# Compactification of Level Maps of Moduli Spaces of Drinfeld Shtukas

Patrick Bieker  Technische Universität Darmstadt, Department of Mathematics,
Schlossgartenstrasse 7, 64289 Darmstadt, Germany<EMAIL_ADDRESS>darmstadt.de

###### Abstract.

We define Drinfeld level structures for Drinfeld shtukas of any rank and show
that their moduli spaces are regular and admit finite flat level maps. In
particular, the moduli spaces of Drinfeld shtukas with Drinfeld
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures provide a good integral model
and relative compactification of the moduli space of shtukas with naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level defined using shtukas for dilated group
schemes.

###### Contents

  1. 1 Introduction
  2. 2 Moduli spaces of shtukas and naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures
  3. 3 Group schemes attached to Drinfeld shtukas
  4. 4 Drinfeld $\Gamma_{1}(\mathfrak{p}^{n})$-level structures on shtukas
  5. 5 Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structures on shtukas
  6. 6 Comparison with naive level structures and Bruhat-Tits theory

##  1\. Introduction

Moduli spaces of shtukas are function field analogues of Shimura varieties and
thus play a central role in the Langlands programme over function fields.
While a lot of progress has been made in understanding the geometry of moduli
spaces of shtukas for general reductive groups with parahoric level, compare
for example [AH14], [AH19], [Bre18], [YZ19] and [Zhu14], little is known for
deeper level structures.

The goal of this work is to study deeper level structures in the case of
Drinfeld shtukas (that means $\operatorname{GL}_{r}$-shtukas for some fixed
$r\geq 1$ with two legs bounded by the minuscule cocharacters
$(0,\ldots,0,-1)$ and $(1,0,\ldots,0)$). More explicitly, we define for all
$n\geq 0$ an analogue of $\Gamma_{0}(p^{n})$-level structures on elliptic
curves that behaves well at the place of bad reduction. Let us explain the
construction in more detail.

Let $X$ be a smooth, projective and geometrically connected curve over a
finite field ${\mathbb{F}_{q}}$. Let us fix an ${\mathbb{F}_{q}}$-rational
point $\infty$ of $X$ and let us denote $X^{\prime}=X\setminus\\{\infty\\}$ A
_Drinfeld shtuka_ of rank $r$ (with pole fixed at $\infty$) over a scheme $S$
is given by the data $\underline{\mathcal{E}}=(x,\mathcal{E},\varphi),$ where

  * •

$x\in X^{\prime}(S)$ is the characteristic section (also called leg or paw),

  * •

$\mathcal{E}$ is a rank $r$ vector bundle on $X_{S}$ and

  * •

$\varphi\colon\sigma^{\ast}\mathcal{E}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{\infty})}\xrightarrow{\cong}\mathcal{E}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{\infty})}$
is an isomorphism of $\mathcal{O}_{X_{S}}$-modules away from the graphs
$\Gamma_{x}$ of $x$ and $\Gamma_{\infty}$ of $\infty$, such that $\varphi$
extends to a map
$\varphi\colon\sigma^{\ast}\mathcal{E}|_{X^{\prime}}\to\mathcal{E}|_{X^{\prime}}$
with $\operatorname{coker}(\varphi)$ supported on $\Gamma_{x}$ and invertible
on its support, and $\varphi^{-1}$ extends to a map
$\varphi^{-1}\colon\mathcal{E}|_{X\setminus\Gamma_{x}}\to\sigma^{\ast}\mathcal{E}|_{X\setminus\Gamma_{x}}$
with $\operatorname{coker}(\varphi^{-1})$ supported on $\infty$ and invertible
on its support.

We denote by $\operatorname{Sht}_{r}$ the stack of Drinfeld shtukas of rank
$r$, it is a Deligne-Mumford stack locally of finite type over
${\mathbb{F}_{q}}$. The projection to the characteristic section defines a map
$\operatorname{Sht}_{r}\to X^{\prime}$, which is smooth of relative dimension
$(2r-2)$. Moduli spaces of Drinfeld shtukas are used with great success in
[Laf02] to establish the Langlands correspondence for $\operatorname{GL}_{r}$.

Drinfeld shtukas in this sense are a generalisation of Drinfeld modules
(compare Proposition 3.3 for a precise statement). Moduli spaces of rank 2
Drinfeld modules can be thought of as a function field analogue of the moduli
space of elliptic curves.

In the case of modular curves with $\Gamma_{1}$\- and $\Gamma_{0}$-level
structures, the reduction modulo $p$ was studied by [DR73] using a
normalisation procedure. [KM85] gave an explicit moduli description of an
integral model using Drinfeld level structures. This notion goes back to
[Dri76], who first introduced this notion of level structure to study full
level structures of Drinfeld modules. The analogy to the modular curve
suggests a strategy to construct good integral models in our shtuka setting,
in other words to define a good notion of level structure that also behaves as
desired at a place of bad reduction: to use Drinfeld level structures.

Let us fix a ${\mathbb{F}_{q}}$-rational point $0$ of $X$ and denote by
$\mathcal{O}_{0}$ (respectively $\mathfrak{p}=\mathfrak{p}_{0}$) the
completion of the local ring of $X$ and $0$ (respectively its maximal ideal).
In order to define Drinfeld level structures for shtukas, we associate to a
Drinfeld shtuka its _scheme of $\mathfrak{p}^{n}$-torsion points_
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$. This was essentially constructed
in [Dri87] and shares similar properties with the scheme of
$\mathfrak{p}^{n}$-torsion points of a Drinfeld module (respectively the
scheme of $p^{n}$-torsion points of an elliptic curve). It is a finite locally
free $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module scheme of rank $q^{2}$ over
$S$. Moreover, we show that étale locally on $S$ we find an embedding of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ as a Cartier divisor into
$\mathbb{A}^{1}_{S}$ by adapting a similar result for the $p^{n}$-torsion of
one-dimensional $p$-divisible groups of [Fri19], compare Proposition 3.13 and
Remark 3.14.

This allows us to define Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level
structures on Drinfeld shtukas as follows.

######  Definition 1.1 (compare Definition 5.2 for general rank).

A _$\Gamma_{0}(\mathfrak{p}^{n})$ -level structure_ on a rank 2 shtuka
$\underline{\mathcal{E}}$ is a finite locally free closed subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank $q^{n}$
that admits a generator fppf-locally on $S$, that means an
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-linear map
$\iota\colon(\mathfrak{p}^{-n}/\mathcal{O}_{0})\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$
such that after the choice of an embedding
$\mathcal{E}[\mathfrak{p}^{n}]\hookrightarrow\mathbb{A}^{1}_{S}$ we have

$\sum_{\alpha\in\mathfrak{p}^{-n}/\mathcal{O}_{0}}[\iota(\alpha)]=\mathbf{H}\qquad\text{and}\qquad\sum_{\alpha\in\mathfrak{p}^{-1}/\mathcal{O}_{0}}[\iota(\alpha)]\subseteq\underline{\mathcal{E}}[\mathfrak{p}]$

as Cartier divisors in $\mathbb{A}^{1}_{S}$.

Note the subtle difference compared to the definition of
$\Gamma_{0}(p^{n})$-level structures in [KM85]: In the setting of elliptic
curves the second condition is automatic. However, in our setting the second
condition is in particular necessary to get well-defined level maps, see
Remark 4.4 for an explicit counterexample.

Adapting the theory of Drinfeld level structures for elliptic curves in
[KM85], we obtain the following.

######  Theorem 1.2 (compare Theorem 5.3).

Let $r\geq 1$ and $n\geq 0$ be integers .

  1. (1)

The moduli stack $\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ of rank
$r$ Drinfeld shtukas with Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level
structures is representable by a regular Deligne-Mumford stack locally of
finite type over ${\mathbb{F}_{q}}$.

  2. (2)

The level map
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
is schematic, finite and flat. Moreover, it is finite étale away from
$\mathfrak{p}$.

As in the case of elliptic curves in [KM85, Chapter 5], we first show the
corresponding results for $\Gamma_{1}(\mathfrak{p}^{n})$-level structures. The
main step in the proof of the $\Gamma_{1}(\mathfrak{p}^{n})$ case is the study
of the deformation theory at supersingular points, where we rely on results of
[Dri76].

Using the flatness of the moduli space, we construct a compatible system of
level maps

$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{m})}$

that are finite and flat for all $m\leq n$. The level maps allow us to
interpret our construction in the following way in terms of the combinatorics
of the Bruhat-Tits building of $\operatorname{GL}_{r}$ over the fraction field
$K_{0}$ of $\mathcal{O}_{0}$. Let us denote by $\operatorname{Sht}_{r,\Omega}$
the moduli stack of shtukas for the Bruhat-Tits group scheme
$\operatorname{GL}_{r,\Omega}$ bounded by $(0,\ldots,0,1)$ and
$(1,0,\ldots,0)$ corresponding to the standard simplex $\Omega$ of sidelength
$n$ in the standard appartment of the Bruhat-Tits building. To such a bounded
$\operatorname{GL}_{r,\Omega}$-shtuka we can associate a Drinfeld shtuka with
Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structure, this is explained in
more detail below. Moreover, using the level maps, we get compatible system of
maps
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\mathfrak{f}}$
to the moduli space of shtukas for Bruhat-Tits group schemes for all facets
$\mathfrak{f}\prec\Omega$ contained in the closure of $\Omega$.

######  Theorem 1.3 (compare Theorem 6.7).

The map
$\operatorname{Sht}_{r,\Omega}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
is a quasi-compact open immersion and an isomorphism away from $0$. Its
schematic image in the sense of [EG21] is
$\overline{\operatorname{Sht}}_{r,\Omega}=\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$
via the maps

$\operatorname{Sht}_{r,\Omega}\hookrightarrow\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\hookrightarrow\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$

constructed above. In the parahoric case $n=1$, the map
$\operatorname{Sht}_{r,\Omega}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p})}$
is an isomorphism.

Note that this also shows that
$\overline{\operatorname{Sht}}_{r,\Omega}=\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$
is the flat closure of the generic fibre inside
$\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$.
Theorem 1.3 suggests that a candidate for a good integral model for the moduli
spaces of shtukas for a general reductive group with deep Bruhat-Tits level
structure (i.e. level structures generalising
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures in the
$\operatorname{GL}_{r}$-case) might be the closure of the moduli stack of
shtukas for the Bruhat-Tits group scheme inside the limit of all moduli stacks
with corresponding parahoric level. We study this more general setup in an
upcoming subsequent paper.

We can also interpret this result as follows. We denote for $n\in\mathbb{N}$
by $D_{n}=n[0]\subseteq X$ the effective Cartier divisor in $X$ defined by
(multiples of) the point $0$. Then
$D_{n}=\operatorname{Spec}(\mathcal{O}_{0}/\mathfrak{p}^{n})$. For a Drinfeld
shtuka $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ over a
(${\mathbb{F}_{q}}$-)scheme $S$, we denote by
$\underline{\mathcal{E}}|_{D_{n,S}}$ its pullback to $D_{n,S}$. A _naive_
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure on a rank 2 shtuka
$\underline{\mathcal{E}}$ is given by a quotient of
$\mathcal{O}_{D_{n,S}}$-modules $\mathcal{E}\twoheadrightarrow\mathcal{L}$
such that $\mathcal{L}$ is finite locally free of rank 1 on
$\mathcal{O}_{D_{n,S}}$ and such that $\varphi$ descends to a map
$\sigma^{*}\mathcal{L}\to\mathcal{L}$. Following [MRR20] we can view Drinfeld
shtukas with naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures as shtukas
for the Bruhat-Tits group scheme $\operatorname{GL}_{r,\Omega}$ bounded by
$\underline{\mu}=((0,\ldots,0,-1),(1,0,\ldots,0))$. For a precise definition
see Section 2.5 below.

By the analogy with the modular curve, in the fibre of the moduli space of
shtukas with $\Gamma_{0}(\mathfrak{p}^{n})$-level structures over $0$ we
should expect to find $n+1$ components intersecting at supersingular points.
However, it can be shown that in the non-parahoric case (in other words when
$n\geq 2$), the moduli space of Drinfeld shtukas with naive level structures
as above only has two components which moreover do not intersect in its fibre
over $0$. In particular, its supersingular points are missing (compare Remark
2.20).

Another way to phrase Theorem 1.3 is that we compactified the level map
$\operatorname{Sht}_{r,\Omega}\to\operatorname{Sht}_{r}$, which we saw in the
example above is not proper in general, by factoring it in an open immersion
with dense image followed by a finite (hence proper) and surjective map

$\operatorname{Sht}_{r,\Omega}\hookrightarrow\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}.$

In particular, in the rank 2 example
$\operatorname{Sht}_{2,\Gamma_{0}(\mathfrak{p}^{n})}$ acquires the
supersingular points missing in $\operatorname{Sht}_{2,\Omega}$.

### Organisation.

This paper is organised as follows. In Section 2, we recall some facts on
shtukas (in particular Drinfeld shtukas) and define naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures. In Section 3, we explain the
comparison with Drinfeld modules. This provides us with a way to associate to
(global, local and finite) shtukas group schemes, which is what makes it
possible to define Drinfeld level structures in the first place. In
particular, we construct the scheme of $\mathfrak{p}^{n}$-division points of a
Drinfeld shtuka and study its properties. In Sections 4 and 5, we define our
Drinfeld ($\Gamma_{1}$\- and $\Gamma_{0}$-type) level structures and prove the
regularity of their moduli spaces. For this, we follow [KM85]. In Section 6,
we show that the Drinfeld level structures actually provide a good (relative)
compactification of the moduli space with naive level structure
$\operatorname{Sht}_{r,\Omega}$.

### Acknowledgements.

First of all I thank my advisor Timo Richarz for introducing me to this topic,
his steady encouragement and his interest in my work. I thank Paul Hamacher,
João Lourenço, Lenny Taelman and Torsten Wedhorn for helpful conversations
surrounding this work. I thank Gebhard Böckle and Urs Hartl for their comments
on a previous version of this paper. I thank Tasho Kaletha for sharing a
preliminary version of [KP] with us.

This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized
Structures, project number 444845124.

### Notation.

We fix the following notation. Let ${\mathbb{F}_{q}}$ be a finite field with
$q$ elements, let $p$ be the characteristic of ${\mathbb{F}_{q}}$. All schemes
will be ${\mathbb{F}_{q}}$-schemes unless otherwise specified. Let $X$ be a
smooth projective and geometrically connected curve over ${\mathbb{F}_{q}}$
with function field $K$. For 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. Moreover, we denote by $K_{x}$ the completion of $K$ at $x$.

We fix two distinct ${\mathbb{F}_{q}}$-rational points $\infty$ and $0$ of $X$
and denote by $\mathfrak{p}=\mathfrak{p}_{0}$ the maximal ideal in the
complete local ring $\mathcal{O}_{0}$ at $0$. Let us also fix a uniformiser
$\varpi$ of $\mathfrak{p}$.

We denote by $\sigma$ the (absolute) $q$-Frobenius endomorphism
$\operatorname{Frob}_{S}$ of some ${\mathbb{F}_{q}}$-scheme $S$, and also the
map $\sigma=id_{X}\times\operatorname{Frob}_{S}\colon X_{S}\to X_{S}$. It is
always clear from context which map $\sigma$ is meant.

##  2\. Moduli spaces of shtukas and naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures

Drinfeld Shtukas were introduced in [Dri87] as _elliptic sheaves_ and were
vastly generalised to arbitrary reductive groups or even general smooth affine
group schemes by [Var04] and [AH14], respectively.

We introduce naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures on Drinfeld
shtukas, present how to encode these level structures in terms of Bruhat-Tits
group schemes following [MRR20] and explain, why the naive definition is not
appropriate for deeper level (that means for $n>1$).

###  2.1. Global shtukas.

We recall the definitions of global shtukas and isogenies of Drinfeld shtukas.
We restrict ourselves to shtukas with two legs with one leg fixed at the point
$\infty$.

######  Definition 2.1 ([AH14]).

Let $G$ be a smooth affine group scheme on $X$. A _global $G$-shtuka_ over a
scheme $S$ is given by the data

$\underline{\mathcal{E}}=(x,\mathcal{E},\varphi\colon\sigma^{\ast}\mathcal{E}\dashrightarrow\mathcal{E}),$

where

  * •

$x\in X^{\prime}(S)$ is a section of $X^{\prime}=X\setminus\\{\infty\\}$,

  * •

$\mathcal{E}$ is a $G$-bundle on $X_{S}$ and

  * •

$\varphi\colon\sigma^{\ast}\mathcal{E}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{\infty})}\xrightarrow{\cong}\mathcal{E}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{\infty})}$
is an isomorphism of $G$-bundles away from the graphs $\Gamma_{x}$ of $x$ and
$\Gamma_{\infty}$ of $\infty$.

The point $x$ is called _characteristic_ or _leg_ of
$\underline{\mathcal{E}}$. A map of $G$-shtukas is a tuple of maps of
$G$-bundles compatible with the maps $\varphi$ and $\varphi^{\prime}$.

Note that there are several ways to bound the zeros (and poles, respectively)
of $G$-shtukas, and in general they are not equivalent (compare Remark 2.7).
We will mostly be interested in the case of Drinfeld shtukas, that means we
consider $G=\operatorname{GL}_{r}$ (or corresponding Bruhat-Tits group
schemes) and bounds given by the minuscule coweights
$\underline{\mu}=((0,\ldots,0,-1),(1,0,\ldots,0))$. These admit the following
explicit description.

######  Definition 2.2 ([Dri87]).

A _Drinfeld shtuka_ of rank $r$ over a scheme $S$ is given by the data

$\underline{\mathcal{E}}=(x,\mathcal{E},\varphi),$

where

  * •

$x\in X^{\prime}(S)$ is the characteristic section,

  * •

$\mathcal{E}$ is a rank $r$ vector bundle on $X_{S}$ and

  * •

$\varphi\colon\sigma^{\ast}\mathcal{E}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{\infty})}\xrightarrow{\cong}\mathcal{E}|_{X_{S}\setminus(\Gamma_{x}\cup\Gamma_{\infty})}$
is an isomorphism of $\mathcal{O}_{X_{S}}$-modules away from the graphs
$\Gamma_{x}$ of $x$ and $\Gamma_{\infty}$ of $\infty$, such that $\varphi$
extends to a map
$\varphi\colon\sigma^{\ast}\mathcal{E}|_{X^{\prime}}\to\mathcal{E}|_{X^{\prime}}$
with $\operatorname{coker}(\varphi)$ supported on $\Gamma_{x}$ and invertible
on its support, and $\varphi^{-1}$ extends to a map
$\varphi^{-1}\colon\mathcal{E}|_{X\setminus\Gamma_{x}}\to\sigma^{\ast}\mathcal{E}|_{X\setminus\Gamma_{x}}$
with $\operatorname{coker}(\varphi^{-1})$ supported on $\infty$ and invertible
on its support.

We denote by $\operatorname{Sht}_{r}$ the stack of Drinfeld shtukas of rank
$r$.

It is well known that $\operatorname{Sht}_{r}$ is a Deligne-Mumford stack
locally of finite type over ${\mathbb{F}_{q}}$. It has a forgetful map
$\operatorname{Sht}_{r}\to X^{\prime}$ which is smooth of relative dimension
$(2r-2)$, see [Dri87, Proposition 3.2 and 3.3].

In the context of Drinfeld shtukas, the characteristic section $x$ is often
called the _zero_ of $\mathcal{E}$ while the second leg (that we fixed to be
$\infty$) is the _pole_ of $\underline{\mathcal{E}}$. By a slight abuse of
notation we say that $\underline{\mathcal{E}}$ is _in characteristic
$\mathfrak{p}$_ if its characteristic section factors through $0$.

######  Remark 2.3.

Note that once zero and pole of the shtuka do not intersect, we can glue
$\mathcal{E}$ and $\sigma^{*}\mathcal{E}$ along the isomorphism $\varphi$ over
$X_{S}\setminus(\Gamma_{x}\cup\Gamma_{x^{\prime}})$ and obtain a vector bundle
$\mathcal{E}^{\prime}$ together with maps

$\varphi^{\prime}\colon\mathcal{E}\hookrightarrow\mathcal{E}^{\prime}\hookleftarrow\sigma^{\ast}\mathcal{E}\colon\varphi$

of $\mathcal{O}_{X_{S}}$-modules that satisfy the analogous conditions on the
cokernels as in our definition of Drinfeld shtukas. This notion is used in the
original definition of Drinfeld shtukas in [Dri87] and does not require the
two legs of the shtuka to be disjoint. We denote by
$\operatorname{Sht}_{r,X^{2}}\to X^{2}$ the stack of Drinfeld shtukas in this
sense. Then
$\operatorname{Sht}_{r}=\operatorname{Sht}_{r,X^{2}}\times_{X^{2}}(\\{\infty\\}\times
X^{\prime})$.

For $n\in\mathbb{N}$ we denote by $D_{n}=n[0]\subseteq X$ the effective
Cartier divisor in $X$. Note that
$D_{n}=\operatorname{Spec}(\mathcal{O}_{0}/\mathfrak{p}^{n})$.

######  Definition 2.4.

A map $f\colon\underline{\mathcal{E}}_{1}\to\underline{\mathcal{E}}_{2}$ of
Drinfeld shtukas is an _isogeny_ if $f$ is injective and
$\operatorname{coker}(f)$ is finite locally free as $\mathcal{O}_{S}$-module.
Moreover, we say that $f$ is a $\mathfrak{p}^{n}$-isogeny, if the
$\mathcal{O}_{X_{S}}$-module structure on $\operatorname{coker}(f)$ factors
through $\mathcal{O}_{D_{n,s}}$, in other words, if $\operatorname{coker}(f)$
is $\mathfrak{p}^{n}$-torsion.

In order to give a criterion which $\mathcal{O}_{S}$-modules can arise as
cokernels of $\mathfrak{p}^{n}$-isogenies, we use the following notion of a
_$\mathfrak{p}^{n}$ -torsion shtuka_, which are an
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-linear analogue of the $\varphi$-sheaves
introduced by [Dri87].

######  Definition 2.5.

A _$\mathfrak{p}^{n}$ -torsion shtuka_ over $S$ is a pair
$\underline{\mathcal{F}}=(\mathcal{F},\varphi)$ consisting of a quasi-coherent
$\mathcal{O}_{D_{n,S}}$-module $\mathcal{F}$ which is finite locally free as
$\mathcal{O}_{S}$-module and an $\mathcal{O}_{D_{n,S}}$-module homomorphism
$\varphi\colon\sigma^{*}\mathcal{F}\to\mathcal{F}$. A map of
$\mathfrak{p}^{n}$-torsion shtukas is a map of the underlying
$\mathcal{O}_{D_{n,S}}$-modules compatible with $\varphi$. We say that a
$\mathfrak{p}^{n}$-torsion shtuka is _étale_ if $\varphi$ is an isomorphism.

In [HS19] Drinfeld’s $\varphi$-sheaves are also called finite shtukas. For our
purposes however, the $\mathcal{O}_{D_{n}}$-module structure is central.

To a rank $r$ Drinfeld shtuka
$\underline{\mathcal{E}}=(x,\mathcal{E},\varphi)$ over $S$ we associate its
$\mathfrak{p}^{n}$-torsion shtuka defined as the pullback of $\mathcal{E}$ to
the divisor $D_{n,S}$, which is more explicitly given by
$\underline{\mathcal{E}}|_{D_{n,S}}=(\mathcal{E}|_{D_{n,S}},\varphi|_{D_{n,S}})$.
Note that its underlying $\mathcal{O}_{S}$-module has rank ${nr}$. A second
important class of examples of $\mathfrak{p}^{n}$-torsion shtukas are
cokernels of $\mathfrak{p}^{n}$-isogenies of Drinfeld shtukas. Note that
$\underline{\mathcal{E}}|_{D_{n,S}}$ is the cokernel of the
$\mathfrak{p}^{rn}$-isogeny
$\underline{\mathcal{E}}(\mathfrak{p}^{n})\hookrightarrow\underline{\mathcal{E}}$,
where we denote by
$\underline{\mathcal{E}}(\mathfrak{p}^{n})=\underline{\mathcal{E}}\otimes\mathcal{O}(D_{n,S})$
the twist of $\underline{\mathcal{E}}$ by the divisor $D_{n}$.

###  2.2. Local shtukas.

We can associate to Drinfeld shtukas its local counterparts called _local
shtukas_ in the same way $p$-divisible groups are local ananlogues of abelian
varieties. Local shtukas are introduced as Dieudonné
${\mathbb{F}_{q}}\left\llbracket\varpi\right\rrbracket$-modules in [Har05] as
analogues of Dieudonné modules of $p$-divisible groups and are studied and
generalised for example by [HV11] and [AH14].

Let us denote by ${\mathbb{F}_{q}}\left\llbracket\zeta\right\rrbracket$ the
ring of formal power series in the formal variable $\zeta$ and by
$\mathcal{N}ilp_{{\mathbb{F}_{q}}\left\llbracket\zeta\right\rrbracket}$ the
category of schemes $S$ over
${\mathbb{F}_{q}}\left\llbracket\zeta\right\rrbracket$ such that $\zeta$ is
locally nilpotent in $S$. For a ring $R$, we denote by
$R\left\llbracket\varpi\right\rrbracket$ the ring of formal power series in
the formal variable $\varpi$ and by $R\left(\\!\left(\varpi\right)\\!\right)$
the ring of formal Laurent series in $\varpi$ on $S$. Note that for
$\operatorname{Spec}(R)\in\mathcal{N}ilp_{\mathbb{F}\left\llbracket\zeta\right\rrbracket}$,
we have
$R\left(\\!\left(\varpi\right)\\!\right)=R\left\llbracket\varpi\right\rrbracket\left[\frac{1}{\varpi-\zeta}\right]$.
We denote by $\sigma$ the endomorphism of
$R\left\llbracket\varpi\right\rrbracket$ (respectively
$R\left(\\!\left(\varpi\right)\\!\right)$) that acts as the identity on
$\varpi$ and as $b\mapsto b^{q}$ on $R$.

######  Definition 2.6.

Let
$S=\operatorname{Spec}(R)\in\mathcal{N}ilp_{{\mathbb{F}_{q}}\left\llbracket\zeta\right\rrbracket}$.
A _local shtuka_ $\underline{\mathcal{G}}=(\mathcal{G},\varphi)$ of rank $r$
over $S$ is a locally free sheaf of
$R\left(\\!\left(\varpi\right)\\!\right)$-modules $\mathcal{G}$ of rank $r$
together with an isomorphism
$\varphi\colon\sigma^{*}\mathcal{G}[\frac{1}{\varpi-\zeta}]\to\mathcal{G}[\frac{1}{\varpi-\zeta}]$
of $R\left(\\!\left(\varpi\right)\\!\right)$-modules. The local shtuka
$\underline{\mathcal{G}}$ is called _effective_ if $\varphi$ comes from a map
$\tilde{\varphi}\colon\sigma^{*}\mathcal{G}\hookrightarrow\mathcal{G}$ of
$R\left\llbracket\varpi\right\rrbracket$-modules and _étale_ if additionally
$\tilde{\varphi}$ is an isomorphism. A _quasi-isogeny_
$f\colon\underline{\mathcal{G}}\to\underline{\mathcal{G}^{\prime}}$ between
local shtukas is an isomorphism
$\mathcal{G}\left[\frac{1}{\varpi}\right]\tilde{\to}\mathcal{G}^{\prime}\left[\frac{1}{\varpi}\right]$
of the underlying $R\left(\\!\left(\varpi\right)\\!\right)$-modules, which is
compatible with $\varphi$ and $\varphi^{\prime}$.

We say a local shtuka $\underline{\mathcal{G}}=(\mathcal{G},\varphi)$ is
_bounded by $(1,0,\ldots,0)$_ if it is effective,
$\operatorname{coker}(\varphi)$ is locally free of rank 1 as an $R$-module and
$(\varpi-\zeta)$ annihilates $\operatorname{coker}(\varphi)$. Similarly, we
say $\underline{\mathcal{G}}$ is bounded by $(0,\ldots,0,-1)$ if
$\varphi^{-1}$ is bounded by $(1,0,\ldots,0)$ in the above sense. More
precisely, $\underline{\mathcal{G}}$ is bounded by $(0,\ldots,0,-1)$ if
$\varphi$ induces a map $\mathcal{G}\hookrightarrow\sigma^{*}\mathcal{G}$ with
a cokernel which is locally free of rank 1 as $R$-module and which is
annihilated by $(\varpi-\zeta)$.

######  Remark 2.7.

There are several ways to define bounds for local shtukas in general, cf.
[HV11, Definition 3.5 and Lemma 4.3.] and [AH14, Defintion 4.8.]. For the
Drinfeld case the bound in the sense of [AH14] is also given more explicitly
in [Bre18, Section 7.2.]. Note that the straightforward generalisation of our
definition above does not produce the correct notion for coweights
$(d,0,\ldots,0)$ with $d>1$ by [HS19, Example 8.3]. In particular [HV11,
Example 4.5] and [Zhu17, Example 2.1.8.] seem to be problematic.

The Newton stratification for local shtukas is defined in [HV11] as an
analogue of the Newton stratification for $F$-isocrystals in [RR96].

######  Definition 2.8.

The _Newton point_ of of a local shtuka $\underline{\mathcal{G}}$ of rank $r$
over an algebraically closed field $\ell$ is
$(u_{1},\ldots,u_{r})\in\mathbb{Q}^{r}$ with $u_{1}\geq\ldots\geq u_{r}$ and
the $u_{i}$ are the slopes associated to the corresponding isoshtuka
$\underline{\mathcal{G}}\left[\varpi^{-1}\right]$ by the Dieudonné-Manin
classification in the function field case [Lau96, Theorem 2.4.5].

We denote by $B(\operatorname{GL}_{r})$ the _Kottwitz set_ of isomorphism
classes of isoshtukas over an algebraically closed field $\ell$, in other
words, the set of $\sigma$-conjugacy classes of invertible $(r\times
r)$-matrices over $\ell\left(\\!\left(\varpi\right)\\!\right)$. The set
$B(\operatorname{GL}_{r})$ does not depend on the choice of $\ell$. Recall
that the Newton map $\nu_{\operatorname{GL}_{r}}\colon
B(\operatorname{GL}_{r})\to\mathbb{Q}^{r}$ is already injective (this fails
for general reductive groups). The Bruhat order on the space of cocharacters
$X_{\ast}(T)\otimes_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{Q}^{r}$ induces a
partial order on $B(\operatorname{GL}_{r})$. It is more explicitly given by

$(u_{1},\ldots,u_{r})\leq(u_{1}^{\prime},\ldots,u_{r}^{\prime})\qquad\text{if}\qquad\sum_{j=1}^{i}u_{j}\leq\sum_{j=1}^{i}u_{j}^{\prime}$

for all $1\leq i\leq r$ with equality in the case $i=r$. Moreover, for a
dominant cocharacter $\mu$, in other words,
$\mu=(\mu_{1},\ldots,\mu_{r})\in\mathbb{Z}^{r}$ with
$\mu_{1}\geq\ldots\geq\mu_{r}$, we denote by
$B(\operatorname{GL}_{r},\mu)=\\{[b]\in
B(\operatorname{GL}_{r})\colon\nu_{\operatorname{GL}_{r}}([b])\leq\mu\\}$.

For a local shtuka $\underline{\mathcal{G}}$ over a scheme
$S=\operatorname{Spec}(R)$ and a geometric point $s$ of $S$ we denote by
$[\underline{\mathcal{G}}_{s}]$ the associated point in
$B(\operatorname{GL}_{r})$ after pullback to $s$. Note that if
$\underline{\mathcal{G}}$ is bounded by $\mu=(1,0,\ldots,0)$, then
$[\underline{\mathcal{G}}_{s}]$ is contained in $B(\operatorname{GL}_{r},\mu)$
for all $s\in S$. The Newton point induces a stratification in the following
way.

######  Proposition 2.9 ([HV11, Theorem 7.3], compare also [RR96, Theorem
3.6]).

Let $S=\operatorname{Spec}(R)$ be an affine ${\mathbb{F}_{q}}$-scheme and
$\underline{\mathcal{G}}$ be a local shtuka over $S$ and $b\in
B(\operatorname{GL}_{r})$. Then the set $\\{s\in
S\colon[\underline{\mathcal{G}}_{s}]\leq b\\}$ is a Zariski-closed subset of
$S$. Furthermore, $\\{s\in S\colon[\underline{\mathcal{G}}_{s}]=b\\}$ is an
open subset of the former.

We denote by $S_{\leq b}$ the closed subscheme of $S$ given by the reduced
subscheme on $\\{s\in S\colon[\underline{\mathcal{G}}_{s}]\leq b\\}$ and
similarly $S_{b}$ the corresponding open subscheme of $S_{\leq b}$. Then
$S_{b}$ is a locally closed subscheme of $S$.

###  2.3. Global-to-local functor and a Serre-Tate theorem.

We explain how to associate local shtukas to global shtukas. We follow the
general construction of [AH14]. This is a generalisation of the construction
of [BH11, Section 8] for abelian sheaves and Anderson motives.

We follow the notation of [AH14, Section 5.2.]. Let $y$ be a closed point of
$X$, which we assume for simplicity to be defined over ${\mathbb{F}_{q}}$.
This is the only case we use later. For the general construction we refer to
[AH14]. Let $\mathcal{O}_{y}$ be the completed local ring at $y$. The choice
of a uniformiser $\varpi_{y}$ at $y$ defines an isomorphism
$\mathcal{O}_{y}\cong{\mathbb{F}_{q}}\left\llbracket\varpi_{y}\right\rrbracket$.
Let $x\in X(\operatorname{Spec}(R))$ be a section that factors through
$\operatorname{Spf}(\mathcal{O}_{y})$. Then $\varpi_{y}$ is nilpotent in $R$.
Let $\mathbb{D}_{y}=\operatorname{Spec}(\mathcal{O}_{y})$ and
$\hat{\mathbb{D}}_{y}=\operatorname{Spf}(\mathcal{O}_{y})$. We denote by
$\hat{\mathbb{D}}_{y,R}$ the $\varpi_{y}$-adic completion of
$\mathbb{D}_{y}\times_{\mathbb{F}_{q}}\operatorname{Spec}(R)$.

By [AH14, Lemma 5.3.], the section $x$ induces a canonical isomorphism of the
formal completion of $X_{R}$ along the graph $\Gamma_{x}$ of $x$ with
$\hat{\mathbb{D}}_{y,R}$. By construction, the formal completion along
$\Gamma_{x}$ has structure sheaf
$R\left\llbracket\varpi_{y}-\zeta\right\rrbracket$, where $\zeta$ is the image
of $\varpi_{y}$ in $R$. As $\zeta$ is nilpotent in $R$,
$R\left\llbracket\varpi_{y}-\zeta\right\rrbracket$ and
$R\left\llbracket\varpi_{y}\right\rrbracket$ are isomorphic.

We fix a pair $\underline{y}=(y_{1},y_{2})$ of (${\mathbb{F}_{q}}$-rational)
closed points of $X$ with $y_{1}\neq y_{2}$. Let $\mathcal{O}_{\underline{y}}$
be the completion of the local ring of $X^{2}$ at $\underline{y}$. We denote
by
$\operatorname{Sht}_{r}^{\underline{y}}=\operatorname{Sht}_{r,X^{2}}\times_{X^{2}}\operatorname{Spf}(\mathcal{O}_{\underline{y}})$
the substack of $\operatorname{Sht}_{r,X^{2}}$ such that the legs factor
through $\operatorname{Spf}(\mathcal{O}_{y_{1}})$ and
$\operatorname{Spf}(\mathcal{O}_{y_{2}})$, respectively. In particular, for
points of $\operatorname{Sht}_{r}^{\underline{y}}$ the graphs of its legs are
disjoint. Let
$\underline{\mathcal{E}}=(x^{\prime},x,\mathcal{E},\varphi)\in\operatorname{Sht}_{r}^{\underline{y}}(R)$.
The local shtuka associated to $\underline{\mathcal{E}}$ at $y_{i}$ is then
its pullback to $\hat{\mathbb{D}}_{y_{i},R}$ for $i=1,2$.

######  Definition 2.10.

The _global-to-local functor_ that associates to a global shtuka
$\operatorname{Sht}_{r}^{\underline{y}}$ a pair of local shtukas (at $y_{1}$
and $y_{2}$, respectively) is defined for
$\underline{\mathcal{E}}\in\operatorname{Sht}_{r}^{\underline{y}}(R)$ as

$\widehat{\underline{\mathcal{E}}_{y_{i}}}:=(\mathcal{E}|_{\hat{\mathbb{D}}_{y_{i},R}},\tilde{\varphi})\qquad\text{and}\qquad\widehat{\underline{\mathcal{E}}_{\underline{y}}}=\prod_{i=1,2}\widehat{\underline{\mathcal{E}}_{y_{i}}}.$

Then, $\widehat{\underline{\mathcal{E}}_{y_{i}}}$ is called the _local shtuka_
of $\underline{\mathcal{E}}$ at $y_{i}$.

By definition of $\operatorname{Sht}_{r,X^{2}}$, the local shtuka at $y_{2}$
is bounded by $(1,0,\ldots,0)$ as the condition that
$(\varpi_{y_{2}}-\zeta_{y_{2}})$ annihilates the cokernel in the local case
directly corresponds to the fact that the cokernel is supported on the graph
in the global case. Similarly, the local shtuka at $y_{1}$ is bounded by
$(0,\ldots,0,-1)$.

The global-to-local functor also gives rise to a Serre-Tate theorem relating
the deformation theory of global shtukas with the deformation theory of their
associated local shtukas. Let
$S=\operatorname{Spec}(R)\in\mathcal{N}ilp_{\mathcal{O}_{\underline{y}}}$ and
let $i\colon\overline{S}=\operatorname{Spec}(R/I)\hookrightarrow S$ be a
closed subscheme defined by a nilpotent ideals $I$. Let
$\underline{\bar{\mathcal{E}}}\in\operatorname{Sht}_{r}^{\underline{y}}(\overline{S})$.
The category $\operatorname{Def}_{\underline{\bar{\mathcal{E}}}}(S)$ is the
category of deformations of $\underline{\mathcal{E}}$ to $S$, i.e. the
category of pairs $(\underline{\mathcal{E}},f\colon
i^{*}\underline{\mathcal{E}}\tilde{\rightarrow}\underline{\bar{\mathcal{E}}})$
where $\mathcal{E}\in\operatorname{Sht}_{r}^{\underline{y}}(S)$ and $f$ is an
isomorphism of shtukas over $\overline{S}$. Similarly, for a local shtuka
$\underline{\bar{\mathcal{G}}}$ bounded by $(1,0,\ldots,0)$ we define
$\operatorname{Def}^{\leq(1,0,\ldots,0)}_{\underline{\bar{\mathcal{G}}}}(S)$
as the category of deformations of $\underline{\bar{\mathcal{G}}}$ to $S$,
i.e. the category of pairs $(\underline{\mathcal{G}},g\colon
i^{*}\underline{\mathcal{G}}\tilde{\rightarrow}\underline{\bar{\mathcal{G}}})$
where $\mathcal{G}$ is a local shtuka on $S$ bounded by $(1,0,\ldots 0)$ and
$g$ is an isomorphism of local shtukas over $\overline{S}$. Similarly, we
define deformations of local shtukas bounded by $(0,\ldots,0,-1)$.

######  Proposition 2.11 (Serre-Tate Theorem for shutkas, [AH14, Theorem
5.10.]).

Let
$\underline{\bar{\mathcal{E}}}\in\operatorname{Sht}_{r}^{\underline{y}}(\overline{S})$.
Then the functor

$\widehat{(-)_{\underline{y}}}\colon\operatorname{Def}_{\underline{\bar{\mathcal{E}}}}(S)\to\operatorname{Def}^{\leq(0,\ldots,0,-1)}_{\widehat{\underline{\overline{\mathcal{E}}}_{y_{1}}}}(S)\times\operatorname{Def}^{\leq(1,0,\ldots,0)}_{\widehat{\underline{\overline{\mathcal{E}}}_{y_{2}}}}(S),\qquad(\underline{\mathcal{E}},f)\mapsto\prod_{i=1,2}(\widehat{\underline{\mathcal{E}}_{y_{i}}},\widehat{f_{y_{i}}})$

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

###### Proof.

As before, this follows directly from the unbounded case in [AH14, Theorem
5.10.] as a global $\operatorname{GL}_{r}$-shtuka is bounded by
$(0,\ldots,0,-1),(1,0,\ldots,0)$ if and only if the associated local shtukas
are. ∎

The Newton stratification induces also a stratification on the special fibre
of the stack of (global) Drinfeld shtukas in the sense of [Bre18, Section 4].
We continue to restrict ourselves to the case of Drinfeld shtukas with one leg
fixed at $\infty$ as this is the only case of interest to us in the following.
The following has obvious analogues also for $\operatorname{Sht}_{r,X^{2}}$.
For a closed point $y$ of $X$ with residue field $F_{y}$ different from
$\infty$ we set
$\operatorname{Sht}_{r,\mathbb{F}_{y}}=\operatorname{Sht}_{r}\times_{X^{\prime},y}\mathbb{F}_{y}$.

######  Definition 2.12 (compare [Neu16, Proposition 4.1.4.], [Bre18,
Definition 4.12.] for the general definition).

Le $b_{\infty}\in B(\operatorname{GL}_{r,K_{\infty}},(0,\ldots,0,-1))$. The
locally closed and reduced substack of $\operatorname{Sht}_{r}$ where the
associated local shtuka at $\infty$ has Newton point $b_{\infty}$ is called
the _Newton stratum_ associated to $b_{\infty}$ and is denoted by
$\operatorname{Sht}_{r,b_{\infty}}$. In a similar fashion, for a closed point
$y$ of $X$ and a pair

$\underline{b}_{\underline{y}}=(b_{\infty},b_{y})\in
B(\operatorname{GL}_{r,K_{\infty}},(0,\ldots,0,-1))\times
B(\operatorname{GL}_{r,K_{y}},(1,0,\ldots,0))$

the locally closed and reduced substack of
$\operatorname{Sht}_{r,\mathbb{F}_{y}}$ where the associated local shtuka at
$\infty$ has Newton point $b_{\infty}$ and the associated local shtuka at $y$
has Newton point $b_{y}$ is denoted by
$\operatorname{Sht}_{r,\underline{b}_{\underline{y}}}$.

###  2.4. Isogenies of Drinfeld shtukas.

We study isogenies of Drinfeld shtukas in more detail. We consider the
following moduli problem of Drinfeld shtukas with chains of isogenies.

######  Definition 2.13.

Let $m,r_{1},\ldots,r_{m}\geq 1$ be positive integers such that
$\sum_{j=1}^{m}r_{j}\leq r$. A _chain of $\mathfrak{p}^{n}$-isogenies of type
$(r_{1},\ldots,r_{m})$_ on a Drinfeld shtuka $\underline{\mathcal{E}}$ over a
scheme $S$ is a flag of quotients of $\mathfrak{p}^{n}$-torsion shtukas

$\underline{\mathcal{E}}|_{D_{n,S}}=\underline{\mathcal{F}}_{m+1}\twoheadrightarrow\underline{\mathcal{F}}_{m}\twoheadrightarrow\ldots\twoheadrightarrow\underline{\mathcal{F}}_{1}\twoheadrightarrow
0$

over $S$ such that $\mathcal{F}_{i}$ has rank $n\cdot(r_{1}+\ldots+r_{i})$ as
$\mathcal{O}_{S}$-module. We denote the stack of Drinfeld shtukas with chains
of $\mathfrak{p}^{n}$-isogenies of type $(r_{1},\ldots,r_{m})$ by
$\operatorname{Sht}_{r,(r_{1},\ldots,r_{m})-\mathfrak{p}^{n}\text{-chain}}$.

We show below that a chain of $\mathfrak{p}^{n}$-isogenies of type
$(r_{1},\ldots,r_{m})$ in the sense of the definition is the same as giving a
chain of actual $\mathfrak{p}^{n}$-isogenies of Drinfeld shtukas

$\underline{\mathcal{E}}(\mathfrak{p}^{n})=\underline{\mathcal{E}}_{m+1}\xrightarrow{f_{m+1}}\underline{\mathcal{E}}_{m}\xrightarrow{f_{m}}\ldots\xrightarrow{f_{2}}\underline{\mathcal{E}}_{1}\xrightarrow{f_{1}}\underline{\mathcal{E}}_{0}=\underline{\mathcal{E}}$

such that the composition $f_{m+1}\circ\ldots\circ f_{1}$ is the inclusion
$\underline{\mathcal{E}}(\mathfrak{p}^{n})\to\underline{\mathcal{E}}$.

######  Proposition 2.14.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ be a Drinfeld
shtuka. Every quotient
$\underline{\mathcal{E}}|_{D_{n,S}}\twoheadrightarrow\underline{\mathcal{F}}$,
where $\underline{\mathcal{F}}$ is a $\mathfrak{p}^{n}$-torsion shtuka, is the
cokernel of a $\mathfrak{p}^{n}$-isogeny.

Moreover, for two $\mathfrak{p}^{n}$-isogenies
$f_{1}\colon\underline{\mathcal{E}}_{1}\hookrightarrow\underline{\mathcal{E}}$
and
$f_{2}\colon\underline{\mathcal{E}}_{2}\hookrightarrow\underline{\mathcal{E}}$
such that the cokernels factor as successive quotients
$\underline{\mathcal{E}}|_{D_{n,S}}\twoheadrightarrow\operatorname{coker}(f_{1})\twoheadrightarrow\operatorname{coker}(f_{2})$,
there exists a unique $\mathfrak{p}^{n}$-isogeny
$f\colon\underline{\mathcal{E}_{1}}\hookrightarrow\underline{\mathcal{E}_{2}}$
such that $f_{1}=f_{2}\circ f$ and
$\operatorname{coker}(f)\cong\ker(\operatorname{coker}(f_{1})\twoheadrightarrow\operatorname{coker}(f_{2}))$.

###### Proof.

Let $\underline{\mathcal{F}}$ be a $\mathfrak{p}^{n}$-torsion shtuka as in the
statement of the proposition. Let us denote by
$\mathcal{E}^{\prime}=\ker(\mathcal{E}\twoheadrightarrow\mathcal{F})$. As a
first step, we want to show that $\mathcal{E}^{\prime}$ is finite locally free
of rank $r$ on $X_{S}$. In order to do so, we may by reduction to the
universal case assume that $S=\operatorname{Spec}R$ is affine and noetherian.
As $\mathcal{E}^{\prime}\hookrightarrow\mathcal{E}$ is an isomorphism away
from $\mathfrak{p}$, it then suffices by fpqc-descent to show that the
completion at $0$ is finite locally free of rank $r$. As we assumed $R$ to be
noetherian, completion at $0$ is exact. The completion
$\widehat{\mathcal{E}^{\prime}_{0}}=\mathcal{E}^{\prime}\otimes\mathcal{O}_{0}$
of $\mathcal{E}^{\prime}$ at $0$ is hence given by the kernel of
$\widehat{\mathcal{E}_{0}}\twoheadrightarrow\underline{\mathcal{F}}$. The
assertion now follows from [Gen96, Lemma 2.2.8].

By the right exactness of the tensor product, the cokernel of the induced map
$\sigma^{*}\mathcal{E}^{\prime}\to\sigma^{*}\mathcal{E}$ is given by
$\sigma^{*}\mathcal{F}$. By [Har19, Lemma 2.2], the map
$\sigma^{*}\mathcal{E}^{\prime}\to\sigma^{*}\mathcal{E}$ is thus injective,
and
$\sigma^{*}\mathcal{E}^{\prime}=\ker(\sigma^{*}\mathcal{E}\twoheadrightarrow\sigma^{*}\mathcal{F})$.
In particular, we obtain an induced map
$\tilde{\varphi}\colon\sigma^{*}\mathcal{E}^{\prime}\dashrightarrow\mathcal{E}^{\prime}$
defined away from $\Gamma_{x}$ and $\Gamma_{\infty}$. As the map
$\mathcal{E}^{\prime}\to\mathcal{E}$ is an isomorphism away from $0$, locally
around $\infty$ we obtain a map
$\mathcal{E}^{\prime}\to\sigma^{*}\mathcal{E}^{\prime}$ with cokernel
supported at $\infty$ and of rank 1 as $\mathcal{O}_{S}$-module. It follows
also that
$\varphi^{\prime}|_{X^{\prime}_{S}}\colon\sigma^{*}\mathcal{E}^{\prime}|_{X^{\prime}_{S}}\to\mathcal{E}^{\prime}|_{X^{\prime}_{S}}$
is a well-defined and injective map (as $\varphi$ is). Note that
$(\mathcal{E}^{\prime}|_{{X^{\prime}_{S}}},\varphi^{\prime}|_{{X^{\prime}_{S}}})$
is the associated $A$-motive in the sense of [Har19] where
$A=\Gamma(X\setminus\\{\infty\\},\mathcal{O}_{X})$. By [Har19, Proposition
2.3], the $A$-motive
$(\mathcal{E}^{\prime}|_{X^{\prime}_{S}},\varphi^{\prime}|_{X^{\prime}_{S}})$
is effective, this means that
$\operatorname{coker}(\varphi^{\prime}|_{X^{\prime}_{S}})$ is annihilated by
$\mathcal{J}^{n}$ for some positive integer $n$, where $\mathcal{J}$ is the
quasi-coherent sheaf of ideals defining $\Gamma_{x}\subseteq X_{S}$. Using
[Har19, Proposition 5.8] we obtain that
$\operatorname{coker}(\varphi^{\prime}|_{X^{\prime}_{S}})$ has rank 1 as
$\mathcal{O}_{S}$-module. Thus,
$\operatorname{coker}(\varphi^{\prime}|_{X^{\prime}_{S}})$ is already
annihilated by $\mathcal{J}$, which means that
$(\mathcal{E}^{\prime},\varphi^{\prime})$ defines a point of
$\operatorname{Sht}_{r}(S)$.

For the second part let
$f_{1}\colon\underline{\mathcal{E}}_{1}\hookrightarrow\underline{\mathcal{E}}$
and
$f_{2}\colon\underline{\mathcal{E}}_{2}\hookrightarrow\underline{\mathcal{E}}$
be two $\mathfrak{p}^{n}$-isogenies as in the assertion. It follows
essentially by assumption that there is a unique injective homomorphism of
shtukas $f\colon\underline{\mathcal{E}}_{2}\to\underline{\mathcal{E}}_{1}$
such that $f_{2}=f_{1}\circ f$. It remains to check that $f$ is a
$\mathfrak{p}^{n}$-isogeny. We have the short exact sequence of $R$-modules

$0\to\mathcal{E}_{1}/f(\mathcal{E}_{2})=\operatorname{coker}(f)\hookrightarrow\mathcal{E}/f_{2}(\mathcal{E}_{2})=\operatorname{coker}(f_{2})\twoheadrightarrow\mathcal{E}/f_{1}(\mathcal{E}_{1})=\operatorname{coker}(f_{1})\to
0,$

where the first map is $f_{1}$ and the second map is well-defined by
assumption. As both $f_{1}$ and $f_{2}$ are isogenies, their cokernels are
finite locally free $R$-modules. It follows that $\operatorname{coker}(f)$ is
finite locally free as well, and thus $f$ is an isogeny. That it is a
$\mathfrak{p}^{n}$-isogeny is also clear. ∎

######  Remark 2.15.

  1. (1)

Note that in the proof we really used that $0$ is ${\mathbb{F}_{q}}$-rational.
It would be desirable to have an analogous statement in general.

  2. (2)

Using the comparison [Har19, Theorem 5.8] with isogenies of Drinfeld modules,
we get as immediate corollaries that any finite locally free closed submodule
scheme with strict ${\mathbb{F}_{q}}$-action of the $\mathfrak{p}^{n}$-torsion
of a Drinfeld module is the kernel of an isogeny and a factorisation property
as in the second part of the proposition. Both of these facts seem to be only
proven in the literature when the base is a field in [Leh09, 2, Lemma 3.1 and
Lemma 3.2].

  3. (3)

This also shows that giving a point of
$\operatorname{Sht}_{r,(r_{1},\ldots,r_{m})-\mathfrak{p}^{n}\text{-chain}}$ is
the same as giving a chain of actual $\mathfrak{p}^{n}$-isogenies of Drinfeld
shtukas

$\underline{\mathcal{E}}(\mathfrak{p}^{n})=\underline{\mathcal{E}}_{m+1}\xrightarrow{f_{m+1}}\underline{\mathcal{E}}_{m}\xrightarrow{f_{m}}\ldots\xrightarrow{f_{2}}\underline{\mathcal{E}}_{1}\xrightarrow{f_{1}}\underline{\mathcal{E}}_{0}=\underline{\mathcal{E}}$

such that $\operatorname{coker}(f_{i})$ has rank $n\cdot r_{i}$ and such that
the composition $f_{m+1}\circ\ldots\circ f_{1}$ is the inclusion
$\underline{\mathcal{E}}(\mathfrak{p}^{n})\to\underline{\mathcal{E}}$.

###  2.5. Naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures and shtukas
for Bruhat-Tits group schemes.

We introduce naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures on Drinfeld
shtukas and explain how to interpret them as shtukas for certain Bruhat-Tits
group schemes. These naive level structures seem inadequate in the non-
parahoric case (that means when $n>1$), as their moduli spaces are missing
points in the fibre over $0$. In other words, the level map to
$\operatorname{Sht}_{r}$ is not proper, compare Remark 2.20 below. The
interpretation of naive level structures in terms of Bruhat-Tits group schemes
allows us to give a candidate for a compactification of the level map: We can
take the closure of the stack of shtukas with naive level in the product of
the stacks of Drinfeld shtukas with corresponding parahoric level.

######  Definition 2.16.

A _naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structure_ on a Drinfeld shtuka
$\underline{\mathcal{E}}=(\mathcal{E},\varphi)\in\operatorname{Sht}_{r}(S)$ of
rank $r$ is a flag of quotients of $\mathfrak{p}^{n}$-torsion finite shtukas

$\underline{\mathcal{E}}|_{D_{n,S}}=\underline{\mathcal{L}}_{r}\twoheadrightarrow\underline{\mathcal{L}}_{r-1}\twoheadrightarrow\ldots\twoheadrightarrow\underline{\mathcal{L}}_{1}\twoheadrightarrow\underline{\mathcal{L}}_{0}=0$

such that $\mathcal{L}_{i}$ is finite locally free of rank $i$ as
$\mathcal{O}_{D_{n,S}}$-module (and hence of rank $in$ as
$\mathcal{O}_{S}$-module).

######  Remark 2.17.

By Proposition 2.14, a naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structure is
equivalently given as a chain of isogenies

$\underline{\mathcal{E}}(\mathfrak{p}^{n})=\underline{\mathcal{E}}_{r}\stackrel{{\scriptstyle
f_{r}}}{{\rightarrow}}\underline{\mathcal{E}}_{r-1}\stackrel{{\scriptstyle
f_{r-1}}}{{\rightarrow}}\underline{\mathcal{E}}_{r-2}\rightarrow\ldots\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}\underline{\mathcal{E}}_{0}=\underline{\mathcal{E}}$

such that $\operatorname{coker}(f_{i})$ is finite locally free of rank 1 as
$\mathcal{O}_{D_{n,S}}$-module for all $1\leq i\leq r$.

We interpret naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures on Drinfeld
shtukas as shtukas for certain Bruhat-Tits group schemes in the following
sense.

######  Definition 2.18.

A _Bruhat-Tits group scheme_ on $X$ is a smooth affine group scheme $G\to X$
such that

  1. (1)

all fibres of $G$ are connected,

  2. (2)

the generic fibre of $G$ is a reductive group over $K$ and

  3. (3)

for every closed point $x$ of $X$ the base change
$G_{\mathcal{O}_{x}}=G\times_{X}\operatorname{Spec}(\mathcal{O}_{x})$ is a
Bruhat-Tits group scheme in the sense that there is a non-empty bounded subset
$\Omega$ in some appartment in the Bruhat-Tits building of $G_{K_{x}}$ such
that $G(\mathcal{O}_{x})\subseteq G(K_{x})$ is the connected fixator of
$\Omega$ in the sense of [BT84, (4.6.26)].

A Bruhat-Tits group scheme is _parahoric_ , if the subgroups
$G(\mathcal{O}_{x})\subseteq G(K_{x})$ in (3) are parahoric for all places of
$X$.

######  Remark 2.19.

Of particular relevance to our situation is the case where $\Omega$ is the
stabiliser of a regular $(r-1)$-simplex $\Omega$ in the standard appartment of
the (reduced) Bruhat-Tits building of $\operatorname{GL}_{r,K_{0}}$ with side-
length $n$. We denote by $\operatorname{GL}_{r,\Omega}\to X$ the corresponding
Bruhat-Tits group scheme that is isomorphic to $\operatorname{GL}_{r}$ away
from $0$ and such that
$\operatorname{GL}_{r}(\mathcal{O}_{0})\subseteq\operatorname{GL}_{r}(K_{0})$
is the connected stabiliser of $\Omega$.

We can more explicitly describe this subgroup by
$\operatorname{GL}_{r,\Omega}(\mathcal{O}_{0})=\\{M\in\operatorname{GL}_{r}(\mathcal{O}_{0})\colon
M\mod\mathfrak{p}^{n}\in B(\mathcal{O}_{0}/\mathfrak{p}^{n})\\}$. By [MRR20,
Lemma 3.1 and Theorem 3.2], the group scheme $\operatorname{GL}_{r,\Omega}$
can thus also be interpreted as the Néron blowup of $\operatorname{GL}_{r}$ in
its subgroup $B$ of upper triangular matrices along the divisor $D_{n}$ in the
sense of [MRR20, Section 3.1].

By [MRR20, Theorem 4.8], giving a $\operatorname{GL}_{r,\Omega}$-torsor on $X$
is equivalent to giving a $\operatorname{GL}_{r}$-torsor $\mathcal{E}$ on $X$
together with a reduction of $\mathcal{E}$ to an $B$-torsor over the divisor
$D_{n}$ of $X$. More explicitly, a point of
$\operatorname{Bun}_{\operatorname{GL}_{r,\Omega}}(S)$ is given by a rank $r$
vector bundle $\mathcal{E}$ on $X_{S}$ together a flag of quotients of
$\mathcal{E}|_{D_{n,S}}$ as in the definition of naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures. In this sense, a naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure on a Drinfeld shtuka
$\underline{\mathcal{E}}$ defines a $(B,D_{n})$-level structure on
$\underline{\mathcal{E}}$ in the sense of [MRR20, Section 4.2.2].

A $\operatorname{GL}_{r,\Omega}$-shtuka is called _bounded by
$\underline{\mu}=((0,\ldots,0,-1),(1,0,\ldots,0))$_ if its underlying
$\operatorname{GL}_{r}$-shtuka $(x,\mathcal{E},\varphi)$ is bounded by
$(0,\ldots,0,-1),(1,0,\ldots,0)$, and if the flag of quotients given by the
$(B,D_{n})$-structure on the underlying vector bundle $\mathcal{E}$ is
$\varphi$-stable. In other words, the $\operatorname{GL}_{r,\Omega}$-shtukas
bounded by $\underline{\mu}$ in this sense are exactly the Drinfeld shtukas
with naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structures in the sense above.
We denote this stack of bounded $\operatorname{GL}_{r,\Omega}$-shtukas (or
equivalently the stack of Drinfeld shtukas with naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures) by
$\operatorname{Sht}_{r,\Omega}$.

For a facet $\mathfrak{f}$ in the Bruhat-Tits building of
$\operatorname{GL}_{r,K_{0}}$ we write $\mathfrak{f}\prec\Omega$ if
$\mathfrak{f}$ is contained in the closure of $\Omega$. In a similar fashion
to the construction above, for such a facet $\mathfrak{f}\prec\Omega$ we write
$\operatorname{GL}_{r,\mathfrak{f}}$ for the corresponding Bruhat-Tits group
scheme and $\operatorname{Sht}_{r,\mathfrak{f}}$ for the stack of
$\operatorname{GL}_{r,\mathfrak{f}}$-shtukas bounded by $\underline{\mu}$ in
the sense above. By Bruhat-Tits theory, for any facet $\mathfrak{f}$ contained
in $\Omega$ there is a map of group schemes
$\operatorname{GL}_{r,\Omega}\to\operatorname{GL}_{r,\mathfrak{f}}$ that is
the identity away from $0$. By [Bre18, Corollary 3.16], we get maps
$\operatorname{Sht}_{r,\Omega}\to\operatorname{Sht}_{r,\mathfrak{f}}$.

In particular, in the case $n=1$ the set $\Omega$ is just given by the base
alcove (corresponding to the standard Iwahori subgroup of matrices that upper
triangular mod $\mathfrak{p}$). Hence, for an alcove $\mathfrak{f}$ its
corresponding moduli space of shtukas $\operatorname{Sht}_{r,\mathfrak{f}}$
parametrises chains of isogenies of Drinfeld shtukas as in Definition 2.16.
For a facet $\mathfrak{f}^{\prime}$ of the alcove $\mathfrak{f}$ the map
$\operatorname{Sht}_{r,\mathfrak{f}}\to\operatorname{Sht}_{r,\mathfrak{f}^{\prime}}$
is then given by projection to some subchain of isogenies, depending on the
position of $\mathfrak{f}^{\prime}$. In particular, when
$\mathfrak{f}^{\prime}$ is a vertex,
$\operatorname{Sht}_{r,\mathfrak{f}^{\prime}}$ parametrises single Drinfeld
shtukas and when $\mathfrak{f}^{\prime}$ is an edge,
$\operatorname{Sht}_{r,\mathfrak{f}^{\prime}}$ parametrises pairs of Drinfeld
shtukas with a certain isogeny between them.

In order to describe the maps
$\operatorname{Sht}_{r,\Omega}\to\operatorname{Sht}_{r,\mathfrak{f}}$ for
facets $\mathfrak{f}\prec\Omega$ more explicitly, we label the vertices in
$\Omega$ by tuples $\underline{m}=(m_{1},\ldots,m_{r-1})$ such that $n\geq
m_{1}\geq\ldots\geq m_{r-1}\geq 0$, edges are between vertices $\underline{m}$
and $\underline{m}^{\prime}$ if and only if $0\leq m_{i}-m_{i}^{\prime}\leq 1$
for all $i$ or $0\leq m_{i}-m_{i}^{\prime}\leq 1$ for all $i$. The base alcove
corresponds to the simplex defined by the vertices
$(0,\ldots,0),(1,0,\ldots,0),(1,1,0,\ldots,0),\ldots,(1,1,\ldots,1)$. The
vertex $(0,\ldots,0)$ corresponds to the constant group scheme
$\operatorname{GL}_{r}$.

Note that every facet $\mathfrak{f}\prec\Omega$ has a unique base point
$\underline{m}$ (such that $m_{i}\leq x_{i}$ for all $i$ and points
$\underline{x}\in\mathfrak{f}$), and an orientation that we encode by an
element $\tau\in\operatorname{Sym}_{r-1}$ of the symmetric group on $(r-1)$
elements. The orientation $\tau$ is chosen such that the vertices of
$\mathfrak{f}$ are given by $\underline{m}+\tau(\mathbf{1}_{r-1}^{(i)})$ for
$0\leq i\leq r-1$, where
$\mathbf{1}_{r-1}^{(i)}=(1,\ldots,1,0,\ldots,0)\in\mathbb{Z}^{r-1}$ has
exactly $i$ many entries equal to 1. For a given pair $(\underline{m},\tau)$
there clearly exists a unique alcove $\mathfrak{f}_{\underline{m},\tau}$ in
the standard apartment of the Bruhat-Tits building with base point
$\underline{m}$ and orientation $\tau$.

Starting from a $\operatorname{GL}_{r,\Omega}$-shtuka
$(\underline{\mathcal{E}},(\underline{\mathcal{L}}_{i}))\in\operatorname{Sht}_{r,\Omega}(S)$,
we construct a Drinfeld shtuka $\underline{\mathcal{E}}_{\underline{m}}$ for a
vertex $\underline{m}\prec\Omega$ as follows. Assume that
$S=\operatorname{Spec}(R)$ is affine and that all $\mathcal{L}_{i}$ are finite
free as $R[\varpi]/(\varpi^{n})$-modules. In this case, we can choose a basis
$(e_{1},\ldots,e_{r-1})$ of
$\mathcal{L}_{r-1}=\left(R[\varpi]/(\varpi^{n})\right)^{r-1}$ such that
$(e_{1},\ldots,e_{i})$ is a basis for $\mathcal{L}_{i}$ for all $1\leq i\leq
r-1$. We consider the quotient

$\mathcal{L}_{r-1}\twoheadrightarrow\mathcal{L}_{\underline{m}}:=R[\varpi]/(\varpi^{m_{1}})e_{1}\oplus\ldots\oplus
R[\varpi]/(\varpi^{m_{r-1}})e_{r-1}.$

As all the $\mathcal{L}_{i}$ are $\varphi$-stable quotients of
$\mathcal{L}_{r-1}$, the matrix representation of $\varphi$ with respect to
$(e_{1},\ldots,e_{r-1})$ is upper-triangular. This shows that also
$\mathcal{L}_{\underline{m}}$ is $\varphi$-stable as $m_{1}\geq\ldots\geq
m_{r-1}$ by assumption. By a similar argument, $\mathcal{L}_{\underline{m}}$
does not depend on the choice of basis (any base change matrix is again upper-
triangular). Thus, we can glue to obtain a $\varphi$-stable quotient
$\mathcal{L}_{\underline{m}}$ also in the general case. We then associate to
the vertex $\underline{m}$ the Drinfeld shtuka corresponding to the kernel
$\underline{\mathcal{E}}_{\underline{m}}=\ker(\underline{\mathcal{E}}\twoheadrightarrow\underline{\mathcal{L}}_{\underline{m}})$
by Proposition 2.14. Moreover, by the second part of Proposition 2.14, there
are also canonical isogenies associated to the edges in the Bruhat-Tits
building.

Using this construction, for an alcove
$\mathfrak{f}_{\underline{m},\tau}\prec\Omega$ the level map
$\operatorname{Sht}_{r,\Omega}\to\operatorname{Sht}_{r,\mathfrak{f}_{\underline{m}}}$
associates to $(\underline{\mathcal{E}},(\underline{\mathcal{L}}_{i}))$ the
chain of isogenies

$\underline{\mathcal{E}}_{\underline{m}}(\mathfrak{p}^{n})\hookrightarrow\underline{\mathcal{E}}_{\underline{m}+\tau(\mathbf{r-1}^{(1)}_{r-1})}\hookrightarrow\ldots\hookrightarrow\underline{\mathcal{E}}_{\underline{m}+\tau(\mathbf{1}^{(1)}_{r-1})}\hookrightarrow\underline{\mathcal{E}}_{\underline{m}}.$

This means that the induced map
$\operatorname{Sht}_{r,\Omega}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
associates to a Drinfeld shtuka with naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure a diagram
$(\underline{\mathcal{E}}_{\underline{m}})_{\underline{m}}$ with the canonical
isogenies as constructed above.

######  Remark 2.20.

For parahoric level (in our case that means $n\leq 1$) [Bre18] shows that the
level maps are proper and surjective. An explicit calculation for deeper level
(that is for $n>1$) shows that this is false already in the
$\operatorname{GL}_{2}$-case over $X=\mathbb{P}^{1}$ in general. Namely, we
study the fibre over $0$ using the local model of [AH19]. An explicit
calculation in the local model shows that for $n=1$ we get the familiar local
picture of two copies of $\mathbb{P}^{1}$ intersecting transversally at
supersingular points.

However, for any $n>1$ the special fibres of the corresponding local models
only contain two copies of $\mathbb{A}^{1}$ that do not intersect. This means
in particular that $\operatorname{Sht}_{r,\Omega}$ is missing the
supersingular points in the special fibre. Moreover, from the comparison with
the modular curve, we might expect to find $n+1$-components two of which are
reduced by [KM85, Theorem 13.4.7]. The two components we see using the naive
level structure correspond to the two reduced components, but we do not get
the non-reduced ones.

It turns out that requiring the quotients in the definition of naive level
structures to be locally free as $\mathcal{O}_{D_{n,S}}$-modules is too
restrictive and we rather should allow in the special fibre also degenerations
to certain $\mathfrak{p}^{n}$-torsion finite shtukas which are not locally
free as $\mathcal{O}_{D_{n,S}}$-modules.

The goal of this paper is to explain one way to remedy this. We show that we
can explicitly describe the schematic image of the map
$\operatorname{Sht}_{r,\Omega}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
in terms of _Drinfeld level structures_ and that this provides a natural
compactification of the level map.

##  3\. Group schemes attached to Drinfeld shtukas

In order to define Drinfeld level structures for Drinfeld shtukas, we explain
how to construct a (finite locally free
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module) scheme of
$\mathfrak{p}^{n}$-torsion points $\underline{\mathcal{E}}$ of a Drinfeld
shtuka. This scheme of $\mathfrak{p}^{n}$-torsion points serves as an analogue
of the scheme of $p^{n}$-torsion points of an elliptic curve and behaves
similarly in many ways. In order to study properties of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ we use an explicit comparison of
Drinfeld shtukas and Drinfeld modules.

###  3.1. Comparison with Drinfeld modules.

We recall some facts about Drinfeld modules and show how to construct Drinfeld
shtukas from them. Let $A=\Gamma(X\setminus\\{\infty\\},\mathcal{O}_{X})$. The
point $0$ of $X$ then corresponds to a maximal ideal of $A$, which by a slight
abuse of notation we denote by $\mathfrak{p}$ as well.

Roughly speaking, a Drinfeld $A$-module is an $A$-module structure on a
(geometric) line bundle. Drinfeld modules were first introduced in [Dri76] in
order to construct a Langlands correspondence in the cohomology of their
moduli spaces. In this sense, Drinfeld modules (of rank 2) are function field
analogues of elliptic curves in the number field case. For a more detailed
treatment also compare [Leh09], [BS97] or [Lau96].

Let $\mathcal{L}$ be an invertible sheaf on $S$. The corresponding geometric
line bundle is denoted by
$\mathbb{G}_{a,\mathcal{L}}=\underline{\operatorname{Spec}}_{S}(\operatorname{Sym}(\mathcal{L}^{-1}))$.
If $S=\operatorname{Spec}(R)$ is an affine scheme such that $\mathcal{L}$ is
trivial, the corresponding line bundle is given by
$\mathbb{G}_{a,R}=\operatorname{Spec}R[t]$. Locally, the ring of endomorphisms
of a line bundle is then given by the skew-polynomial ring $R\\{\tau\\}$ with
the commutation relation $\tau c=c^{q}\tau$ for $c\in R$.

######  Definition 3.1.

A _Drinfeld $A$-module_ $\mathbf{E}=(\mathbb{G}_{a,\mathcal{L}},e)$ of rank
$r$ over a scheme $S$ consists of an additive group scheme
$\mathbb{G}_{a,\mathcal{L}}$ and a ring homomorphism $e\colon
A\to\operatorname{End}(\mathbb{G}_{a,\mathcal{L}}),a\mapsto e_{a}$ such that
$e_{a}$ is finite for all $a\neq 0\in A$ of degree $|a|^{r}$, where $|\cdot|$
is the normalised absolute value on $K$ corresponding to $\infty$. The
composition $\partial\circ e$ with the differential induces a map
$S\to\operatorname{Spec}(A)$ called the _characteristic_ of $\mathbf{E}$.

We denote by $\operatorname{Dr-Mod}_{r}$ the moduli stack of Drinfeld modules
of rank $r$. It is a Deligne-Mumford stack of finite type over
${\mathbb{F}_{q}}$, which is smooth of relative dimension $r-1$ over
$X^{\prime}=\operatorname{Spec}(A)$.

When $S=\operatorname{Spec}\ell$ is the spectrum of a field (or more generally
when the line bundle $\mathcal{L}$ is trivial), a Drinfeld module as a ring
homomorphism $e\colon A\to\ell\\{\tau\\}$. As for Drinfeld shtukas, in a
slight abuse of notation, we say $\mathbf{E}$ has characteristic
$\mathfrak{p}$ if the the characteristic of $\mathbf{E}$ factors through $0$,
or in other words, if the kernel of the induced map $A\to\mathcal{O}_{S}(S)$
is $\mathfrak{p}$. We say that a Drinfeld module over a field $\ell$ in
characteristic $\mathfrak{p}$ has height $h$, if the smallest non-vanishing
coefficient in $e_{\varpi}\in\ell\\{\tau\\}$ has degree $h$, where
$\varpi\in\mathfrak{p}$ is a uniformiser.

There are several ways to associate vector bundles to Drinfeld modules, for
example the so-called _elliptic sheaves_ due to [Dri77], for a more detailed
treatment also compare [BS97], [Har05] or [Wie04], or $t$-motives [And86] and
their generalisations, see for example [Har19]. However, a precise comparison
to Drinfeld shtukas, which is certainly well-known to the experts, does not
seem to be part of the literature yet. We explain how to construct Drinfeld
shtukas from Drinfeld modules.

Recall that an _elliptic sheaf_ $\underline{\mathcal{E}}$ over $S$ of rank $r$
is given by the data
$(x,(\mathcal{E}_{i})_{i\in\mathbb{Z}},(s_{i})_{i\in\mathbb{Z}},(t_{i})_{i\in\mathbb{Z}})$
where $x\colon S\to X^{\prime}=X\setminus\\{\infty\\}$ is a map of schemes,
$\mathcal{E}_{i}$ is a rank $r$ vector bundle on $X\times S$ for every
$i\in\mathbb{Z}$ and $s_{i}\colon\mathcal{E}_{i}\to\mathcal{E}_{i+1}$ and
$t_{i}\colon\sigma^{*}\mathcal{E}_{i}\to\mathcal{E}_{i+1}$ are injective maps
that satisfy some further properties. In particular,
$\operatorname{coker}(s_{i})$ and $\operatorname{coker}(t_{i})$ are supported
on $\infty$ and $\Gamma_{x}$, respectively and invertible as
$\mathcal{O}_{S}$-modules on their support. We denote by
$\mathcal{E}\ell\ell_{r}$ the moduli stack of elliptic sheaves. We have a
well-defined map $\mathcal{E}\ell\ell_{r}\to\operatorname{Sht}_{r}$ given by
the projection

$(x,(\mathcal{E}_{i}),(s_{i}),(t_{i}))\mapsto(x,\mathcal{E}_{0},s^{-1}_{0}|_{X\setminus(\Gamma_{x}\cup\Gamma_{\infty})}\circ
t_{0}|_{X\setminus(\Gamma_{x}\cup\Gamma_{\infty})}),$

or by Remark 2.3 equivalently by projection to
$(x,\mathcal{E}_{0},\mathcal{E}_{1},s_{0},t_{0})$. We use this second
perspective for the remainder of this section as it more convenient in this
context. We define a functor $\mathbb{Z}\times\operatorname{Dr-
Mod}_{r}\to\operatorname{Sht}_{r}$ by composing the equivalence
$\mathbb{Z}\times\operatorname{Dr-Mod}_{r}\to\mathcal{E}\ell\ell_{r}$ of
[Dri77] with this projection.

######  Lemma 3.2.

The projection $\mathcal{E}\ell\ell_{r}\to\operatorname{Sht}_{r}$ is fully
faithful.

###### Proof.

Let $\underline{\mathcal{E}}_{\bullet}=(x,(\mathcal{E}_{i}),(s_{i}),(t_{i}))$
and
$\underline{\tilde{\mathcal{E}}}_{\bullet}=(x,(\tilde{\mathcal{E}}_{i}),(\tilde{s}_{i}),(\tilde{t}_{i}))$
be two elliptic sheaves over $S$. Assume that we have a map of the
corresponding shtukas, in other words a pair of maps
$f_{0}\colon\mathcal{E}_{0}\to\tilde{\mathcal{E}}_{0}$ and
$f_{1}\colon\mathcal{E}_{1}\to\tilde{\mathcal{E}}_{1}$ that commute with
$s_{0}$ and $t_{0}$ in the obvious way. By [Wie04, Corollary 5.4] we may then
glue $f_{1}$ and $\sigma^{*}f_{1}$ to get a map
$f_{2}\colon\mathcal{E}_{2}\to\tilde{\mathcal{E}}_{2}$ again commuting with
$s_{1}$ and $t_{1}$. Such a map is necessarily unique. We continue inductively
to define maps in higher degrees. The maps in degrees smaller than $0$ can be
constructed as twists. ∎

Let us denote by $b_{\infty}=(-1/r,\ldots,-1/r)\in
B(\operatorname{GL}_{r,K_{\infty}},(0,\ldots,0,-1))$ the basic Newton polygon.
Recall that we defined $\operatorname{Sht}_{r,b_{\infty}}$ to be the (reduced)
locus in $\operatorname{Sht}_{r}$ where the local shtuka at $\infty$ has
Newton polygon $b_{\infty}$. Note that $\operatorname{Sht}_{r,b_{\infty}}$ is
a closed substack of $\operatorname{Sht}_{r}$ as $b_{\infty}$ is basic.

######  Proposition 3.3.

The functor $\mathbb{Z}\times\operatorname{Dr-
Mod}_{r}\to\operatorname{Sht}_{r}$ is schematic and a closed immersion which
factors through an isomorphism

$\mathbb{Z}\times\operatorname{Dr-
Mod}_{r}\xrightarrow{\cong}\operatorname{Sht}_{r,b_{\infty}}.$

###### Proof.

As a first step we show that the locus where a Drinfeld shtuka can be extended
to an elliptic sheaf is closed. Let
$\underline{\mathcal{E}}=(x,\mathcal{E}_{-1},\mathcal{E}_{0},s_{-1},t_{-1})\in\operatorname{Sht}_{r}(S)$.
As the zero and pole of $\underline{\mathcal{E}}$ do not intersect, we can
repeatedly glue $\mathcal{E}_{i}$ and $\sigma^{*}\mathcal{E}_{i}$ to obtain a
commutative diagram

${\mathcal{E}_{-1}}$${\mathcal{E}_{0}}$${\mathcal{E}_{2}}$${\ldots}$${\sigma^{*}\mathcal{E}_{-1}}$${\sigma^{*}\mathcal{E}_{0}}$${\sigma^{*}\mathcal{E}_{2}}$${\ldots}$$\scriptstyle{\varphi=s_{-1}}$$\scriptstyle{s_{0}}$$\scriptstyle{s_{2}}$$\scriptstyle{\varphi=t_{-1}}$$\scriptstyle{t_{0}}$$\scriptstyle{t_{2}}$

If the diagram comes from an elliptic sheaf, we have by definition that
$\mathcal{E}_{0}\hookrightarrow\mathcal{E}_{r}$ identifies $\mathcal{E}_{0}$
with $\mathcal{E}_{r}(-\infty)$. In a similar fashion we get
$\mathcal{E}_{1}=\mathcal{E}_{r+1}(-\infty)$.

We claim that these two conditions are already sufficient for the diagram to
come from an elliptic sheaf. By construction, the cokernel of $s_{i}$ is
supported on $\Gamma_{\infty}$ and the cokernel of $t_{i}$ is supported on
$\Gamma_{x}$, and both are invertible on their respective supports. We first
check that $s_{0}=s_{r}\otimes\operatorname{id}_{\mathcal{O}_{X}(-\infty)}$.
As all $s_{i}$ are isomorphisms away from $\infty$ and the question is fpqc-
local on $S$, it suffices to consider the completion at $\infty$ and we may
assume that $S=\operatorname{Spec}(R)$ is affine and all $\mathcal{E}_{i}$ are
free $R\left\llbracket\varpi_{\infty}\right\rrbracket$-modules of rank $r$.
Thus, the $s_{i}$ are identified with endomorphisms of
$R\left\llbracket\varpi_{\infty}\right\rrbracket^{r}$ such that both
$s_{r-1}\circ\ldots\circ s_{0}$ and $s_{r}\circ\ldots\circ s_{1}$ are
multiplication by $\varpi_{\infty}$ by assumption, where $\varpi_{\infty}$ is
a uniformiser at $\infty$. But as multiplication by $\varpi_{\infty}$ is
injective and lies in the centre of the endomorphism ring, this implies that
$s_{0}=s_{r}$ as desired. Moreover, the $s_{i}$ induce isomorphism
$\operatorname{coker}(t_{i-1})\xrightarrow{\cong}\operatorname{coker}(t_{i})$
for all $i\geq 1$, hence
$t_{r}=t_{-1}\otimes\operatorname{id}_{\mathcal{O}_{X}(\infty)}$. Hence, we
get inductively that
$\mathcal{E}_{i+r}=\mathcal{E}_{i}\otimes\mathcal{O}_{X}(\infty)$ for all
$i\geq 1$. The data for indices $i\leq 0$ is then obtained by twisting. This
shows the claim.

It remains to check that the conditions of the claim are closed conditions. In
order to see that the locus where $\mathcal{E}_{0}=\mathcal{E}_{r}(-\infty)$
is closed, we argue as follows. As $\mathcal{E}_{r}/\mathcal{E}_{0}$ is
supported on $\Gamma_{\infty}$, the uniformiser $\varpi_{\infty}$ at $\infty$
acts on $\mathcal{E}_{r}/\mathcal{E}_{0}$ and we have
$\mathcal{E}_{0}=\mathcal{E}_{r}(-\infty)$ if and only if $\varpi_{\infty}=0$
in $\operatorname{End}_{\mathcal{O}_{S}}(\mathcal{E}_{r}/\mathcal{E}_{0})$.
Hence, the locus where $\mathcal{E}_{0}=\mathcal{E}_{r}(-\infty)$ is
represented by the vanishing locus $V(\mathcal{I})$ of the quasi-coherent
ideal
$\mathcal{I}=\mathrm{image}(\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{E}_{r}/\mathcal{E}_{0})^{\vee}\xrightarrow{\varpi_{\infty}^{\vee}}\mathcal{O}_{S})$.
In a similar fashion, the locus where
$\mathcal{E}_{1}=\mathcal{E}_{r+1}(-\infty)$ is representable by a closed
subscheme of $S$ given by the vanishing locus of a quasi-coherent sheaf of
ideals $\mathcal{I}^{\prime}$ in $\mathcal{O}_{S}$. Thus, the locus where
$\underline{\mathcal{E}}$ defines a (necessarily unique) elliptic sheaf is
representable by the closed subscheme
$S^{\prime}=V(\mathcal{I}+\mathcal{I}^{\prime})$ of $S$. In particular,
$\mathbb{Z}\times\operatorname{Dr-Mod}_{r}\to\operatorname{Sht}_{r}$ is
schematic and a closed immersion.

Moreover, it is clear that both stacks have the same geometric points, as one
can easily see by the classification of bounded local shtukas over
algebraically closed fields that a Drinfeld shtuka over an algebraically
closed field $\ell$ comes from a Drinfeld module if and only if the local
shtuka at $\infty$ is

$\left(\ell\left\llbracket\varpi_{\infty}\right\rrbracket^{r},\sigma\cdot\begin{pmatrix}0&&&\varpi_{\infty}^{-1}\\\
1&&&\\\ &\ddots&&\\\ &&1&0\end{pmatrix}\right)$

The second part of the assertion follows as $\operatorname{Dr-Mod}_{r}$ and
the Newton stratum $\operatorname{Sht}_{r,b_{\infty}}$ both are reduced. ∎

######  Remark 3.4.

  1. (1)

For the general case (when $\infty$ is not defined over ${\mathbb{F}_{q}}$)
giving a correct definition of the truncation is more subtle, as the pole
might not be supported at $\infty$ but at some Frobenius twist of $\infty$. In
order to remedy this, one should include a Frobenius twist in the action of
$\mathbb{Z}$ by shifts and still obtain a well-defined closed immersion
$\mathbb{Z}\times\operatorname{Dr-Mod}_{r}\to\operatorname{Sht}_{r}$.

  2. (2)

It follows that we get an essentially surjective functor
$\operatorname{Sht}_{r,b_{\infty}}\to\operatorname{Dr-Mod}_{r}$ that agrees
with the construction from [Bre18, Proposition 7.8] (up to forgetting the
level structure).

  3. (3)

By [Har19] and [HS19], the comparison is compatible with local and finite
objects. More precisely, the local equivalence of [HS19, Theorem 8.3] and
[Har19, Theorem 7.6] identifies the $\mathfrak{p}$-divisible module associated
to a Drinfeld module $\mathbf{E}$ over
$S\in\mathcal{N}ilp_{\mathbb{F}\left\llbracket\varpi\right\rrbracket}$ the
local shtuka at $0$ of any Drinfeld shtuka associated to $\mathbf{E}$ by the
comparison. We call this local shtuka the _local shtuka at $0$_ (or the _local
shtuka at $\mathfrak{p}$_) of the Drinfeld module $\mathbf{E}$. By [Dri76,
Proposition 1.7], the Newton polygon of the local shtuka at $\mathfrak{p}$
associated to a Drinfeld module of height $h$ over an algebraically closed
field in characteristic $\mathfrak{p}$ is given by
$(1/h,\ldots,1/h,0,\ldots,0)$.

###  3.2. Strong stratification property of the Newton stratification.

The existence of supersingular Drinfeld modules then implies the non-emptiness
of the basic Newton stratum in $\operatorname{Sht}_{r}$.

######  Proposition 3.5.

Let $\underline{b}_{\underline{0}}=((-1/r,\ldots,-1/r),(1/r,\ldots,1/r))\in
B(\operatorname{GL}_{r,K_{\infty}})\times B(\operatorname{GL}_{r,K_{0}})$.
Then the basic Newton stratum
$\operatorname{Sht}_{r,\underline{b}_{\underline{0}}}\subseteq\operatorname{Sht}_{r,\mathbb{F}_{0}}$
is non-empty.

###### Proof.

By Remark 3.4 and Proposition 3.3 a basic Drinfeld module in characteristic
$\mathfrak{p}$, that is, a Drinfeld module of both rank and height $r$,
defines a point in $\operatorname{Sht}_{r,\underline{b}_{\underline{{0}}}}$.
But basic Drinfeld modules in characteristic $\mathfrak{p}$ exist by [KY20,
Proposition 7.4.1]. ∎

As a next step, we study closure relations among the Newton strata. The result
may be well-known to experts. The author was unable to track down a precise
reference. The corresponding statement for Shimura varieties in the Siegel
case is due to [Oor01] and has been generalised to the PEL case by [Ham15].

For a pair of closed points $\underline{y}=(y_{1},y_{2})$ of $X$ we define a
partial order on $B(\operatorname{GL}_{r,K_{y_{1}}})\times
B(\operatorname{GL}_{r,K_{y_{2}}})$ (and also on
$B(\operatorname{GL}_{r,K_{y_{1}}},\mu_{1})\times
B(\operatorname{GL}_{r,K_{y_{2}}},\mu_{2})$ for a pair of cocharacters
$\mu_{1},\mu_{2}$ of $\operatorname{GL}_{r}$) by
$b_{\underline{y}}=(b_{y_{1}},b_{y_{2}})\leq
b^{\prime}_{\underline{y}}=(b^{\prime}_{y_{1}},b^{\prime}_{y_{2}})$ if
$b_{y_{1}}\leq b^{\prime}_{y_{1}}$ and $b_{y_{2}}\leq b^{\prime}_{y_{2}}$. Let
us fix the cocharacters $\mu_{1}=(0,\ldots,0,-1)$ and $\mu_{2}=(1,0,\ldots,0)$
of $\operatorname{GL}_{r}$.

######  Theorem 3.6.

The Newton stratification on $\operatorname{Sht}_{r,\mathbb{F}_{0}}$ satisfies
the strong stratification property. In other words, for all $\underline{b}\in
B(\operatorname{GL}_{r,\infty},\mu_{1})\times
B(\operatorname{GL}_{r,0},\mu_{2})$ we have

$\overline{\operatorname{Sht}_{r,\underline{b}}}=\bigcup_{{\underline{b}^{\prime}}\leq\underline{b}}\operatorname{Sht}_{r,\underline{b}^{\prime}}=\operatorname{Sht}_{r,\leq{\underline{b}}}.$

Moreover, all the Newton strata $\operatorname{Sht}_{r,\underline{b}}$ for
$\underline{b}\in B(\operatorname{GL}_{r,\infty},\mu_{1})\times
B(\operatorname{GL}_{r,0},\mu_{2})$ are non-empty.

Viehmann in [Vie20, Remark 5.6.] remarks that the assertion should follow by a
similar argument as in [Ham15] for Shimura varieties of PEL type.

###### Proof.

Let $\underline{b}^{0}$ correspond to $((1/r,\ldots,1/r),(-1/r,\ldots,-1/r))$,
which is the unique basic point in
$B(\operatorname{GL}_{r,K_{\infty}},\mu_{1})\times
B(\operatorname{GL}_{r,K_{0}},\mu_{2})$. By Proposition 3.5 the Newton stratum
$\operatorname{Sht}_{r,\underline{b}^{0}}$ is non-empty. The non-emptiness of
the other strata will follow from the closure relations.

Now, let $\underline{b}\in B(\operatorname{GL}_{r,K_{\infty}})\times
B(\operatorname{GL}_{r,K_{0}})$ and assume that
$\operatorname{Sht}_{r,\underline{b}}$ is non-empty. We fix a point
$s\in\operatorname{Sht}_{r,\underline{b}}$ and let $R$ be its universal
deformation ring. Then $s$ lies in the closure of some
$\operatorname{Sht}_{r,\underline{b^{\prime}}}$ for
${\underline{b^{\prime}}}\leq{\underline{b}}$ if and only if the same is true
in the Newton stratification on $\operatorname{Spec}R$. By the Serre-Tate
Theorem (Proposition 2.11) the universal deformation ring factors as
$\operatorname{Spec}R=\operatorname{Spec}R_{\infty}\times\operatorname{Spec}R_{0}$,
where $R_{\ast}$ is the universal deformation ring of the corresponding local
shtuka at $\ast=\infty,0$. Under this isomorphism we have
$\operatorname{Spec}(R)_{\underline{b}}=\operatorname{Spec}(R_{1})_{{b}_{{\infty}}}\times\operatorname{Spec}(R_{2})_{{b}_{{0}}}$,
where we denote by $\operatorname{Spec}(R_{\ast})_{{b}_{\ast}}$ the
corresponding Newton strata in $\operatorname{Spec}R_{\ast}$ for
$\ast=\infty,0$. On $\operatorname{Spec}R_{\ast}$ the closure properties hold
by [Vie13, Theorem 2, Lemma 21 (2)], and thus they hold on
$\operatorname{Spec}R$. This proves the assertion. ∎

######  Remark 3.7.

In a similar fashion, there is a Newton stratification on the moduli space of
Drinfeld modules in characteristic $\mathfrak{p}$ defined via the local
shtukas as defined in Remark 3.4 (3). The Newton stratifications are clearly
compatible with the projection
$\operatorname{Sht}_{r,b_{\infty},\mathbb{F}_{0}}\to\operatorname{Dr-
Mod}_{r,\mathbb{F}_{0}}$ in the fibre over $0$ from Remark 3.4 (2). Thus, the
Newton stratification on Drinfeld modules retains the strong stratification
property as in the theorem above.

######  Corollary 3.8.

Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka over a complete local
noetherian ring $R$ with algebraically closed residue field $\ell$ such that
the characteristic of $\underline{\mathcal{E}}_{\ell}$ factors through $0$.
Then there exists a Drinfeld module $\mathbf{E}$ over $R$ such that the local
shtukas at $0$ of $\underline{\mathcal{E}}$ and $\mathbf{E}$ are isomorphic.

###### Proof.

The case where $R=\ell$ is a field directly follows from the non-emptiness of
Newton strata of Theorem 3.6 and Remark 3.7. The case that $R$ is local
artinian then follows from the Serre-Tate Theorem 2.11, and the general case
that $R$ is a complete noetherian local ring with algebraically closed residue
field follows from the fact that $\operatorname{Dr-Mod}_{r}$ is of finite type
over ${\mathbb{F}_{q}}$ and [HV11, Proposition 3.16]. ∎

###  3.3. The $\mathfrak{p}^{n}$-torsion scheme of a Drinfeld shtuka.

We briefly explain how to construct a scheme of $\mathfrak{p}^{n}$-torsion
points of a shtuka, which will play the role of the $p^{n}$-torsion points of
an elliptic curve. The construction goes back to [Dri87]. Let
$\underline{\mathcal{F}}$ be a $\varphi$-sheaf (for example a
$\mathfrak{p}^{n}$-torsion shtuka) over $S$. We set

$\operatorname{Dr}_{q}(\underline{\mathcal{F}})=\underline{\operatorname{Spec}}\left(\operatorname{Sym}^{\bullet}\mathcal{F}\right)/\mathcal{I},$

where $\mathcal{I}$ is the ideal locally generated by the sections $v^{\otimes
q}-\varphi(\sigma^{*}v)$. It induces a contravariant functor from the category
of $\varphi$-sheaves to the category of finite locally free group schemes with
${\mathbb{F}_{q}}$-action over $S$. Assume $S=\operatorname{Spec}R$ is affine,
$\mathcal{F}=R^{r}$ is trivial and $\varphi$ is given by the Matrix
$(a_{ij})$. Then

$\operatorname{Dr}_{q}(\underline{\mathcal{F}})=\operatorname{Spec}\left(R[Y_{1},\ldots,Y_{r}]/\left(Y_{1}^{q}-\sum_{i=1}^{r}a_{i1}Y_{i},\ldots,Y_{r}^{q}-\sum_{i=1}^{r}a_{ir}Y_{i}\right)\right).$

######  Proposition 3.9 ([Dri87, Proposition 2.1], [Abr06, Theorem 2] and
[HS19, Theorem 5.2]).

Let $\underline{\mathcal{F}}=(\mathcal{F},\varphi)$ be a finite shtuka of rank
$r$ on $S$. Then the group scheme
$\operatorname{Dr}_{q}(\underline{\mathcal{F}})$ is finite locally free of
rank $q^{r}$ over $S$, étale over $S$ if and only if $\varphi$ is an
isomorphism, and radicial over $S$ if and only if $\varphi$ is locally
nilpotent on $S$. Moreover, the functor $\operatorname{Dr}_{q}$ is
${\mathbb{F}_{q}}$-linear and exact. Its essential image is characterised by
the property that the ${\mathbb{F}_{q}}$-action is strict in the sense of
[Fal02].

Note that the notion of a strict ${\mathbb{F}_{q}}$-action is a condition on
the ${\mathbb{F}_{q}}$ action on the co-Lie complex of a certain deformation
of the group scheme. We do not need the exact definition here and refer to
[Fal02] or [HS19] for more details. In our setting the strictness of the
${\mathbb{F}_{q}}$-action will usually be automatic.

######  Definition 3.10.

Let $\underline{\mathcal{E}}$ be a rank $r$ Drinfeld shtuka over $S$. We
denote by

$\underline{\mathcal{E}}[\mathfrak{p}^{n}]=\operatorname{Dr}_{q}(\underline{\mathcal{E}}|_{D_{n,S}})$

the _scheme of $\mathfrak{p}^{n}$-division points_ of
$\underline{\mathcal{E}}$.

The previous proposition implies that
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is a finite locally free $S$-group
scheme of rank $q^{nr}$ with strict ${\mathbb{F}_{q}}$-action. The
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module structure on
$\underline{\mathcal{E}}|_{D_{n,S}}$ gives rise to a canonical
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module structure on
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$. The finite shtuka equivalence in
particular induces an equivalence of quotients of
$\underline{\mathcal{E}}|_{D_{n,S}}$ as $\mathfrak{p}^{n}$-torsion shtukas and
finite locally free closed $\mathcal{O}/\mathfrak{p}^{n}$-module subschemes
with strict ${\mathbb{F}_{q}}$-action of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$.

By comparison with Drinfeld modules, we get the following explicit description
of the $\mathfrak{p}^{n}$-torsion in characteristic $\mathfrak{p}$.

######  Corollary 3.11.

Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka over a complete local
noetherian ring $R$ with algebraically closed residue field $\ell$ such that
the characteristic of $\underline{\mathcal{E}}_{\ell}$ factors through $0$.
Then there exists a Drinfeld module $\mathbf{E}$ over $R$ such that

$\underline{\mathcal{E}}[\mathfrak{p}^{n}]\cong\mathbf{E}[\mathfrak{p}^{n}]$

as $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module schemes over $R$ for all
$n\in\mathbb{N}$.

###### Proof.

This follows directly from the corresponding assertion for the local shtukas
in Corollary 3.8. ∎

######  Proposition 3.12.

The scheme of $\mathfrak{p}^{n}$-division points of a shtuka
$\underline{\mathcal{E}}$ of rank $r$ over an algebraically closed $\ell$ is
given by the $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module scheme

$\underline{\mathcal{E}}[\mathfrak{p}^{n}]=\alpha_{q^{h}}\times(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-h},$

where the operation of $\varpi$ on $\alpha_{q^{h}}$ is given by $t\mapsto
t^{q^{h}}$, and where $h$ is the height of $\underline{\mathcal{E}}$ (we use
the convention $h=0$ when the characteristic of $\underline{\mathcal{E}}$ is
away from $0$).

###### Proof.

We first consider the case that $\mathfrak{p}$ is away from the characteristic
of $\underline{\mathcal{E}}$. Then,
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is a finite étale scheme by the
finite shtuka equivalence. It follows that étale locally on $S$ the
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module scheme
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is constant. Over geometric
points, we have that
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]\cong(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}$
as the corresponding étale local shtuka is trivial by [AH14, Corollary 2.9].
In characteristic $\mathfrak{p}$, by the previous Corollary 3.11 it suffices
to check the assertion for Drinfeld modules, which then follows from [Leh09,
3, Proposition 1.5], [Leh09, 3, Proposition 1.5] and [Leh09, 2, Corollary
2.4.]. ∎

Even more generally, we can embed the scheme of $\mathfrak{p}$-torsion points
as a closed subscheme of a smooth curve. However, this smooth curve will not
be a Drinfeld module in general.

######  Proposition 3.13.

Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka over $S$. Then étale-
locally on $S$, the scheme of $\mathfrak{p}^{n}$-division points of
$\underline{\mathcal{E}}$ can be embedded as a closed subscheme of a smooth
curve over $S$. More precisely, we can étale-locally on $S$ embed
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ as a closed subscheme of
$\mathbb{A}^{1}_{S}$.

######  Remark 3.14.

For one-dimensional $p$-divisible groups a similar statement is discussed in
[Fri19, Lemma 5.2.1], building on arguments from [HT01]. However, [Fri19]
claims that an embedding even exists Zariski-locally on $S$, this seems to be
problematic to us for the following reason. Let us assume that $S$ is the
spectrum of a finite field. We assume that the étale part of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is non-trivial and constant over
$S$. Then the number of rational points of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ tends to infinity as $n\to\infty$.
However, the number of $S$-rational points on $\mathbb{A}^{1}_{S}$ is bounded.
In particular, it cannot be possible to embed
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ into $\mathbb{A}^{1}_{S}$ for all
$n\in\mathbb{N}$.

For the proof of the proposition, we essentially adapt the proof of [Fri19,
Lemma 5.2.1], but we allow finite extensions on the residue fields in order to
circumvent the issue discussed above. So we only get an étale local statement.

###### Proof.

We adapt the proof of [Fri19, Lemma 5.2.1]. We first consider the case that
$S=\operatorname{Spec}\ell$ is the spectrum of an algebraically closed field.
In this case, the assertion follows from the explicit description of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ in Proposition 3.12.

For the general case, we may by reduction to the universal case assume that
$S$ is locally of finite type over ${\mathbb{F}_{q}}$. As the statement is
local on $S$, we may further assume that $S=\operatorname{Spec}(R)$ is affine
and of finite type over ${\mathbb{F}_{q}}$. Then
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]=\operatorname{Spec}(B)$ is affine
as well. We fix a closed point $s\in S$. By the argument above, there exists a
finite extension $\mathbb{F}$ of the residue field $k(s)$ (which is finite by
assumption) at $s$ such that there exists a closed immersion
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{\mathbb{F}}\hookrightarrow\mathbb{A}^{1}_{\mathbb{F}}$
over $\mathbb{F}$, in other words a surjection
$\mathbb{F}[t]\twoheadrightarrow B\otimes\mathbb{F}$. By [Stacks, Tag 00UD]
there exists an étale neighbourhood
$\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R$ of $s$ and a point
$s^{\prime}$ over $s$ such that the extension of residue fields $k(s)\to
k(s^{\prime})$ is given by $k(s)\to\mathbb{F}$. We can thus lift the
surjection to a map $R^{\prime}[t]\to B\otimes R^{\prime}$ by choosing a lift
of the image of $t$. By Nakayma’s lemma this is a surjection over some
$R^{\prime}_{f}$, where $f\in R^{\prime}$ is not contained in the maximal
ideal at $s^{\prime}$. In other words,
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{\mathbb{F}}\hookrightarrow\mathbb{A}^{1}_{\mathbb{F}}$
extends to
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{R^{\prime}_{f}}\hookrightarrow\mathbb{A}^{1}_{R^{\prime}_{f}}$
over the étale neighbourhood $\operatorname{Spec}({R^{\prime}_{f}})$ of $s$. ∎

We conclude this section by collecting some consequences on isogenies of
Drinfeld shtukas. Using the finite shtuka equivalence we see that a chain of
$\mathfrak{p}^{n}$-isogenies of type $(r_{1},\ldots,r_{m})$ on a Drinfeld
shtuka $\underline{\mathcal{E}}$ is equivalent to the data of a flag

$0\subseteq\mathbf{H}_{1}\subseteq\mathbf{H}_{2}\subseteq\ldots\subseteq\mathbf{H}_{m}\subseteq\underline{\mathcal{E}}_{0}[\mathfrak{p}^{n}]$

of finite locally free submodule schemes
$\mathbf{H}_{i}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank
$q^{n\cdot(r_{1}+\ldots+r_{i})}$ over $S$ with strict
${\mathbb{F}_{q}}$-action. In particular, $\mathbf{H}_{i}/\mathbf{H}_{i-1}$
has rank $q^{nr_{i}}$ and has an induced strict ${\mathbb{F}_{q}}$-action.

######  Proposition 3.15.

The stack
$\operatorname{Sht}_{r,(r_{1},\ldots,r_{m})-\mathfrak{p}^{n}\text{-chain}}$ is
a Deligne-Mumford stack locally of finite type over ${\mathbb{F}_{q}}$. The
forgetful map to $\operatorname{Sht}_{r}$ given by projection to
$\underline{\mathcal{E}}$ is schematic and finite. Moreover, the forgetful map
$\operatorname{Sht}_{r,(r_{1},\ldots,r_{m})-\mathfrak{p}^{n}\text{-chain}}\to\operatorname{Sht}_{r}$
is finite étale away from $0$.

###### Proof.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$. The functor on
$S$-schemes

$T\mapsto\left\\{\begin{array}[]{l}\text{flags of quotients
}\underline{\mathcal{E}}|_{D_{n,T}}\twoheadrightarrow\underline{\mathcal{F}}_{1}\twoheadrightarrow\ldots\twoheadrightarrow\underline{\mathcal{F}}_{m}\twoheadrightarrow
0\text{ of $\mathfrak{p}^{n}$-torsion}\\\ \text{finite shtukas such that
$\mathcal{F}_{i}$ has rank $n(r_{1}+\ldots+r_{i})$ as
$\mathcal{O}_{T}$-module}\end{array}\right\\}$

is representable by the closed subscheme of a certain flag variety of
quotients of $\underline{\mathcal{E}}|_{D_{n,S}}$ (as
$\mathcal{O}_{S}$-module) where both the map
$\sigma^{*}\underline{\mathcal{E}}|_{D_{n,S}}\to\underline{\mathcal{E}}|_{D_{n,S}}$
and the $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module structure descend to all the
$\mathcal{F}_{i}$. As the flag variety is projective, we see that
$\operatorname{Sht}_{r,(r_{1},\ldots,r_{m})-\mathfrak{p}^{n}\text{-chain}}\to\operatorname{Sht}_{r}$
is schematic and projective.

In order to show finiteness of the map we proceed as in the proof of [KM85,
Proposition 6.5.1]. By [EGA4, Corollaire 18.12.4] it suffices to show that the
map has finite fibres. Let $\ell$ be an algebraically closed field and let
$\underline{\mathcal{E}}$ be a rank $r$ shtuka over $\ell$. It suffices to
show that $\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ only has finitely many
submodule schemes. We know by Proposition 3.12 that for some $h\geq 0$ we have

$\underline{\mathcal{E}}[\mathfrak{p}^{n}]\cong\alpha_{q^{h}}\times\left(\mathfrak{p}^{-n}/\mathcal{O}_{0}\right)^{r-h}.$

As $\ell$ is in particular perfect, any
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-submodule scheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ factors as
$\mathbf{H}\cong\mathbf{H}^{\text{conn}}\times\mathbf{H}^{\text{\'{e}t}}$ but
for both factors (which are necessarily submodule schemes of $\alpha_{q^{h}}$
and $\left(\mathfrak{p}^{-n}/\mathcal{O}_{0}\right)^{r-h}$, respectively)
there are only finitely many possibilities.

The étaleness away from $0$ follows for example from [Var04, Lemma 3.3 a)]. ∎

##  4\. Drinfeld $\Gamma_{1}(\mathfrak{p}^{n})$-level structures on shtukas

In this section, we introduce $\Gamma_{1}$-type (Drinfeld-) level structures
on Drinfeld shtukas adapting similar constructions for Drinfeld modules and
elliptic curves. We show that the moduli space of Drinfeld shtukas with these
level structures is regular following the arguments of [KM85]. For Drinfeld
modules, full Drinfeld level structures were studied extensively starting with
[Dri76], compare for example also [Leh09]. For other kinds of level structures
some results are known, [Sha07] considers $\Gamma_{1}(\mathfrak{p})$-level
structures in the rank 2 case and [KY20] study level structures for arbitrary
torsion modules and higher rank Drinfeld modules.

We propose a slightly different generalisation of a $M$-level structure on
Drinfeld shtukas for a $\mathfrak{p}$-torsion $\mathcal{O}_{0}$-module $M$. In
this notation $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}$-structures are full
level structures and in the rank 2 case
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})$-strucutres are
$\Gamma_{1}(\mathfrak{p}^{n})$-level structures. For us, it does not seem to
be a priori clear that our definition and the analogue of [KY20] agree, even
for full level, as is claimed in [KY20, (4.1.2.)]. For full level structures
on Drinfeld modules, this follows from a deep result on the deformation theory
of [Leh09, Proposition 3.3]. We show that the two definitions agree in general
in a similar fashion. One could also directly adapt the definition of [KM85],
as does for example [Tae06]. However, it seems to us that this definition does
not give the correct moduli space, see Remark 4.4.

Moreover, we define analogues of balanced level structures of [KM85] and use
this notion of balanced level structure to give a definition of
$\Gamma_{1}(\mathfrak{p}^{n})$-level structure for Drinfeld shtukas of
arbitrary rank and arbitrary $n\in\mathbb{N}$ in Definition 4.15.

###  4.1. $M$-Structures on Drinfeld shtukas.

In order to define Drinfeld level structures we use the notion of _full sets
of sections_ , compare [KM85, Section 1.8]. When working with closed
subschemes of a smooth curve, this can be expressed in terms of Cartier
divisors by [KM85, Theorem 1.10.1]. Katz and Mazur hoped that the notion of
"full sets of sections" might be useful to define level structures for higher
dimensional abelian varietes. However, this notion gives rise to a moduli
problem which is not even flat over $\mathbb{Z}$ in general (compare [CN90]).
Nevertheless, these issues do not appear in our setting, as Proposition 3.13
allows us to locally work with Cartier divisors in $\mathbb{A}^{1}$. Note that
in a similar fashion Drinfeld level structures are well-behaved when working
with one-dimensional $p$-divisible groups, as do [HT01] and [Sch13].

Let $M$ be a finitely generated $\mathfrak{p}^{n}$-torsion
$\mathcal{O}_{0}$-module. In order to define $M$-structures on Drinfeld
shtukas, we would like for an $\mathcal{O}_{0}$-module homomorphism
$\iota\colon M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$ to induce a
(unique) scheme _generated_ by $\iota$, similar to the Cartier divisor
generated by an $\Gamma_{1}(p^{n})$-Drinfeld level structure on elliptic
curves. In other words, we are looking for a unique finite locally free scheme
over $S$ such that the image of $\iota$ forms a full set of sections for $S$
in the sense of [KM85, Section 1.8]. This notion is defined as follows. Let
$Z$ be a finite locally free $S$-scheme of rank $N$. A set of sections
$P_{1},\ldots,P_{N}\in Z(S)$ is called _full set of sections_ of $Z$ if for
every affine $S$-scheme $\operatorname{Spec}(R)\to S$ and every
$f\in\Gamma(Z_{R},\mathcal{O}_{Z_{R}})$ we have
$\mathrm{Norm}(f)=\prod_{i=1}^{N}f(P_{i})$. By [KM85, Theorem 1.10.1], when
$Z\hookrightarrow C$ is embedded as a relative effective Cartier divisor in a
smooth curve $C$ over $S$, the set $P_{1},\ldots,P_{N}\in Z(S)$ is a full set
of sections of $Z$ if and only if $Z=\sum_{i=1}^{N}[P_{i}]$ as Cartier
divisors in $C$.

Recall that in general, that given a set of sections
$P_{1},\ldots,P_{N^{\prime}}\in Z(S)$ we can neither expect that a finite
locally free subscheme $Z^{\prime}$ of $Z$ of rank $N^{\prime}$ such that
$P_{1},\ldots,P_{N^{\prime}}$ forms a full set of sections of $Z^{\prime}$
exists nor that it is unique when it exists (compare [KM85, Remark 1.10.4]).
However, Proposition 3.13 allows us to construct such a unique scheme in the
cases we are interested in.

######  Lemma 4.1.

Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka over $S$ and let $M$ be a
$\mathfrak{p}^{n}$-torsion module. Let $\iota\colon
M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$ be an
$\mathcal{O}_{0}$-linear map.

  1. (1)

Assume there exists a closed immersion
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]\hookrightarrow C$ into a smooth
curve $C$ over $S$. Then there exists a unique finite locally free closed
subscheme $\mathbf{H}$ of $C$ such that the image of $\iota$ (in $C(S)$) forms
a full set of sections for $\mathbf{H}$.

  2. (2)

There exists at most one finite locally free closed subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ such that the
image of $\iota$ forms a full set of sections for $\mathbf{H}$.

  3. (3)

There exists a (by the previous point necessarily unique) finite locally free
closed subscheme $\mathbf{H}$ of $\underline{\mathcal{E}}[\mathfrak{p}^{n}]$
such that $\iota$ gives a full set of sections for $\mathbf{H}$ if and only if
the following equivalent conditions are satisfied:

    1. (a)

For all étale maps $U\to S$ and all closed immersions
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{U}\hookrightarrow C$ into a smooth
curve over $U$, the Cartier divisor defined by the image of $\iota$ in $C(U)$
is a subscheme of $\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{U}$.

    2. (b)

There exists an étale cover $\\{U_{i}\\}_{i\in I}$ of $S$ and for each $i\in
I$ a smooth curve $C_{i}$ over $U_{i}$ together with a closed immersion
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{U_{i}}\hookrightarrow C_{i}$ such
that the Cartier divisor defined by the image of $\iota$ in $C_{i}(U_{i})$ is
a subscheme of $\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{U_{i}}$ for all
$i\in I$.

The existence of such an $\mathbf{H}$ is a closed condition on $S$, defined
locally on $S$ by finitely many equations.

###### Proof.

  1. (1)

This is [KM85, Theorem 1.10.1]. The scheme $\mathbf{H}$ is the Cartier divisor
$\sum_{\alpha\in M}[\iota(\alpha)]$.

  2. (2)

This is clear from the previous point, as étale-locally on $S$,
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ admits an embedding into a smooth
curve over $S$ by Proposition 3.13.

  3. (3)

It is clear that the existence of an $\mathbf{H}$ implies condition (a), and
that condition (a) implies (b) using Proposition 3.13. Let us now assume that
condition (b) is satisfied. We denote by $\mathbf{H}_{i}$ the Cartier divisor
in $C_{i}$ defined by $\iota$. We can glue the $\mathbf{H}_{i}$ to form a
finite locally free scheme $\mathbf{H}$ over $S$ by the uniqueness in the
previous point. It is clear that $\iota$ forms a full set of sections for
$\mathbf{H}$, this can be checked étale-locally on $S$.

In order to check that the locus of existence of $\mathbf{H}$ is closed in
$S$, we may choose an étale cover $\\{U_{i}\\}$ of $S$ together with
embeddings of $\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ into a smooth curve
over $U_{i}$. The assertion on $U_{i}$ follows from [KM85, Key Lemma 1.3.4].

∎

We can now give our definition of $M$-structures for shtukas.

######  Definition 4.2.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ be a rank $r$ shtuka
over $S$. Let $M$ be a finitely generated
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module. A _$M$ -structure_ on
$\underline{\mathcal{E}}$ is an $\mathcal{O}_{0}$-module homomorphism
$\iota\colon M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$ such that there
exists a finite locally free subscheme $\mathbf{H}$ of
$\underline{\mathcal{E}}[\mathfrak{p}]$ of rank $|M[\mathfrak{p}]|$ such that
the image of the restriction $\iota|_{M[\mathfrak{p}]}$ of $\iota$ to the
$\mathfrak{p}$-torsion forms a full set of sections for $\mathbf{H}$ in the
sense of [KM85, Section 1.8].

######  Remark 4.3.

Note that in the theory of Drinfeld modules the modules $M$ and
$\underline{\mathcal{E}}[\mathfrak{p}]$ would usually be considered as
$\mathfrak{p}$-torsion $A$-modules, where
$A=\Gamma(X\setminus\\{\infty\\},\mathcal{O}_{X})$. As
$A/\mathfrak{p}^{n}\cong\mathcal{O}_{0}/\mathfrak{p}^{n}$ this does not give a
different notion of level structures. In our context, working with
$\mathcal{O}_{0}$\- instead of $A$-modules seems more natural and should
stress that the level structure only depends on local data of
$\underline{\mathcal{E}}$ at $0$ and in particular not on the choice of
$\infty$.

######  Remark 4.4.

Our definition is an analogue of Drinfeld’s original definition of full level
structures in [Dri76]. The definition in [KM85] for elliptic curves is
slightly different. The direct analogue of their definition asserts that there
is a finite locally free subgroup scheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank $|M|$
such that $\iota$ is a full set of sections for $\mathbf{H}$ (instead of the
corresponding assertion only for the $\mathfrak{p}$-torsion). We show that
this is implied by our definition in Proposition 4.10. For full level
structures on Drinfeld modules, [Leh09, 3, Proposition 3.1] and [Wie10] show
that the two notions are equivalent. However, this is not true in general as
we can see in the following example. Consider
$X=\mathbb{P}^{1}_{\mathbb{F}_{q}}$, and
$S=D_{2}=2[0]=\operatorname{Spec}({\mathbb{F}_{q}}[\zeta]/(\zeta^{2}))$ viewed
as an $X$-scheme via the canonical inclusion. Then the map

$\varphi\colon\sigma^{*}\mathcal{O}_{X_{S}}^{2}\ext@arrow
0359\xleaders\hbox{$\relbar$}\hfill\rightarrow{\left(\begin{smallmatrix}0&\varpi-\zeta\\\
1&-\zeta\end{smallmatrix}\right)}\mathcal{O}_{X_{S}}^{2}$

defines a rank 2 Drinfeld shtuka over $S$. Its schemes of $\mathfrak{p}$\- and
$\mathfrak{p}^{2}$-torsion points are given by

$\underline{\mathcal{E}}[\mathfrak{p}]=\operatorname{Spec}\left(R[t]/(t^{q^{2}}+\zeta
t^{q}+\zeta
t)\right)\qquad\text{and}\qquad\underline{\mathcal{E}}[\mathfrak{p}^{2}]=\operatorname{Spec}\left(R[t]/(t^{q^{4}}+\zeta
t^{q^{3}}+\zeta t^{q^{2}})\right),\quad\text{respectively}.$

Then the constant zero map $\iota\colon\mathfrak{p}^{-2}/\mathcal{O}_{0}\to
R,\varpi^{-2}\mapsto 0$ defines the subscheme of
$\operatorname{Spec}(R[t]/(t^{q^{2}}))\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{2}]$.
However, the restriction of $\iota$ to $\mathfrak{p}^{-1}/\mathcal{O}_{0}$
induces the subscheme $\operatorname{Spec}(R[t]/(t^{q^{2}}))$ of
$\underline{\mathcal{E}}[\mathfrak{p}^{2}]$, which is not a subscheme of
$\underline{\mathcal{E}}[\mathfrak{p}]$. Thus, $\iota$ is not an
$\mathfrak{p}^{-2}/\mathcal{O}_{0}$-structure in the sense of our definition.
Hence, the definition of [KM85] does not yield well-defined level maps in our
setting and thus does not seem to be adequate here.

We also do not require the subscheme defined by $\iota$ to be a subgroup
scheme as we show this is already automatic below in Proposition 4.10.
Moreover, we show that it is even automatically an $\mathcal{O}_{0}$-module
subscheme.

######  Proposition 4.5.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ be a Drinfeld shtuka
over $S$ and let $M$ be a finitely generated
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module. Let $\iota\colon
M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$ be a $M$-structure on
$\underline{\mathcal{E}}$ and let $M^{\prime}\subseteq M$ be a submodule. The
restriction of $\iota$ to $M^{\prime}$ defines a $M^{\prime}$-structure on
$\underline{\mathcal{E}}$.

###### Proof.

Étale locally, the Cartier divisor defined by the restriction of $\iota$ to
$M^{\prime}[\mathfrak{p}]$ is a closed subscheme of the Cartier divisor
defined by $\iota|_{M[\mathfrak{p}]}$, which in turn is a closed subscheme of
$\underline{\mathcal{E}}[\mathfrak{p}]$ by assumption. The assertion follows
from Lemma 4.1. ∎

In the étale case, we have the following descriptions of $M$-structures.

######  Proposition 4.6.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ be a Drinfeld shtuka
over $S$ and let $M$ be a finitely generated
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module. Let $\iota\colon
M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$ be an
$\mathcal{O}_{0}$-module homomorphism. The following are equivalent:

  1. (1)

For every geometric point $\operatorname{Spec}\ell\to S$, the induced
homomorphism

$\iota_{\ell}\colon M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](\ell)$

is injective.

  2. (2)

The map $\iota$ defines a locally free closed subscheme of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ which is finite étale over $S$.

  3. (3)

The map $\iota$ defines a closed immersion of
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module $S$-schemes

$M_{S}\hookrightarrow\underline{\mathcal{E}}[\mathfrak{p}^{n}]$

and $\iota$ is a full set of sections for the image of this map (as subscheme
of $\underline{\mathcal{E}}[\mathfrak{p}^{n}]$).

If the equivalent conditions (1)-(3) are satisfied, $\iota$ is a $M$-structure
on $\underline{\mathcal{E}}$. Moreover, when $S$ is connected, these
conditions are equivalent to saying that $M\to\mathbf{H}(S)$ is an isomorphism
of $\mathcal{O}_{0}$-modules for some constant closed finite locally free
$\mathcal{O}_{0}$-module subscheme $\mathbf{H}$ of $\underline{\mathcal{E}}$.

Moreover, these conditions are automatically satisfied when the characteristic
of $\underline{\mathcal{E}}$ is away from $0$ and $\iota$ is a $M$-structure
on $\underline{\mathcal{E}}$.

###### Proof.

We adapt the proof of an analogous assertion for elliptic curves in [KM85,
Lemma 1.4.4].

(2) $\Leftrightarrow$ (3): This follows directly from the set-theoretic
analogue in [KM85, Proposition 1.8.3].

(1) $\Leftrightarrow$ (3): The map $\iota$ defines a map of
$\mathcal{O}_{0}$-module schemes
$M_{S}\rightarrow\underline{\mathcal{E}}[\mathfrak{p}^{n}]$. We may work
étale-locally on $S$ and assume that we can embed
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ into a smooth curve $C$ over $S$.
Let us denote by $D$ the Cartier divisor in $C$ defined by $\iota$. We can
check that the natural map $M_{S}\to D$ is an isomorphism on geometric points
as in the proof of [KM85, Lemma 1.4.4.], and this is clearly satisfied if and
only if $\iota$ is injective on geometric points.

Let us now assume that (1)-(3) are satisfied. By (3) the restriction of
$\iota$ to $M[\mathfrak{p}]$ defines a subscheme of
$\underline{\mathcal{E}}[\mathfrak{p}]$. Thus, $\iota$ is a $M$-structure on
$\underline{\mathcal{E}}$.

Now assume that $\mathfrak{p}$ is away from the characteristic. Let $\iota$ be
a $M$-structure on $\underline{\mathcal{E}}$. We check that condition (1) is
satisfied. Let $\operatorname{Spec}\ell\to S$ be a geometric point of $S$. By
Proposition 3.12, we have an $\mathcal{O}_{0}$-linear isomorphism
$\underline{\mathcal{E}}[\mathfrak{p}^{n}](\ell)\cong(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}.$
Now $\iota_{\ell}$ is injective if and only if the restriction
$\iota_{\ell}|_{M[\mathfrak{p}]}$ is injective, as multiplying a non-trivial
element $m$ in the kernel of $\iota_{\ell}$ by the maximal power of $\varpi$
that does not kill $m$ produces a non-trivial element in the kernel of
$\iota_{\ell}|_{M[\mathfrak{p}]}$. The injectivity of
$\iota_{\ell}|_{M[\mathfrak{p}]}$ follows by assumption and the implication
(2) $\Rightarrow$ (1) in the $n=1$ case. ∎

######  Remark 4.7.

If the characteristic is away from $0$, Drinfeld level structures agree with
corresponding classical level structures. Let
$\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ be such that the
characteristic is away from $0$. By the previous Proposition 4.6, a full level
structure is an isomorphism of étale $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module
schemes over $S$

$\left(\mathfrak{p}^{-n}/\mathcal{O}_{0}\right)^{r}_{S}\xrightarrow{\cong}\underline{\mathcal{E}}[\mathfrak{p}^{n}]$

by. By [Dri87, Proposition 2.2], this is the same as giving a trivialisation
of $\underline{\mathcal{E}}|_{D_{n,S}}$.

###  4.2. Regularity of the moduli stack of shtukas with $M$-structures.

We show the main result on $M$-structures: the corresponding moduli problem
gives rise to a Deligne-Mumford stack, which we show to be regular following
the corresponding result on elliptic curves in [KM85].

######  Proposition 4.8.

Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka over $S$ and let $M$ be a
finitely generated $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module. The functor on
$S$-schemes

$T\mapsto\\{M\text{-structures on $\underline{\mathcal{E}}_{T}$}\\}$

is representable by a finite $S$-scheme locally of finite presentation.
Moreover, it is finite étale over $S$ if $\mathfrak{p}$ is away from the
characteristic of $\underline{\mathcal{E}}$ and can in this case étale locally
on $S$ be represented by the constant $S$-scheme

$S\times\\{\text{injective $\mathcal{O}_{0}$-module homomorphisms
}M\hookrightarrow(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}\\}.$

###### Proof.

We proceed as in the proof of the corresponding assertions for elliptic curves
in [KM85, Proposition 1.6.2, Proposition 1.6.4 and Corollary 3.7.2]. By the
classification of finitely generated modules over principal ideal domains,
there exists an isomorphism of $\mathcal{O}_{0}/\mathfrak{p}^{n}$-modules
$M\cong(\mathcal{O}_{0}/\mathfrak{p}^{n_{1}})\oplus\ldots\oplus(\mathcal{O}_{0}/\mathfrak{p}^{n_{m}})$
for some $m\geq 0$ and integers $1\leq n_{i}\leq n$ for $1\leq i\leq m$. Using
this isomorphism, we find for a scheme $T$ over $S$ that

$\operatorname{Hom}_{\mathcal{O}_{0}}(M,\underline{\mathcal{E}}[\mathfrak{p}^{n}](T))=\prod_{i}\operatorname{Hom}_{\mathcal{O}_{0}/\mathfrak{p}^{n_{i}}}(\mathcal{O}_{0}/\mathfrak{p}^{n_{i}},\underline{\mathcal{E}}[\mathfrak{p}^{n_{i}}](T))=\prod_{i}\underline{\mathcal{E}}[\mathfrak{p}^{n_{i}}](T).$

The functor of $M$-structures on $\underline{\mathcal{E}}$ is clearly
represented by the closed subscheme of
$\prod_{i}\underline{\mathcal{E}}[\mathfrak{p}^{n_{i}}](T)$ over which the
universal homomorphism defines a $M$-structure on $\underline{\mathcal{E}}$.
This is a closed subscheme defined locally by finitely many equations by Lemma
4.1.

Now assume that the characteristic of $\underline{\mathcal{E}}$ is away from
$0$. By the above, it suffices to show that the scheme is formally étale.
Therefore, let $T_{0}\subseteq T$ be a closed subscheme defined by a locally
nilpotent sheaf of ideals. Let $\iota_{0}$ be a $M$-structure on
$\underline{\mathcal{E}}_{T_{0}}$. Then $\iota_{0}$ factors through
$\underline{\mathcal{E}}[\mathfrak{p}^{n}](T_{0})$. As $\mathfrak{p}$ is away
from the characteristic of $\underline{\mathcal{E}}$, the scheme
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is étale over $S$. We construct a
map $\iota\colon M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](T)$ where we
associate to $m\in M$ the unique lift of $\iota_{0}(m)$ to
$\underline{\mathcal{E}}[\mathfrak{p}^{n}](T)$. As lifts are unique, the
$\mathcal{O}_{0}$-linearity of $\iota_{0}$ implies that $\iota$ is an
$\mathcal{O}_{0}$-linear map as well. We check that $\iota$ defines a
$M$-structure on $\underline{\mathcal{E}}$. Using Proposition 4.6 this can be
done on geometric points, but the geometric points of $T_{0}$ and $T$ agree.

For the second claim, we may assume that
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]\cong(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}_{S}$
by Proposition 3.12. So, the claim follows from Proposition 4.6 (3). ∎

We denote by $\operatorname{Sht}_{r,M}$ the stack parametrising shtukas of
rank $r$ with a $M$-structure as defined above.

######  Theorem 4.9.

Let $M$ be a submodule of $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}$. The stack
$\operatorname{Sht}_{r,M}$ is a regular Deligne-Mumford stack locally of
finite type over ${\mathbb{F}_{q}}$. Moreover, the forgetful map
$\operatorname{Sht}_{r,M}\to\operatorname{Sht}_{r}$ is schematic and finite
flat. It is finite étale away from $0$.

###### Proof.

By Proposition 4.8, the forgetful map to $\operatorname{Sht}_{r}$ is schematic
and finite. In particular, $\operatorname{Sht}_{r,M}$ is a Deligne-Mumford
stack locally of finite type over ${\mathbb{F}_{q}}$ (since
$\operatorname{Sht}_{r}$ is a DM-stack locally of finite type over
${\mathbb{F}_{q}}$). Also by Proposition 4.8, the forgetful map is finite
étale away from $0$.

We proceed as in the proof of [KM85, Theorems 5.1.1 and 5.2.1]. Again since
$\operatorname{Sht}_{r}$ is a smooth DM-stack of dimension $(2r-1)$ over
${\mathbb{F}_{q}}$, we find an étale presentation $S\to\operatorname{Sht}_{r}$
by a $(2r-1)$-dimensional smooth scheme $S$ over ${\mathbb{F}_{q}}$. We denote
by $T=S\times_{\operatorname{Sht}_{r}}\operatorname{Sht}_{r,M}$.

We denote by $U\subseteq S$ the set of points in $s\in S$ such that the local
rings at all points in $T$ over $s$ are regular. Then $U$ is open in $S$, as
its complement is the image under a finite (hence closed) map of the non-
regular locus in $T$, which is closed in $T$ as $T$ is locally of finite type
over a perfect field.

In order to show that $U=S$, it suffices to show that all closed points of $S$
are contained in $U$. As the map $T\to S$ is étale away from $0$, clearly all
points away from $0$ are contained in $U$. It remains to check that all closed
points in the fibre over $0$ are in $U$. By passing to the completion of the
strict henselisation we are reduced to showing that for all $\ell$-valued
points $\bar{s}$ of $S$ in the fibre over $0$, where $\ell$ is an algebraic
closure of ${\mathbb{F}_{q}}$, the complete local rings at all $\ell$-valued
points of $T$ over $\bar{s}$ are regular. Note that the completion of the
strict henselisation at a closed point $s\in S$ over $0$ is then given by
$\widehat{\mathcal{O}_{S,s}^{\mathrm{sh}}}=\widehat{\mathcal{O}}_{S_{\ell\left\llbracket\varpi\right\rrbracket},\overline{s}}$,
where $S_{\ell\left\llbracket\varpi\right\rrbracket}$ denotes the base change
$S\times_{X}\operatorname{Spec}(\ell\left\llbracket\varpi\right\rrbracket)$.

Let us fix some $\ell$-valued point $\bar{s}$ of $S$ over $0$ and let
$\underline{\mathcal{E}}_{0}\in\operatorname{Sht}_{r}(\ell)$ be the
corresponding shtuka. By [Stacks, Tag 07N9], the disjoint union of the spectra
of the completions of all local rings at $\ell$-valued points of
$T_{\ell\left\llbracket\varpi\right\rrbracket}$ over $\overline{s}$ is given
by the scheme

$T^{\prime}=T_{\ell\left\llbracket\varpi\right\rrbracket}\times_{S_{\ell\left\llbracket\varpi\right\rrbracket}}\operatorname{Spec}(\widehat{\mathcal{O}}_{S_{\ell\left\llbracket\varpi\right\rrbracket},\overline{s}}).$

As $S$ is étale over $\operatorname{Sht}_{r}$, we get by [Dri76, Proposition
3.3] that
$\widehat{\mathcal{O}}_{S_{\ell\left\llbracket\varpi\right\rrbracket},\overline{s}}\cong\ell\left\llbracket\varpi,T_{1},\ldots,T_{2r-2}\right\rrbracket$,
which identifies the pullback of the universal shtuka to
$\operatorname{Spec}(\widehat{\mathcal{O}}_{S_{\ell\left\llbracket\varpi\right\rrbracket}})$
with the universal deformation $\underline{\mathcal{E}}$ of
$\underline{\mathcal{E}}_{0}$. In particular,
$T^{\prime}=(\operatorname{Sht}_{r,M})_{\underline{\mathcal{E}}}:=\operatorname{Sht}_{r,M}\times_{\operatorname{Sht}_{r}}\operatorname{Spec}(\widehat{\mathcal{O}}_{S_{\ell\left\llbracket\varpi\right\rrbracket},\overline{s}})$,
where we interpret $\underline{\mathcal{E}}$ as the corresponding
$\operatorname{Spec}(\widehat{\mathcal{O}}_{S_{\ell\left\llbracket\varpi\right\rrbracket},\overline{s}})$-valued
point of $\operatorname{Sht}_{r}^{\prime}$.

Thus, by construction, $T^{\prime}$ only depends on the scheme of
$\mathfrak{p}^{n}$-torsion points of $\underline{\mathcal{E}}$ (and thus on
its local shtuka at $0$ by [Har19, Theorem 7.6]), which in turn by the Serre-
Tate Theorem (Proposition 2.11) only depends on the local shtuka at $0$ of
$\underline{\mathcal{E}}_{0}$, which are classified up to isomorphism by their
Newton polygons. In particular, $\bar{s}$ is contained in $U$ if and only if
$U$ contains all points in the fibre over $0$ in the same Newton stratum. By
Theorem 3.6 it thus suffices to show that $U$ contains a basic point.

Let thus $\bar{s}$ be a basic point in characteristic $\mathfrak{p}$ (recall
that such a point exists by Proposition 3.5) corresponding to a shtuka
$\underline{\mathcal{E}}_{0}$. By Proposition 3.12, we get that
$\underline{\mathcal{E}}_{0}[\mathfrak{p}^{n}](\ell)=\\{0\\}$ for all
$n\in\mathbb{N}$, so in particular the only possible $M$-structure is the zero
map (which is readily checked to be a $M$-structure as $M$ is a submodule of
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}$), so there is exactly one point
lying over $\bar{s}$. This means that $T^{\prime}$ is the spectrum of the
complete local ring pro-representing the deformation functor of the unique
point lying over $\bar{s}$. Note that since $\underline{\mathcal{E}}$ is
basic, the associated divisible module at $0$ is formal (in the sense of
[Dri76, §1] or [HS19, Definition 1.1]). As $M$-structures only depend on the
local shtukas, the Serre-Tate Theorem is also compatible with $M$-structures
and we are thus reduced to showing that the deformation functor of formal
modules with $M$-structures is representable by a $r$-dimensional regular
local ring.

This can be shown as in [Dri76, Proposition 4.3]. We sketch the argument. We
write
$M=(\mathfrak{p}^{-n_{1}}/\mathcal{O}_{0})\oplus\ldots\oplus(\mathfrak{p}^{-n_{r^{\prime}}}/\mathcal{O}_{0})$.
Note that by assumption $r^{\prime}\leq r$. The lemma in the proof of [Dri76,
Proposition 4.3] shows that the deformation functor on formal modules with
$(\mathfrak{p}^{-1}/\mathcal{O}_{0})^{r^{\prime}}$-structure is pro-
represented by a complete regular local ring $R_{1}$ finite flat over
$R_{0}=\ell\left\llbracket\varpi,T_{1},\ldots,T_{r-1}\right\rrbracket$ whose
maximal ideal is generated by
$\iota(e_{1}),\ldots\iota(e_{r^{\prime}}),T_{r^{\prime}},\ldots,T_{r-1}$,
where $e_{1},\ldots,e_{r^{\prime}}$ is a basis of
$(\mathfrak{p}^{-1}/\mathcal{O}_{0})^{r^{\prime}}$ and $\iota$ is the
universal $(\mathfrak{p}^{-1}/\mathcal{O}_{0})^{r^{\prime}}$-structure. This
settles the case that $M$ is $\mathfrak{p}$-torsion. As a next step we show
the claim for $M[\mathfrak{p}^{m}]$ by induction on $m$. Let us assume that a
complete regular local ring $R_{m}$ finite flat over $R_{m-1}$ pro-represents
the deformation functor of formal modules with $M[\mathfrak{p}^{m}]$-level and
that
$\iota_{m}(\varpi^{-\min\\{n_{1},m\\}}),\ldots,\iota_{m}(\varpi^{-\min\\{n_{r^{\prime}},m\\}}),T_{r^{\prime}},\ldots,T_{r}$
forms a system of local parameters for $R_{m}$, where $\iota_{m}$ is the
universal $M[\mathfrak{p}^{m}]$-level structure. Let us denote by
$i_{1},\ldots,i_{j}$ the indices such that $n_{i}>m$. Then

$R_{m+1}=R_{m}\left\llbracket\tilde{T}_{i_{1}},\ldots,\tilde{T}_{i_{j}}\right\rrbracket/(e_{\varpi}(\tilde{T}_{i_{1}})-\iota_{m}(\varpi^{-\min\\{n_{i_{1}},m\\}}),\ldots,e_{\varpi}(\tilde{T}_{i_{j}})-\iota_{m}(\varpi^{-\min\\{n_{i_{j}},m\\}})).$

This also shows that $R_{m+1}$ is regular and finite flat over $R_{m}$.
Moreover, it has a system of local parameters as desired. ∎

The regularity allows us to collect some consequences. We show that a
$M$-structure defines an $\mathcal{O}_{0}$-module subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$.

######  Proposition 4.10.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ and let $\iota\colon
M\to\underline{\mathcal{E}}[\mathfrak{p}^{n}](S)$ be a $M$-structure on
$\underline{\mathcal{E}}$ for some submodule
$M\subseteq(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}$.

  1. (1)

There exists a (necessarily unique) finite locally free closed subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank $|M|$
such that the image of $\iota$ forms a full set of sections for $\mathbf{H}$.

Moreover, for each submodule $M^{\prime}$ of $M$ the restriction of $\iota$ to
$M^{\prime}$ defines a $M^{\prime}$-structure on $\underline{\mathcal{E}}$ and
in particular there exists a finite locally free closed subscheme
$\mathbf{H}_{M^{\prime}}$ of $\underline{\mathcal{E}}$ of rank $M^{\prime}$
such that $\iota|_{M^{\prime}}$ forms a full set of sections for
$\mathbf{H}_{M^{\prime}}$.

  2. (2)

$\mathbf{H}$ is an $\mathcal{O}_{0}$-module subscheme of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ such that the
${\mathbb{F}_{q}}$-module structure on $\mathbf{H}$ is strict.

We call $\mathbf{H}$ the _subscheme generated by $\iota$_ and the map $\iota$
a _$M$ -generator of $\mathbf{H}$_. To be more precise, when we say that a map
$\iota$ is a $M$-generator of a finite locally free closed subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank $|M|$,
we really mean both that $\iota$ gives a full set of sections of $\mathbf{H}$
_and_ that $\iota$ is a $M$-structure on $\underline{\mathcal{E}}$ (recall
that the first condition does not imply the second one, compare Remark 4.4).

For full level structures on Drinfeld modules the assertion is essentially
shown in [Leh09, 3, Proposition 3.3.]. The proposition also implies that for
general $M$-structures on Drinfeld modules our definition agrees with the one
given in [KY20].

###### Proof.

That the restriction to $M^{\prime}$ defines a $M^{\prime}$-structure is
Proposition 4.5 and thus, the second statement in (1) follows from the first.
In order to show both the first part of (1) and (2), we may assume by
reduction to the universal case that $S$ is locally noetherian and flat over
$X^{\prime}$. Both assertions are true away from $0$ by Proposition 4.6. It
thus remains to show that the conditions are closed in both cases. For the
first part of (1) this follows from Lemma 4.1.

For (2) we may additionally assume that $S=\operatorname{Spec}(R)$ is affine
and that we can embed $\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ in
$\mathbb{A}^{1}_{R}$ as the assertion is étale local on $S$ (for the
strictness of the ${\mathbb{F}_{q}}$-action this is [Har19, Lemma 4.4]). The
locus where the group structure on $\underline{\mathcal{E}}$ restricts to a
group structure on $\mathbf{H}$ is closed by the argument of [KM85, Corollary
1.3.7].

By the discussion above, we can write
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]=\operatorname{Spec}(R[t]/(f))$ for
some monic polynomial $f\in R[t]$ and
$\mathbf{H}=\operatorname{Spec}(R[t]/(g))$ for some monic polynomial $g\in
R[t]$ such that $f\in(g)$. Then the $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module
structure restricts to $\mathbf{H}$ if and only if for each
$a\in\mathcal{O}_{0}/\mathfrak{p}^{n}$ the map $e_{a}$ induces a map

${R[t]/(f)}$${R[t]/(f)}$${R[t]/(g)}$${R[t]/(g),}$$\scriptstyle{e_{a}}$

in other words, that $e_{a}(g)(t)=g(e_{a}^{\sharp}(t))\equiv 0\mod g$, where
$e_{a}^{\sharp}(t)\in R[t]$ is a polynomial defining the map $e_{a}$. Thus,
the locus where $\mathbf{H}$ is an $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module
subscheme is the closed subscheme of $\operatorname{Spec}(R)$ where all the
coefficients of the remainders of $g(e_{a}^{\sharp}(t))$ modulo $g$ vanish.
Note that this is clearly independent of the choice of $e_{a}^{\sharp}$.

It remains to show that the locus where the ${\mathbb{F}_{q}}$-action is
strict is closed. As $\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ carries a
strict ${\mathbb{F}_{q}}$-action by construction, we have a lift of the
${\mathbb{F}_{q}}$-action to
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]^{\flat}=\operatorname{Spec}(R[t]/(tf))$
by [HS19, Lemma 4.4]. By the same argument as for the
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module structure, the
${\mathbb{F}_{q}}$-action restricts to a map on the deformation
$\mathbf{H}^{\flat}=\operatorname{Spec}(R[t]/(tg))\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]^{\flat}$.
That it induces the correct operation on the co-Lie complex of
$(\mathbf{H},\mathbf{H}^{\flat})$ is again a closed condition. ∎

We now define $M$-cyclic isogenies.

######  Definition 4.11.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ and let $M$ be a
finitely generated $\mathfrak{p}^{n}$-torsion module.

  1. (1)

A _$M$ -generator_ of a finite locally free subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is a
$M$-structure $\iota$ on $\underline{\mathcal{E}}_{S^{\prime}}$ such that the
subscheme of $\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{S^{\prime}}$ defined
by $\iota$ is $\mathbf{H}_{S^{\prime}}$.

  2. (2)

A finite locally free subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is called _$M$
-cyclic_ if there is an fppf cover $S^{\prime}\to S$ such that
$\mathbf{H}_{S^{\prime}}$ admits a $M$-generator.

  3. (3)

A $\mathfrak{p}^{n}$-isogeny of Drinfeld shtukas
$f\colon\underline{\mathcal{E}}\to\underline{\mathcal{E}}^{\prime}$ is called
_$M$ -cyclic_ if $\operatorname{Dr}_{q}(\operatorname{coker}(f))$ is
$M$-cyclic.

Note that a $M$-cyclic subscheme necessarily has rank $|M|$. We also use the
term _$\mathfrak{p}^{n}$ -cyclic_ as abbreviation for
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})$-cyclic submodule schemes or isogenies,
respectively.

######  Lemma 4.12.

Let $M$ be a submodule of $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}$. Every
$M$-cyclic subscheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ is an
$\mathcal{O}_{0}$-module subscheme with strict ${\mathbb{F}_{q}}$-action.

###### Proof.

All of the assertions can be checked fppf-locally on the base, where they
follow from Lemma 4.10. ∎

We collect two representability results.

######  Proposition 4.13.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ and assume that its
characteristic is away from $0$. Let $M$ be a finitely generated
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module. Then the functor on $S$-schemes

$T\mapsto\\{M\text{-cyclic subgroups of
}\underline{\mathcal{E}}[\mathfrak{p}^{n}]_{T}\\}.$

is representable by a finite étale $S$-scheme. Moreover, étale locally on $S$
the functor is representable by the constant $S$-scheme

$S\times\\{\text{submodules of }(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}\text{
isomorphic to }M\\}.$

###### Proof.

We proceed as in the proof of [KM85, Theorem 3.7.1]. By descent for finite
locally free schemes and closed immersions, and the fact that cyclicity is
local for the fppf-topology by definition, the functor is a fppf (and hence an
étale) sheaf. By étale descent, it thus suffices to show the second statement.
We may assume that
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]\cong(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r}_{S}$
by Proposition 3.12. By the argument in the proof of [KM85, Theorem 3.7.1],
over a connected base $T$ any finite locally free closed subgroup scheme of a
constant scheme is itself constant. The claim follows from the explicit
description in Proposition 4.6. ∎

######  Proposition 4.14.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$, let $M$ be a
finitely generated $\mathcal{O}_{0}/\mathfrak{p}^{n}$-module and let
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ be a finite
locally free closed $\mathcal{O}_{0}$-module subscheme of rank $|M|$. We
consider its functor of generators, i.e. the functor on $S$-schemes

$T\mapsto\\{M\text{-generators of }\mathbf{H}_{T}\text{ in the sense of
Definition \ref{defnIcycSht}}\\}.$

It is representable by a finite scheme of finite presentation over $S$ denoted
by $\mathbf{H}^{\times}$. Moreover, it is finite étale when $\mathbf{H}$ is
étale (in particular, when the characteristic of $\underline{\mathcal{E}}$ is
away from $0$).

###### Proof.

We adapt the proof of [KM85, Proposition 1.6.5]. The functor clearly is
representable by the closed subscheme of $\operatorname{Hom}(M,\mathbf{H})$
over which the universal homomorphism is a $M$-structure on
$\underline{\mathcal{E}}$ (which is a closed condition locally defined by
finitely many equations by Proposition 4.8) and over which the subscheme
defined by the universal homomorphism is $\mathbf{H}$, which is also a closed
condition given by finitely many equations by Lemma 4.1 and [KM85, Corollary
1.3.5].

If $\mathbf{H}$ is étale, we show as in the proof of Proposition 4.8 that
$\mathbf{H}^{\times}$ is formally étale. ∎

###  4.3. Balanced level structures for shtukas.

######  Definition 4.15.

Let $m\in\mathbb{N}$ and let $r_{1},\ldots,r_{m}$ be positive integers such
that $\sum_{i=1}^{m}r_{i}\leq r$. A _balanced $\mathfrak{p}^{n}$-level
structure of type $(r_{1},\ldots,r_{m})$_ on a Drinfeld shtuka
$\underline{\mathcal{E}}$ over $S$ is a chain of isogenies

$\underline{\mathcal{E}}(\mathfrak{p}^{n})=\underline{\mathcal{E}}_{m+1}\xrightarrow{f_{m+1}}\underline{\mathcal{E}}_{m}\xrightarrow{f_{m}}\ldots\xrightarrow{f_{2}}\underline{\mathcal{E}}_{1}\xrightarrow{f_{1}}\underline{\mathcal{E}}_{0}=\underline{\mathcal{E}}$

such that the composition $f_{m+1}\circ\ldots\circ f_{1}$ is the inclusion
$\underline{\mathcal{E}}(\mathfrak{p}^{n})\to\underline{\mathcal{E}}$,
together with $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{i}}$-generators of
$\operatorname{Dr}_{q}(\operatorname{coker}(f_{i}))\subseteq\underline{\mathcal{E}}_{i}[\mathfrak{p}^{n}]$
for all $1\leq i\leq m$ in the sense of Definition 4.2. We denote by
$\operatorname{Sht}_{r,\mathfrak{p}^{n}\text{-bal-}(r_{1},\ldots,r_{m})}$ the
stack parametrising Drinfeld shtukas together with a balanced
$\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$.

A _$\Gamma_{1}(\mathfrak{p}^{n})$ -level structure_ on a Drinfeld shtuka of
rank $r$ is a balanced $\mathfrak{p}^{n}$-level structure of type
$\mathbf{1}_{r}=(1,\ldots,1)\in\mathbb{Z}^{r}$. We denote by
$\operatorname{Sht}_{r,\Gamma_{1}(\mathfrak{p}^{n})}=\operatorname{Sht}_{r,\mathfrak{p}^{n}\text{-bal-}\mathbf{1}_{r}}$
the stack of Drinfeld shtukas with a $\Gamma_{1}(\mathfrak{p}^{n})$-level
structure.