\to
X^{I}\times\operatorname{Bun}_{\mathcal{G}}$ is a
$\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}$-torsor.

We also need the following versions of both the Hecke stack and the moduli
space of shtukas. We define
$\widetilde{\operatorname{Hecke}}_{\mathcal{G}}=\widetilde{\operatorname{Hecke}}_{\mathcal{G},X}^{I_{\bullet}}=\operatorname{Hecke}_{\mathcal{G}}\times_{X^{I}\times\operatorname{Bun}_{\mathcal{G}}}\operatorname{Bun}_{\mathcal{G},\hat{\Gamma}^{I}}$,
it parametrises the same data as $\operatorname{Hecke}_{\mathcal{G}}$ together
with a trivialisation of $\mathcal{E}_{0}$ over
$\hat{\Gamma}_{\underline{x}}$. Let $\mathcal{Z}$ be a bound such that the
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$-action on $\mathcal{Z}$ factors through
$\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}$. In this situation, let
$\widetilde{\operatorname{Hecke}}^{(r),\mathcal{Z}}_{\mathcal{G}}=\widetilde{\operatorname{Hecke}}^{(r),I_{\bullet},\mathcal{Z}}_{\mathcal{G},X}=\operatorname{Hecke}^{\mathcal{Z}}_{\mathcal{G}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}\times\operatorname{Bun}_{\mathcal{G}}}\widetilde{X}_{\mathcal{Z}}^{I}\times\operatorname{Bun}_{\mathcal{G},r\Gamma^{I}}$.
In a similar fashion we define $\widetilde{\operatorname{Sht}}_{\mathcal{G}}$
and $\widetilde{\operatorname{Sht}}_{\mathcal{G}}^{(r),\mathcal{Z}}$.

######  Lemma 3.4.1.

The map

$\displaystyle\widetilde{\operatorname{Hecke}}_{\mathcal{G}}$
$\displaystyle\xrightarrow{\cong}\operatorname{Gr}_{\mathcal{G}}\times_{X^{I}}\operatorname{Bun}_{\mathcal{G},\hat{\Gamma}_{I}}$
$\displaystyle\left(\mathcal{E}_{m}\overset{\varphi_{m}}{\dashrightarrow}\mathcal{E}_{m-1}\overset{\varphi_{m-1}}{\dashrightarrow}\ldots\overset{\varphi_{1}}{\dashrightarrow}\mathcal{E}_{0}\overset{\alpha}{\rightarrow}\mathcal{G}\right)$
$\displaystyle\mapsto\left(\mathcal{E}_{m}|_{\hat{\Gamma}_{\underline{x}}}\overset{\varphi_{m}}{\dashrightarrow}\mathcal{E}_{m-1}|_{\hat{\Gamma}_{\underline{x}}}\overset{\varphi_{m-1}}{\dashrightarrow}\ldots\overset{\varphi_{1}}{\dashrightarrow}\mathcal{E}_{0}|_{\hat{\Gamma}_{\underline{x}}}\overset{\alpha}{\rightarrow}\mathcal{G}\right),(\underline{x},\mathcal{E}_{0},\alpha)$

is an isomorphism and for a bound $\mathcal{Z}$ restricts to an isomorphism

$\widetilde{\operatorname{Hecke}}_{\mathcal{G}}^{(r),\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}\xrightarrow{\cong}Z_{\underline{F}^{\prime}}\times_{\widetilde{X}_{\underline{F}^{\prime}}^{I}}\left(\operatorname{Bun}_{\mathcal{G},r\Gamma_{I}}\times_{X^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}\right),$

where $\underline{F}^{\prime}$ is an $I$-tuple of finite extensions of
$\underline{F}$ such that $\mathcal{Z}$ admits a representative
$Z_{\underline{F}^{\prime}}$ over $\underline{F}^{\prime}$.

Note that in [Laf18] the former isomorphism is used to define the bounded
version of the Hecke stack.

###### Proof.

The inverse to the map is constructed by Beauville-Laszlo glueing. The bounded
version follows directly from the definition of
$\operatorname{Hecke}_{\mathcal{G}}^{\mathcal{Z}}$. ∎

We get the following generalisation of [Laf18, Proposition 2.8] (and [Laf18,
Proposition 2.9]). Let $\mathcal{Z}$ be a bound such that the
$\mathcal{L}^{+}_{X^{I}}\mathcal{G}$-action factors through
$\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}$ with representative
$Z_{\underline{F}^{\prime}}$ defined over $\underline{F}^{\prime}$. Then by
construction we get a local model roof

${\widetilde{\operatorname{Sht}}^{(r),\mathcal{Z}}_{\mathcal{G}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}}$${\operatorname{Sht}^{\mathcal{Z}}_{\mathcal{G}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}}$${Z_{\underline{F}^{\prime}},}$

where the maps are given by the obvious projections.

######  Proposition 3.4.2.

Let $\mathcal{Z}$ be a bound and $\underline{F}^{\prime}$ an $I$-tuple of
finite extensions of $\underline{F}$ such that $\mathcal{Z}$ admits a
representative over $\underline{F}^{\prime}$. The map
$\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}\to\left[\mathcal{L}^{(r)}_{X^{I}}\mathcal{G}\backslash
Z_{\underline{F}^{\prime}}\right]$ is a smooth map of Artin stacks.

Moreover, assume that $\mathcal{Z}$ arises as from bounds
$\mathcal{Z}^{(I_{j})}$ defined in
$\operatorname{Gr}_{\mathcal{G},X}^{(I_{j})}$ via Construction 2.2.4. Then the
induced map
$\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}\to\prod_{j=1}^{m}\left[\mathcal{L}^{(r)}_{X^{I_{j}}}\mathcal{G}\backslash
Z^{(I_{j})}_{\underline{F}^{\prime}}\right]$ is a smooth map of Artin stacks.

###### Proof.

The proofs of [Laf18, Proposition 2.8 and 2.9] still apply to our situation. ∎

We also get an étale-local version of the local model theorem, generalising
[AH19, Theorem 3.2.1], [Laf18, Proposition 2.11] and [Var04, Theorem 2.20].

######  Proposition 3.4.3.

Let $\mathcal{Z}$ be a bound and $\underline{F}^{\prime}$ a finite extension
of $F$ such that $\mathcal{Z}$ admits a representative over
$\underline{F}^{\prime}$. Then
$\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}$
and $Z_{\underline{F}^{\prime}}$ are étale-locally isomorphic. More precisely,
there is an étale cover
$U\to\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}$
admitting an étale map $U\to Z_{\underline{F}^{\prime}}$ over
$\widetilde{X}^{I}_{\underline{F}^{\prime}}$.

Moreover, assume that $\mathcal{Z}$ arises as from bounds
$\mathcal{Z}^{(I_{j})}$ defined in
$\operatorname{Gr}_{\mathcal{G},X}^{(I_{j})}$ via Construction 2.2.4. Then
$\operatorname{Sht}_{\mathcal{G}}^{\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}$
is étale-locally isomorphic to
$\prod_{j=1}^{m}Z^{(I_{j})}_{\underline{F}^{\prime}}$.

###### Proof.

We sketch how to adapt the argument of [Laf18, Proposition 2.11]. By the usual
reductions (notably bounding the Harder-Narasimhan slopes appearing and
introducing an auxiliary level structure, compare also [AH21, Remark 2.9] for
the generalisation to arbitrary smooth affine group schemes $\mathcal{G}$),
for example as at the beginning of the proof of [Laf18, Proposition 2.11], we
may treat $\operatorname{Bun}_{\mathcal{G}}$ (and thus
$\operatorname{Hecke}_{\mathcal{G}}^{\mathcal{Z}}$) as a scheme of finite type
over ${\mathbb{F}_{q}}$.

For every $j$ we denote by $(\mathcal{E}^{\mathrm{univ}},(x_{i})_{i\in I})$
the universal object of $\operatorname{Bun}_{\mathcal{G}}\times X^{I}$. Let
$U\to\operatorname{Bun}_{\mathcal{G}}\times X^{I}$ be an étale cover
trivialising $\mathcal{E}^{\mathrm{univ}}$ over
$\hat{\Gamma}_{\underline{x}}$. By [AH21, Proposition 3.9], the map
$\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet},\mathcal{Z}}\to\widetilde{X}_{\mathcal{Z}}^{I}\times\operatorname{Bun}_{\mathcal{G}}$
is schematic and of finite type. Hence, the base change
$W=\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet},\mathcal{Z}}\times_{\widetilde{X}_{\mathcal{Z}}^{I}\times\prod_{j}\operatorname{Bun}_{\mathcal{G}}}\widetilde{X}_{\mathcal{Z}}^{I}\times_{X^{I}}U$
is a scheme of finite type over ${\mathbb{F}_{q}}$, and the projection
$W\to\operatorname{Hecke}_{\mathcal{G},X}^{I_{\bullet},\mathcal{Z}}$ is étale.
By construction of $U$, the scheme $W$ admits a natural map
$W\to\operatorname{Bun}_{\mathcal{G}}\times\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$,
that factors through $\operatorname{Bun}_{\mathcal{G}}\times
Z_{\underline{F}^{\prime}}$ and is étale by Lemma 3.4.1.

The first statement for now follows from [Laf18, Lemme 2.13]. As in [Laf18,
Proposition 2.11], the second statement can be shown inductively using a
similar argument. ∎

###  3.5. Functoriality results for moduli spaces of shtukas

We study functoriality properties of moduli spaces of shtukas under
homomorphisms of group schemes. As the main result in this section we show
that for a generic isomorphism $\mathcal{G}\to\mathcal{G}^{\prime}$ the
induced map on the moduli stacks of (iterated, bounded) shtukas is schematic,
separated and of finite type. Moreover, when $\mathcal{G}$ is parahoric and
the bound is generically defined, the map is proper and surjective. The result
is an extension of [Bre18, Theorem 3.20] to our setting. In particular, the
use of global bounds allows us to work on the whole curve $X$ and we need not
restrict the legs to a formal neighbourhood of a fixed point as in [Bre18].
Moreover, we show that the level maps are generically finite étale, which is
not part of [Bre18].

In our setting, the notion of a shtuka datum (respectively a map of shtuka
data) in the sense of [Bre18, Definitions 3.1 and 3.9] restricts to the
following.

######  Definition 3.5.1.

A _shtuka datum_ $(\mathcal{G},\mathcal{Z})$ is a pair of a smooth affine
group scheme $\mathcal{G}\to X$ and a bound $\mathcal{Z}$ in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}$. Let
$(\mathcal{G},\mathcal{Z})$ and $(\mathcal{G}^{\prime},\mathcal{Z}^{\prime})$
be two shtuka data such that for every $i\in I$ the $i$-th components of
$\mathcal{Z}$ and $\mathcal{Z}^{\prime}$ have the same type. A _map of shtuka
data_
$f\colon(\mathcal{G},\mathcal{Z})\to(\mathcal{G}^{\prime},\mathcal{Z}^{\prime})$
is a map of group schemes $f\colon\mathcal{G}\to\mathcal{G}^{\prime}$ such
that for all $I$-tuples of finite extensions $\underline{F}^{\prime}$ such
that both $\mathcal{Z}$ and $\mathcal{Z}^{\prime}$ admit representatives over
$\underline{F}^{\prime}$ the map

$Z_{\underline{F}^{\prime}}\hookrightarrow\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}\xrightarrow{f_{\ast}}\operatorname{Gr}_{\mathcal{G}^{\prime},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}^{I}_{\underline{F}^{\prime}}$

factors through $Z^{\prime}_{\underline{F}^{\prime}}$.

In analogy to the construction of the reflex scheme we set
$\widetilde{X}^{I}_{\mathcal{Z}.\mathcal{Z}^{\prime}}=\widetilde{X}^{I}_{\underline{F}^{\prime}}/\operatorname{Aut}_{\mathcal{Z}.\mathcal{Z}^{\prime}}(\underline{F}^{\prime})$
where $\underline{F}^{\prime}$ is a componentwise Galois extension such that
both $\mathcal{Z}$ and $\mathcal{Z}^{\prime}$ admit representatives over
$\underline{F}^{\prime}$ and

$\operatorname{Aut}_{\mathcal{Z}.\mathcal{Z}^{\prime}}(\underline{F}^{\prime})=\\{(g_{i})_{i\in
I}\in\operatorname{Gal}(F_{i}^{\prime}/F)\colon(g_{i})_{i\in
I}^{\ast}(\mathcal{Z})=\mathcal{Z},\ (g_{i})_{i\in
I}^{\ast}(\mathcal{Z}^{\prime})=\mathcal{Z}^{\prime}\\}.$

Under mild hypotheses, a map of shtuka data
$f\colon(\mathcal{G},\mathcal{Z})\to(\mathcal{G}^{\prime},\mathcal{Z}^{\prime})$
induces a map on the corresponding moduli stacks of shtukas

$f_{\ast}\colon\operatorname{Sht}^{I_{\bullet},\mathcal{Z}}_{\mathcal{G},X}\times_{\widetilde{X}^{I}_{\mathcal{Z}}}\widetilde{X}^{I}_{\mathcal{Z}.\mathcal{Z}^{\prime}}\to\operatorname{Sht}^{I_{\bullet},\mathcal{Z}^{\prime}}_{\mathcal{G}^{\prime},X}\times_{\widetilde{X}^{I}_{\mathcal{Z}^{\prime}}}\widetilde{X}^{I}_{\mathcal{Z}.\mathcal{Z}^{\prime}}$

by the following lemma that is an analogue of [Bre18, Lemma 3.15].

######  Lemma 3.5.2.

Let
$f\colon(\mathcal{G},\mathcal{Z})\to(\mathcal{G}^{\prime},\mathcal{Z}^{\prime})$
be a map of shtuka data. Assume that $\mathcal{Z}^{\prime}$ admits a
representative over its reflex scheme. Let

$\underline{\mathcal{E}}\in(\operatorname{Sht}^{\mathcal{Z}}_{\mathcal{G}}\times_{X^{I}}\widetilde{X}^{I}_{\mathcal{Z}.\mathcal{Z}^{\prime}})(R)$

for some ${\mathbb{F}_{q}}$-algebra $R$. Then
$f_{\ast}\underline{\mathcal{E}}\in(\operatorname{Sht}_{\mathcal{G}^{\prime}}\times_{X^{I}}\widetilde{X}^{I}_{\mathcal{Z}.\mathcal{Z}^{\prime}})(R)$
is bounded by $\mathcal{Z}^{\prime}$.

###### Proof.

Let $\underline{\mathcal{E}}=((x_{i})_{i\in
I},(\mathcal{E}_{j})_{j=0,\ldots,m},(\varphi_{j})_{j=1,\ldots,m},\theta)\in(\operatorname{Sht}^{\mathcal{Z}}_{\mathcal{G}}\times_{X^{I}}X_{\mathcal{Z}.\mathcal{Z}^{\prime}}^{I})(R)$.
Let $R\to R^{\prime}$ be an étale cover that trivialises
$\mathcal{E}_{m}|_{\hat{\Gamma}_{\underline{x}}}$ and choose a trivialisation
$\alpha\colon\mathcal{E}_{m}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}\xrightarrow{\cong}\mathcal{G}|_{\hat{\Gamma}_{\underline{x}_{R^{\prime}}}}$.
Then $(\underline{\mathcal{E}}_{R^{\prime}},\alpha)$ defines an
$R^{\prime}$-valued point in
$\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}X_{\mathcal{Z}.\mathcal{Z}^{\prime}}^{I}$.
Let $\underline{F}^{\prime}$ be an $I$-tuple of finite extensions of
$\underline{F}$ such that both $\mathcal{Z}$ and $\mathcal{Z}^{\prime}$ admit
representatives over $\underline{F}^{\prime}$. As $\underline{\mathcal{E}}$ is
bounded by $\mathcal{Z}$, the induced point
$\operatorname{Spec}(R^{\prime})\times_{\widetilde{X}_{\mathcal{Z}}^{I}}{\widetilde{X}_{F^{\prime}}^{I}}\to\operatorname{Gr}_{\mathcal{G},X}^{I_{\bullet}}\times_{X^{I}}{\widetilde{X}_{F^{\prime}}^{I}}$
factors through $Z_{\underline{F}^{\prime}}$, hence its image under $f_{\ast}$
factors through $Z^{\prime}_{\underline{F}^{\prime}}$ by assumption. By Lemma
2.2.9 the map
$\operatorname{Spec}(R^{\prime})\times_{\widetilde{X}_{\mathcal{Z}^{\prime}}^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}\to\operatorname{Gr}_{\mathcal{G}^{\prime},X}^{I_{\bullet}}\times_{X^{I}}\widetilde{X}_{\underline{F}^{\prime}}^{I}$
factors through $Z^{\prime}_{\underline{F}^{\prime}}$, too. ∎

######  Remark 3.5.3.

Note that we used the assumption on the bounds in the last step of the proof.
We do not know how to prove the lemma without this assumption, neither in our
setting nor in the setting of local boundedness conditions of [Bre18]. In this
sense [Bre18, Lemma 3.15] which does not make a similar kind of assumption
seems to be problematic in general.

We need the following lemma on twisted flag varieties in the local setting.

######  Lemma 3.5.4.

Let $k\left(\\!\left(t\right)\\!\right)$ be the field of formal Laurent series
over an arbitrary field $k$ and let $\mathcal{O}=k\left\llbracket
t\right\rrbracket$ the subring of formal power series. Let $G$ be a smooth
affine group scheme over $k\left(\\!\left(t\right)\\!\right)$ and let
$\mathcal{G}$ and $\mathcal{G}^{\prime}$ be two smooth integral models of $G$
with geometrically connected fibres. Let
$f\colon\mathcal{G}\to\mathcal{G}^{\prime}$ be a homomorphism of
$\mathcal{O}$-group schemes that is the identity on $G$ over $k$. Then the
induced map on loop groups $L^{+}\mathcal{G}\to L^{+}\mathcal{G}^{\prime}$ is
a closed immersion.

  1. (1)

The corresponding twisted flag variety
$L^{+}\mathcal{G}^{\prime}/L^{+}\mathcal{G}$ is representable by a separated
scheme of finite type over $k$. If $k$ is finite or separably closed, then

$\left(L^{+}\mathcal{G}^{\prime}/L^{+}\mathcal{G}\right)(k)=\mathcal{G}^{\prime}(\mathcal{O})/\mathcal{G}(\mathcal{O}).$

  2. (2)

Assume that $k$ is finite. We equip $G(k\left(\\!\left(t\right)\\!\right))$
with the analytic topology induced by the natural topology on
$k\left(\\!\left(t\right)\\!\right)$ (note that
$k\left(\\!\left(t\right)\\!\right)$ is locally compact in this case). Then
$\mathcal{G}(\mathcal{O})\subseteq\mathcal{G}^{\prime}(\mathcal{O})$ are
compact open subgroups of $G(k\left(\\!\left(t\right)\\!\right))$. In
particular, the quotient
$\mathcal{G}^{\prime}(\mathcal{O})/\mathcal{G}(\mathcal{O})$ is discrete and
finite.

  3. (3)

Let $S$ be an $k$-scheme. Giving a $L^{+}\mathcal{G}$-torsor over $S$ is
equivalent to giving a $L^{+}\mathcal{G}^{\prime}$-torsor
$\mathcal{E}^{\prime}$ over $S$ together with an isomorphism
$\mathcal{E}^{\prime}/L^{+}\mathcal{G}\xrightarrow{\cong}L^{+}\mathcal{G}^{\prime}/L^{+}\mathcal{G}$.

Note that giving an isomorphism
$\mathcal{E}^{\prime}/L^{+}\mathcal{G}\xrightarrow{\cong}L^{+}\mathcal{G}^{\prime}/L^{+}\mathcal{G}$
in (3) is also clearly equivalent to giving a section in
$\left(\mathcal{E}^{\prime}/L^{+}\mathcal{G}\right)(S)$.

###### Proof.

  1. (1)

By the argument in the proof of [Bre18, Lemma 3.17], the quotient stack
$L^{+}\mathcal{G}^{\prime}/L^{+}\mathcal{G}$ is representable by a separated
scheme of finite type over $k$ that is moreover a closed subscheme of the
affine Grassmannian $\operatorname{Gr}_{\mathcal{G}}$. For the second claim,
it suffices to show that $H^{1}(k,L^{+}\mathcal{G})$ is trivial by the moduli
description of the quotient stack. But this is shown in the proof of [Ric20,
Corollary 3.22].

  2. (2)

Clearly, both $\mathcal{G}(\mathcal{O})$ and
$\mathcal{G}^{\prime}(\mathcal{O})$ are compact open subgroups of
$G(k\left(\\!\left(t\right)\\!\right))$ by construction. The existence of the
map $f$ then means that $\mathcal{G}(\mathcal{O})$ is a subgroup of
$\mathcal{G}^{\prime}(\mathcal{O})$. The assertion on the quotient then
directly follows from basic facts from topology.

  3. (3)

Given a $L^{+}\mathcal{G}$-torsor $\mathcal{E}$ on $S$, its associated
$L^{+}\mathcal{G}^{\prime}$-torsor is given by
$\mathcal{E}\times^{L^{+}\mathcal{G}}L^{+}\mathcal{G}^{\prime}$. The map on
sections given by $(e,g)\mapsto g$ then induces an isomorphism
$\mathcal{E}^{\prime}/L^{+}\mathcal{G}\xrightarrow{\cong}L^{+}\mathcal{G}^{\prime}/L^{+}\mathcal{G}.$
This construction is an equivalence.

∎

We are now in a position to prove the main result in this section, an
extension of the functoriality result of [Bre18, Theorem 3.20] to moduli
spaces of shtukas with global and generically bounds.

######  Theorem 3.5.5.

Let $\mathcal{G}$ and $\mathcal{G}^{\prime}$ be two smooth affine group
schemes over $X$ with geometrically connected fibres. Let
$f\colon(\mathcal{G},\mathcal{Z})\to(\mathcal{G}^{\prime},\mathcal{Z}^{\prime})$
be a map of shtuka data such that the map
$f\colon\mathcal{G}\to\mathcal{G}^{\prime}$ is an isomorphism over
$U=X\setminus\left\\{y_{1},\ldots,y_{n}\right\\}$ for a finite set of closed
points $\left\\{y_{1},\ldots,y_{n}\right\\}$ of $X$. Assume that
$\mathcal{Z}^{\prime}$ admits a representative over its reflex scheme.

  1. (1)

The induced map