As for Drinfeld shtukas with chains of isogenies, we see that a balanced
$\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$ on a
Drinfeld shtuka $\underline{\mathcal{E}}$ is equivalent to the data of a flag

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

of finite locally free submodule schemes
$\mathbf{H}_{i}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank
$n\cdot(r_{1}+\ldots+r_{i})$ with strict ${\mathbb{F}_{q}}$-action together
with $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{i}}$-generators of
$\mathbf{H}_{i}/\mathbf{H}_{i-1}$ for all $1\leq i\leq m$.

######  Lemma 4.16.

The stack
$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r_{1},\ldots,r_{m})-\mathrm{bal}}$ is
representable by a Deligne-Mumford stack locally of finite type over
${\mathbb{F}_{q}}$. The projection
$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r_{1},\ldots,r_{m})-\mathrm{bal}}\to\operatorname{Sht}_{r}$
is schematic and finite. Moreover, it is finite étale away from $0$.

###### Proof.

We have a well-defined map of stacks

$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r_{1},\ldots,r_{m})-\mathrm{bal}}\to\operatorname{Sht}_{r,(r_{1},\ldots,r_{m})-\mathfrak{p}^{n}\text{-chain}}.$

This map is schematic, finite and moreover finite étale away from $0$ by
Proposition 4.14. The assertions then follow from Proposition 3.15. ∎

######  Proposition 4.17.

The Deligne-Mumford stack
$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r_{1},\ldots,r_{m})-\mathrm{bal}}$ is
regular.

###### Proof.

As in the proof of Theorem 4.9 it suffices to check that the deformation
functor of the $\mathfrak{p}$-divisible module of a basic point over $0$ with
balanced $\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$ is
pro-representable by a regular local ring. By [KM85, Proposition 5.2.2] it
suffices to show that the maximal ideal is generated by $r$ elements.

Let $(\mathbf{G}_{0},(\mathbf{H}_{0,i},\iota_{0,i})_{1\leq i\leq m})$ be the
$\mathfrak{p}$-divisible module of a basic Drinfeld shtuka of rank $r$ over an
algebraically closed field $\ell$ in the fibre over $0$ together with a
balanced $\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$ on
$\mathbf{G}_{0}$. Note that $\mathbf{G}_{0}$ is automatically formal and the
level structure is unique, all the $\iota_{0,i}$ are the zero map. Then by the
Serre-Tate theorem the deformation functor of
$(\mathbf{G}_{0},(\mathbf{H}_{0,i},\iota_{0,i})_{1\leq i\leq m})$ is
representable by a complete local ring denoted by $B$. Let
$(\mathbf{G},((\mathbf{H}_{i}),(\iota_{i}))_{1\leq i\leq m})$ be the universal
deformation over $B$. For every $1\leq i\leq m$ we choose a basis
$e^{(i)}_{1},\ldots,e^{(i)}_{r_{i}}$ of
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{i}}$. We claim that the maximal ideal
of $B$ is generated by $\iota_{i}(e^{(i)}_{j})$ for $1\leq i\leq m$ and $1\leq
j\leq r_{i}$ and $T_{r_{1}+\ldots+r_{m}},\ldots,T_{r-1}$. We proceed as in the
proof of [KM85, Theorem 5.3.2., (5.3.5.)]. We need to check that for every
artin local $\ell\left\llbracket\varpi\right\rrbracket$-algebra $R$ such that
$T_{r_{1}+\ldots+r_{m}},\ldots,T_{r-1}$ vanish in $R$ every deformation
$(\tilde{\mathbf{G}},((\tilde{\mathbf{H}}_{i}),(\tilde{\iota}_{i}))_{1\leq
i\leq m})$ on $R$ such that all $\tilde{\iota}_{i}$ are the constant zero maps
is itself constant.

Using [KM85, Lemma 1.11.3] we see inductively that the zero map is an
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{1}+\ldots+r_{i}}$-structure on
$\mathbf{H}_{i}$ for all $1\leq i\leq m$. In particular, the zero map is an
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{1}+\ldots+r_{i}}$-structure on
$\tilde{\mathbf{G}}$ and thus $\mathbf{G}$ is constant by the proof of Theorem
4.9. ∎

We collect some consequences. We start by constructing balanced level
structures from $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r^{\prime}}$-structures.
Let $m\in\mathbb{N}$ and let $r_{1},\ldots,r_{m}$ be positive integers such
that $r^{\prime}=\sum_{i=1}^{m}r_{i}\leq r$. Let
$(\underline{\mathcal{E}},\iota)$ be a Drinfeld shtuka together with an
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r^{\prime}}$-structure on $S$. For
$1\leq i\leq m$ the restriction of $\iota$ restricted to the first
$r_{1}+\ldots+r_{i}$ components is an
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{1}+\ldots+r_{i}}$-structure by
Proposition 4.5 and thus defines an
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-submodule scheme $\mathbf{H}_{i}$ of
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ by Proposition 4.10. We denote by
$\iota_{i}$ the induced map
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{i}}\to\mathbf{H}_{i}/\mathbf{H}_{i-1}(S)$.

######  Proposition 4.18.

Let $(\underline{\mathcal{E}},\iota)$ be a rank $r$ Drinfeld shtuka together
with an $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r^{\prime}}$-structure over $S$.
Using the notation as above, the flag of finite locally free closed submodule
schemes
$0=\mathbf{H}_{0}\subseteq\mathbf{H}_{1}\subseteq\ldots\subseteq\mathbf{H}_{m}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$
together with the maps $(\iota_{i})_{1\leq i\leq m}$ defines a balanced
$\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$ on
$\underline{\mathcal{E}}$.

###### Proof.

We follow the proof of [KM85, Theorem 5.5.2.]. By reduction to the universal
case and Theorem 4.9 we may assume that $S$ is flat and affine over
$X^{\prime}$. The assertion is clear when the characteristic of
$\underline{\mathcal{E}}$ is away from $0$. The condition that $\iota_{i}$
generates $\mathbf{H}_{i}/\mathbf{H}_{i-1}$ is closed and thus the assertion
follows by flatness. ∎

The proposition can also be applied in the following more general situation.
Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka together with a balanced
$\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$ denoted by
$((\mathbf{H}_{i}),(\iota_{i}))$. Let $1\leq m^{\prime}\leq m$ and for each
$1\leq j\leq m^{\prime}$ let $i_{j}$ and
$r^{\prime}_{i_{1}+\ldots+i_{j-1}+1},\ldots,r^{\prime}_{i_{1}+\ldots+i_{j}}$
be positive integers such that
$\sum_{i=1}^{i_{j}}r^{\prime}_{{i_{1}+\ldots+i_{j-1}+i}}=r_{j}$. By applying
Proposition 4.18 to each $\iota_{j}$ for $1\leq j\leq m^{\prime}$, we obtain a
well-defined balanced $\mathfrak{p}^{n}$-level structure of type
$(r^{\prime}_{1},\ldots,r^{\prime}_{i_{1}+\ldots+i_{m^{\prime}}})$ on
$\underline{\mathcal{E}}$. This construction thus induces a map of stacks

(1)
$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r_{1},\ldots,r_{m})-\mathrm{bal}}\to\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r^{\prime}_{1},\ldots,r^{\prime}_{i_{1}+\ldots+i_{m^{\prime}}})-\mathrm{bal}}.$

######  Corollary 4.19.

The map (1) is finite locally free of constant rank. In particular, fppf-
locally on the base, any balanced $\mathfrak{p}^{n}$-level structure of type
$(r^{\prime}_{1},\ldots,r^{\prime}_{i_{1}+\ldots+i_{m^{\prime}}})$ on
$\underline{\mathcal{E}}$ can be extended to a balanced
$\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$ on
$\underline{\mathcal{E}}$.

###### Proof.

We follow the proof of [KM85, Corollaries 5.5.3. & 5.5.4.] As both the stacks
$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r_{1},\ldots,r_{m})-\mathrm{bal}}$
and
$\operatorname{Sht}_{r,\mathfrak{p}^{n}-(r^{\prime}_{1},\ldots,r^{\prime}_{i_{1}+\ldots+i_{m^{\prime}}})-\mathrm{bal}}$
are regular $r$-dimensional, the map (1) is necessarily finite flat. The
degree can be computed in the étale case, where it is clearly constant. The
second assertion follows immediatly. ∎

######  Corollary 4.20.

Let $\underline{\mathcal{E}}$ be a Drinfeld shtuka over $S$ together with a
balanced $\mathfrak{p}^{n}$-level structure of type $(r_{1},\ldots,r_{m})$
denoted by $(\mathbf{H}_{i},\iota_{i})_{1\leq i\leq m}$. Then $\mathbf{H}_{i}$
is $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{1}+\ldots+r_{i}}$-cyclic.

###### Proof.

We use induction on $i$. For $i=1$ the assertion is clear by definition. Let
now $i>1$ and let us assume that $\mathbf{H}_{i-1}$ is
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{1}+\ldots+r_{i-1}}$-cyclic. As the
question is fppf-local on $S$, we may assume that $\mathbf{H}_{i-1}$ admits a
generator over $S$. Then
$0\subseteq\mathbf{H}_{i-1}\subseteq\mathbf{H}_{i}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$
together with the generators of $\mathbf{H}_{i-1}$ and
$\mathbf{H}_{i}/\mathbf{H}_{i-1}$ defines a balanced $\mathfrak{p}^{n}$-level
structure of type $(r_{1}+\ldots+r_{i-1},r_{i})$ on $\underline{\mathcal{E}}$.
By Corollary 4.19, it can be completed fppf-locally to an
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r_{1}+\ldots+r_{i}}$-structure. But this
exactly means that $\mathbf{H}_{i}$ admits a generator fppf-locally on $S$. ∎

##  5\. Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structures on shtukas

We are now in a position to discuss $\Gamma_{0}(\mathfrak{p}^{n})$-level
structures on Drinfeld shtukas. We closely follow the exposition of [KM85,
Chapter 6] for elliptic curves and adapt the arguments to suit our situation.

###  5.1. Main theorem on $\mathfrak{p}^{n}$-cyclic submodule schemes.

The goal of this section is to show the following analogue of [KM85, Theorems
6.1.1 and 6.4.1].

######  Theorem 5.1 (Main Theorem on $\mathfrak{p}^{n}$-Cyclic Modules).

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ be a Drinfeld shtuka
of rank $r$ over a scheme $S$. Let
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ be a finite
locally free $\mathcal{O}_{0}/\mathfrak{p}^{n}$-submodule scheme of rank
${q}^{n}$ over $S$.

  1. (1)

Suppose that $\mathbf{H}$ is $\mathfrak{p}^{n}$-cyclic and admits a generator
$\iota$. Let $D\subseteq\mathbf{H}$ be the finite locally free subscheme of
$\mathbf{H}$ of rank ${q}^{n-1}({q}-1)$ defined by the restriction of $\iota$
to
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{\times}=(\mathfrak{p}^{-n}/\mathcal{O}_{0})\setminus(\mathfrak{p}^{-(n-1)}/\mathcal{O}_{0})$.
Then $D=\mathbf{H}^{\times}$ as subschemes of $\mathbf{H}$.

  2. (2)

$\mathbf{H}$ is $\mathfrak{p}^{n}$-cyclic if and only if its scheme of
generators $\mathbf{H}^{\times}$ is finite locally free over $S$ of rank

${q}^{n-1}({q}-1).$

  3. (3)

Cyclicity of $\mathbf{H}$ is a closed condition, in the sense that there is a
closed subscheme $W\subseteq S$ locally of finite presentation over $S$ such
that for any $T\to S$ the pullback $\mathbf{H}_{T}$ is
$\mathfrak{p}^{n}$-cyclic if and only if $T\to S$ factors through $W$.

###### Proof.

Assertion (3) follows from (2) by the flattening stratification as in the
proof of [KM85, Theorem 6.4.1]. We sketch the argument. As a first step we
note that in the case where $S=\operatorname{Spec}(k)$ is the spectrum of a
field and $\mathbf{H}$ is not cyclic, we have $\mathbf{H}^{\times}=\emptyset$.
Namely, any generator of $\mathbf{H}$ can be defined over a finite extension
of $k$, but by assumption $\mathbf{H}$ does not admit a generator after any
finite extension of $k$. Hence, $\mathbf{H}^{\times}$ does not have any field
valued points and is thus the empty scheme.

As the question is Zariski-local on $S$ and both $\mathbf{H}$ and
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ are of finite presentation over
$S$, we may assume that $S=\operatorname{Spec}(R)$ is affine and Noetherian.
By the above argument, the maximal rank of $\mathbf{H}^{\times}$ over $S$ is
${q}^{n-1}({q}-1)$. By the flattening stratification, the locus where
$\mathbf{H}^{\times}$ has rank ${q}^{n-1}({q}-1)$ and hence $\mathbf{H}$ is
cyclic by (2) is closed.

It is also clear that (2) follows from (1) as in [KM85, Theorem 6.1.1].
Namely, if $\mathbf{H}^{\times}$ is finite locally free of rank
${q}^{n-1}({q}-1)$, the diagonal map
$\mathbf{H}^{\times}\to\mathbf{H}^{\times}_{\mathbf{H}^{\times}}$ is a section
of $\mathbf{H}^{\times}$ after base change along $\mathbf{H}^{\times}\to S$.
Hence, $\mathbf{H}$ admits a generator after the fppf base change to
$\mathbf{H}^{\times}$ and is thus $\mathfrak{p}^{n}$-cyclic. Conversely,
assume that $\mathbf{H}$ is $\mathfrak{p}^{n}$-cyclic. The question is fppf-
local on $S$, we may thus assume $\mathbf{H}$ admits a generator. The
assertion in this case follows from (1).

It thus remains to show (1). We adapt the proof of [KM85, Theorem 6.1.1]. The
assertion is certainly true when the characteristic of
$\underline{\mathcal{E}}$ is away from $\mathfrak{p}$ by Proposition 4.6.

As a first step we show that $D\subseteq\mathbf{H}^{\times}$. It is clear by
definition that $D\subseteq\mathbf{H}$. By reduction to the universal case and
using Theorem 4.9, it suffices to consider the case when $S$ is Noetherian and
flat over $X^{\prime}$ and as the question is local on $S$, we may further
assume that $S=\operatorname{Spec}(R)$ is affine. It follows that $D$ is then
also flat over $X^{\prime}$. In order to show that
$D\subseteq\mathbf{H}^{\times}$, we show that the tautological section of
$\mathbf{H}$ over $D$ induced by the inclusion $D\hookrightarrow\mathbf{H}$ is
a generator of $\mathbf{H}_{D}$. This is certainly true away from
$\mathfrak{p}$. The claim follows from the flatness of $D$ over $X^{\prime}$
and the fact that the locus where $D\to\mathbf{H}_{D}$ is a generator is
closed in $D$ by Lemma 4.1. Hence, $D\hookrightarrow\mathbf{H}$ factors over
$\mathbf{H}^{\times}$ and the induced map $D\subseteq\mathbf{H}^{\times}$ is
necessarily a closed immersion.

In order to show that the closed immersion
$D\hookrightarrow\mathbf{H}^{\times}$ is an isomorphism, we introduce the
following two auxiliary moduli problems.

$\displaystyle\mathcal{X}_{1}(S)$
$\displaystyle=\left\langle(\underline{\mathcal{E}},\mathbf{H},\iota_{1},P)\colon\begin{array}[]{l}\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S),\
\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]\text{ an
$\mathfrak{p}^{n}$-cyclic submodule scheme,
}\iota_{1}\in\mathbf{H}^{\times}(S),\\\ P\in D(S)\text{ such that }\\{\alpha
P\colon\alpha\in(\mathcal{O}_{0}/\mathfrak{p}^{n})^{\times}\\}\text{ is a full
set of sections for }D\end{array}\right\rangle$
$\displaystyle\mathcal{X}_{2}(S)$
$\displaystyle=\left\langle(\underline{\mathcal{E}},\mathbf{H},\iota_{1},\iota_{2})\colon\begin{array}[]{l}\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S),\
\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]\text{ an
$\mathfrak{p}^{n}$-cyclic submodule scheme,
}\iota_{1},\iota_{2}\in\mathbf{H}^{\times}(S)\end{array}\right\rangle$

It is clear that $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ are stacks, and both
map to $\operatorname{Sht}_{r,(\mathfrak{p}^{-n}/\mathcal{O}_{0})}$ by
forgetting $P$ and $\iota_{2}$, respectively. This maps are clearly schematic
and finite as they are representable by (a closed subscheme of) the finite
schemes $D$ and $\mathbf{H}^{\times}$, respectively. Note that both
$\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ have a unique point lying over a
supersingular point of $\operatorname{Sht}_{r}$ over an algebraically closed
field in characteristic $\mathfrak{p}$.

Since $D\subseteq\mathbf{H}^{\times}$ there is a natural map
$\mathcal{X}_{1}\to\mathcal{X}_{2}$ over
$\operatorname{Sht}_{r,(\mathfrak{p}^{-n}/\mathcal{O}_{0})}$ which is an
isomorphism away from $\mathfrak{p}$ as noted above. We show that the map is
an isomorphism. By an argument as in the proof of Theorem 4.9 (compare also
[KM85, Theorem 6.2.1]), it suffices to check it is an isomorphism at the
completed local rings at the unique points lying over supersingular points
over algebraically closed fields in characteristic $\mathfrak{p}$.

Let $\underline{\mathcal{E}}_{0}\in\operatorname{Sht}_{r}(\ell)$ be a
supersingular rank $r$ Drinfeld shtuka over some algebraically closed field
$\ell$ in characteristic $\mathfrak{p}$ and let $\underline{\mathcal{E}}$ be
its universal formal deformation over
$\tilde{B}=\ell\left\llbracket\varpi,T_{1},\ldots,T_{r-1},T_{r},\ldots,T_{2r-2}\right\rrbracket$.
We denote by
$B=\ell\left\llbracket\varpi,T_{1},\ldots,T_{r-1}\right\rrbracket$. Note that
$B$ pro-represents the deformation functor of the local shtuka of
$\underline{\mathcal{E}}_{0}$ at $\mathfrak{p}$. Recall that
$(\operatorname{Sht}_{r,(\mathfrak{p}^{-n}/\mathcal{O}_{0})})_{\underline{\mathcal{E}}}=\operatorname{Spec}(\tilde{B}_{0})$
is an affine scheme, which is finite over $\tilde{B}$ and that $\tilde{B}_{0}$
is a complete regular noetherian ring by Theorem 4.9. Then
$\mathcal{X}_{1,\underline{\mathcal{E}}}=\operatorname{Spec}(\tilde{B}_{1})$
and
$\mathcal{X}_{2,\underline{\mathcal{E}}}=\operatorname{Spec}(\tilde{B}_{2})$
are affine schemes finite over $\tilde{B}_{0}$ (and therefore also over
$\tilde{B}$). Thus, $\tilde{B}_{1}$ and $\tilde{B}_{2}$ are complete, local
and noetherian rings. We have to check that the map
$\tilde{B}_{2}\to\tilde{B}_{1}$ is an isomorphism. By the Serre-Tate theorem
(which is clearly compatible with all the relevant level structures as they
only depend on the $\mathfrak{p}^{n}$-torsion), we can write
$\tilde{B}_{i}=B_{i}\otimes_{\ell}\ell\left\llbracket T_{r},\ldots
T_{2r-2}\right\rrbracket$ for some complete, local and noetherian rings
$B_{i}$ finite over $B$. Moreover, $B_{0}$ is regular. It clearly suffices to
check that $B_{2}\to B_{1}$ is an isomorphism. Note that since
$D\subseteq\mathbf{H}^{\times}$ is a closed immersion, we obtain that the map
$B_{2}\rightarrow B_{1}$ is surjective.

By Corollary 3.11, we may assume that we can identify
$\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ in an $A$-linear fashion with the
$\mathfrak{p}^{n}$-torsion of a Drinfeld $A$-module $\mathbf{E}$ with trivial
underlying vector bundle (as the base $B$ is local). We have the following
explicit descriptions of the rings $B_{0},B_{1}$ and $B_{2}$. As $B$-algebras
we find

$B_{0}=B[P]/\mathcal{I},$

where $\mathcal{I}$ is the ideal expressing the fact that the map
$\iota\colon\mathfrak{p}^{-n}/\mathcal{O}_{0}\to B[P]=\mathbf{E}(B[P])$
defined by $\varpi^{-n}\mapsto P$ is a well-defined
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})$-structure. More precisely, $\mathcal{I}$
is generated by $e_{\varpi^{n}}(P),e_{a}(P)$, where
$(\varpi^{n},a)=\mathfrak{p}^{n}$ in $A$, (this implies that
$P\in\underline{\mathcal{E}}[\mathfrak{p}^{n}](B_{0})$ and thus that the map
can be extended $A$-linearly to a well-defined map $\iota$) and the equations
defining the condition that
$\sum_{\alpha\in\mathfrak{p}^{-1}/\mathcal{O}_{0}}[\iota(\alpha)]=\sum_{\alpha\in\mathcal{O}_{0}/\mathfrak{p}}[e_{\alpha\varpi^{n-1}}(P)]$
is a subscheme of $\underline{\mathcal{E}}[\mathfrak{p}]$ (this condition is
defined by finitely many equations by [KM85, Lemma 1.3.4.]). Recall that
$B_{0}$ is a regular local ring with maximal ideal generated by
$P,T_{1},\ldots,T_{r-1}$ by the proof of Theorem 4.9. In a similar fashion the
rings $B_{1}$ and $B_{2}$ are given as $B_{0}$-algebras as

$B_{1}=B_{0}[Q]/\mathcal{J},$

where $\mathcal{J}$ is the principal ideal generated by
$\prod_{\alpha\in\mathcal{O}_{0}/\mathfrak{p}^{n}}(Q-e_{\alpha}(P))$, and

$B_{2}=B_{0}[Q]/\mathcal{K},$

where $\mathcal{K}$ is the ideal expressing the fact that $Q$ defines an
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})$-structure as above and defines the same
submodule scheme as $P$, i.e. $\mathcal{K}$ is generated by
$e_{\varpi^{n}}(Q),e_{a}(Q)$, where $(\varpi,a)=\mathfrak{p}$ in $A$, the
equations defining the condition that
$\sum_{\alpha\in\mathcal{O}_{0}/\mathfrak{p}}[e_{\alpha\varpi^{n-1}}(Q)]$ is a
subscheme of $\underline{\mathcal{E}}[\mathfrak{p}]$ and the coefficients of
the polynomial
$\prod_{\alpha\in\mathcal{O}_{0}/\mathfrak{p}^{n}}(t-e_{\alpha}(Q))-\prod_{\alpha\in\mathcal{O}_{0}/\mathfrak{p}^{n}}(t-e_{\alpha}(P))$.

By [KM85, Lemma 6.3.4.], the multiplication by $Q$ on $B_{1}$ is injective. We
denote by $K$ the kernel of the map $B_{2}\to B_{1}$. Applying the snake lemma
to the diagram

${0}$${K}$${B_{2}}$${B_{1}}$${0}$${0}$${K}$${B_{2}}$${B_{1}}$${0,}$

where the vertical maps are given by multiplication by $Q$, yields the short
exact sequence (using the injectivity of multiplication by $Q$ on $B_{1}$)

$0\to K/QK\to B_{2}/QB_{2}\to B_{1}/QB_{1}\to 0.$

By Nakayama’s Lemma $K$ vanishes if and only if $K/QK$ vanishes. It thus
suffices to show that

$B_{2}/QB_{2}\to B_{1}/QB_{1}$

is an isomorphism.

We proceed as in [KM85, Lemma 6.3.5]. From the explicit description of $B_{1}$
and $B_{2}$ above we get that

$B_{1}/QB_{1}=B_{0}/\overline{\mathcal{J}}\qquad\text{and}\qquad
B_{2}/QB_{2}=B_{0}/\overline{\mathcal{K}},$

where $\overline{\mathcal{J}}$ is the ideal generated by
$\prod_{\alpha}e_{\alpha}(P)$ and $\overline{\mathcal{K}}$ is the ideal
generated by the coefficiens of the polynomial
$t^{q^{n}}-\prod_{\alpha}(t-e_{\alpha}(P))$, and the reductions of
$e_{\varpi^{n}}(Q),e_{a}(Q)$, where $(\varpi,a)=\mathfrak{p}$ and the
equations defining the condition that
$\sum_{\alpha\in\mathcal{O}_{0}/\mathfrak{p}}[e_{\alpha\varpi^{n-1}}(Q)]$ is a
subscheme of $\underline{\mathcal{E}}[\mathfrak{p}]$ modulo $Q$. It suffices
to show that $\prod_{\alpha}e_{\alpha}(P)\in\overline{\mathcal{K}}$. We show
that it is (up to multiplication by a unit in $B_{0}$) the coefficient of the
term of degree $q^{n}-q^{n-1}({q}-1)$ of
$t^{q^{n}}-\prod_{\alpha}(t-e_{\alpha}(P))$. This coefficient is the sum of
all $q^{n-1}({q}-1)$-fold products of distinct elements of the set
$\\{e_{\alpha}(P)\colon\alpha\in\mathcal{O}_{0}/\mathfrak{p}^{n}\\}$.

By the definition of Drinfeld modules it follows that $e_{\alpha}(P)$ is of
the form $(\text{unit in }B)\cdot P$ for
$\alpha\in(\mathcal{O}_{0}/\mathfrak{p}^{n})^{\times}$ and of the form
$(\text{elt in }\max(B))\cdot P$ for $\alpha\in\mathfrak{p}$. Thus, both
$\prod_{\alpha}e_{\alpha}(P)$ and the coefficient of the term of degree
$q^{n}-q^{n-1}({q}-1)$ of are of the form $(\text{unit in }B)\cdot
P^{\phi(\mathfrak{p}^{n})}$. ∎

It would be desirable to have a similar statement also for other types of
level structures.

###  5.2. $\Gamma_{0}(\mathfrak{p}^{n})$-level structures on Drinfeld shtukas.

######  Definition 5.2.

A _$\Gamma_{0}(\mathfrak{p}^{n})$ -level structure_ on a Drinfeld shtuka
$\underline{\mathcal{E}}$ over a scheme $S$ is a chain of
$\mathfrak{p}^{n}$-cyclic isogenies

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

such that the composition $f_{r}\circ\ldots\circ f_{1}$ is the inclusion
$\underline{\mathcal{E}}_{0}(\mathfrak{p}^{n})\hookrightarrow\underline{\mathcal{E}}_{0}$.
We denote the stack of Drinfeld shtukas with
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures by
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$.

Using the finite shtuka equivalence, a $\Gamma_{0}(\mathfrak{p}^{n})$-level
structure on $\underline{\mathcal{E}}$ is equivalently given by a flag

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

of finite locally free submodule schemes
$\mathbf{H}_{i}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$ of rank
$n\cdot i$ with strict ${\mathbb{F}_{q}}$-action such that
$\mathbf{H}_{i}/\mathbf{H}_{i-1}$ is $\mathfrak{p}^{n}$-cyclic for all $1\leq
i\leq r$. In particular, a $\Gamma_{0}(\mathfrak{p}^{n})$-level structure can
fppf-locally on the base be extended to a
$\Gamma_{1}(\mathfrak{p}^{n})$-structure by definition. By Proposition 4.18,
such a level structure can fppf-locally be extended to a
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}$-structure on
$\underline{\mathcal{E}}$. We call such an extension a
_$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}$ -generator_ of the
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure.

We can now show one of our main theorems, that Drinfeld
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures produce a regular moduli
problem.

######  Theorem 5.3.

The stack $\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ is a regular
Deligne-Mumford stacks locally of finite type over ${\mathbb{F}_{q}}$. The
forgetful map
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
is schematic and finite flat. It is finite étale away from $0$. Moreover, the
forgetful map
$\operatorname{Sht}_{r,\Gamma_{1}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$
is schematic, faithfully flat and locally of finite presentation.

###### Proof.

We follow the proof of [KM85, Theorem 6.6.1]. As cyclicity is a closed
condition by Theorem 5.1,
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ is the closed substack
of $\operatorname{Sht}_{r,\Gamma_{1}(\mathfrak{p}^{n})}$ over which the
universal isogeny is cyclic. Thus, the forgetful map
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
is schematic and finite. It follows that
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ is a Deligne-Mumford
stack of finite type over ${\mathbb{F}_{q}}$. Moreover, the forgetful map
$\operatorname{Sht}_{r,\Gamma_{1}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$
is representable by the scheme of generators
$(\mathbf{H}_{1}/\mathbf{H}_{0})^{\times}\times(\mathbf{H}_{2}/\mathbf{H}_{1})^{\times}\ldots\times(\mathbf{H}_{r}/\mathbf{H}_{r-1})^{\times}$
and thus in particular finite flat by Theorem 5.1. Since
$\operatorname{Sht}_{r,\Gamma_{1}(\mathfrak{p}^{n})}$ is regular by Theorem
4.17, it follows that $\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ is
also regular by [AK70, VII, Theorem 4.8]. Thus, also the map
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
is finte flat by Miracle Flatness [Mat86, §23]. It is finite étale away from
$0$ by Proposition 4.14. ∎

######  Remark 5.4.

In a similar fashion we can also show the regularity of the moduli stack of
Drinfeld shtukas together with a chain of $\mathfrak{p}^{n}$-cyclic isogenies
of length $r^{\prime}<r$. In other words, a chain of
$\mathfrak{p}^{n}$-isogenies

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

such that $f_{1},\ldots,f_{r^{\prime}}$ are $\mathfrak{p}^{n}$-cyclic. The
following corollaries have also obvious analogues in this setting. Note that
we could generalise the argument to moduli spaces of other kinds of cyclic
isogenies provided we had an analogue of Theorem 5.1.

Using the flatness of our moduli problems, we show that there are well-defined
level maps.

######  Corollary 5.5.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ and let
$(\mathbf{H}_{i})_{i}$ be a $\Gamma_{0}(\mathfrak{p}^{n})$-level structure on
$\underline{\mathcal{E}}$. Let $\underline{n}=(n_{1},\ldots,n_{r-1})$ with
$0\leq n_{r-1}\leq\ldots\leq n_{1}\leq n$. Then there is a canonical subscheme
$\mathbf{H}_{\underline{n}}\subseteq\mathbf{H}_{r-1}$, which is an
$\mathcal{O}_{0}/\mathfrak{p}^{n}$-module subscheme
$\mathbf{H}_{\underline{n}}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{\max\\{n_{i}\\}}]$.
Fppf-locally on $S$, the scheme $\mathbf{H}_{\underline{n}}$ is defined for
any $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}$-generator
$\iota\colon(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}\to\underline{\mathcal{E}}(S)$
of $(\mathbf{H}_{i})_{i}$ by the restriction
$\iota|_{(\mathfrak{p}^{-n_{1}}/\mathcal{O}_{0})\oplus\ldots\oplus(\mathfrak{p}^{-n_{r-1}}/\mathcal{O}_{0})}$
as in Proposition 4.10.

###### Proof.

We follow the proof of [KM85, Theorem 6.7.2]. It suffices to construct
$\mathbf{H}_{\underline{n}}$ fppf-locally on $S$. We may thus assume that
$(\mathbf{H}_{i})_{i}$ admits a
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}$-generator. Let $\iota$ and
$\iota^{\prime}$ be two such
$(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}$-generators of
$(\mathbf{H}_{i})_{i}$. By Proposition 4.10 the restrictions to
$(\mathfrak{p}^{-n_{1}}/\mathcal{O}_{0})\oplus\ldots\oplus(\mathfrak{p}^{-n_{r-1}}/\mathcal{O}_{0})$
of both $\iota$ and $\iota^{\prime}$ define closed submodule schemes
$\mathbf{H}_{\underline{n}},\mathbf{H}^{\prime}_{\underline{n}}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{m}]$.
We have to check
$\mathbf{H}_{\underline{n}}=\mathbf{H}^{\prime}_{\underline{n}}$.

By reduction to the universal case we may assume that $S$ is noetherian and
flat over $X^{\prime}$, as the moduli space of Drinfeld shtukas with
$\Gamma_{0}(\mathfrak{p}^{n})$-level together with two generators is given by

$\operatorname{Sht}_{r,(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}}\times_{\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}}\operatorname{Sht}_{r,(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}},$

which is flat over $X^{\prime}$ by Theorem 5.3 and Corollary 4.19. In this
case equality of closed subschemes of
$\underline{\mathcal{E}}[\mathfrak{p}^{m}]$ is a closed condition by [KM85,
Lemma 6.7.3]. The assertion is clear away from $0$ and thus follows from the
flatness of $S$ in the general case. ∎

Via the finite shtuka equivalence and Proposition 2.14 the submodule scheme
$\mathbf{H}_{\underline{n}}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$
corresponds to a $\mathfrak{p}^{n}$-isogeny
$\underline{\mathcal{E}}_{\underline{n}}\hookrightarrow\underline{\mathcal{E}}$.
For $1\leq m\leq n$ we denote by $\underline{m}^{(i)}=(m,\ldots,m,0,\ldots,0)$
with $i$ non-zero entries. Then

$0=\mathbf{H}_{\underline{m}^{(0)}}\subseteq\mathbf{H}_{\underline{m}^{(1)}}\subseteq\ldots\subseteq\mathbf{H}_{\underline{m}^{(r-1)}}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{m}]$

is a $\Gamma_{0}(\mathfrak{p}^{m})$ level structure on
$\underline{\mathcal{E}}$. This shows that we have a well-defined level map
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{m})}$
for all $0\leq m\leq n$ which is automatically finite flat. We show that we
can also construct this level map by taking closures without making explicit
reference to the generators.

######  Corollary 5.6.

Let $S$ be a scheme which is flat over $X^{\prime}$. Let
$\underline{\mathcal{E}}$ be a Drinfeld shtuka over $S$ and Let
$(\mathbf{H}_{i})_{i}$ be a $\Gamma_{0}(\mathfrak{p}^{n})$-level structure on
$\underline{\mathcal{E}}$. For every $1\leq i\leq r$ the canonical submodule
scheme $\mathbf{H}_{\underline{m}^{(i)}}\subseteq\mathbf{H}_{i}$ is the
schematic closure of
$\mathbf{H}_{i}|_{S\times_{X^{\prime}}(X^{\prime}\setminus\\{0\\})}[\mathfrak{p}^{m}]$
in $\underline{\mathcal{E}}[\mathfrak{p}^{m}]$.

###### Proof.

From the explicit descriptions away from $\mathfrak{p}$ it is clear that
$\mathbf{H}_{\underline{m}^{(i)}}$ is given by the $\mathfrak{p}^{m}$-torsion
of $\mathbf{H}_{i}$ away from $0$. The assertion then follows from the fact
that $\mathbf{H}_{\underline{m}^{(i)}}$ is flat over $S$ and closed in
$\underline{\mathcal{E}}{[\mathfrak{p}^{m}]}$. ∎

Motivated by the discussion in Section 2.5, we also construct additional level
maps. Recall that the $\Gamma_{0}(\mathfrak{p}^{n})$-level corresponds to a
standard $(r-1)$-simplex $\Omega$ of sidelength $n$ in the standard appartment
of the Bruhat-Tits building of $\operatorname{GL}_{r}$. We want to construct
level maps corresponding to inclusions of sub-$(r-1)$-simplices (of smaller
sidelength). Recall that we enumerated alcoves in the standard apartment by
its basepoint $\underline{m}$ and its orientation given by a permutation
$\tau$. In a similar fashion, a $(r-1)$-subsimplex of $\Omega$ is determined
by its basepoint $\underline{m}=(m_{1},\ldots,m_{r-1})$ with
$m_{1}\geq\ldots\geq m_{r-1}$, a sidelength $n^{\prime}$ and an orientation
given by some $\tau\in\operatorname{Sym}_{r-1}$. Note that the simplex with
basepoint $\underline{m}$, sidelength $\tilde{n}$ and orientation
$\tau\in\operatorname{Sym}_{r-1}$ is contained in $\Omega$ if and only if
$\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}\prec\Omega$. Let us denote
by $\underline{\tilde{n}}_{\tau}^{(i)}\in\mathbb{Z}^{r-1}$ the vector
containig $\tilde{n}$ in entries $\tau(1),\ldots,\tau(i)$ and 0 otherwise.

######  Corollary 5.7.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ and let
$(\mathbf{H}_{i})_{i}$ be a $\Gamma_{0}(\mathfrak{p}^{n})$-level structure on
$\underline{\mathcal{E}}$. Let $\underline{m}=(m_{1},\ldots,m_{r-1})$ with
$0\leq m_{r-1}\leq\ldots\leq m_{1}\leq n$. Let $0\leq\tilde{n}\leq n$ such
that $\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}\prec\Omega$ for all
$i$. Then the flag of quotients

$0\subseteq\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(1)}}/\mathbf{H}_{\underline{m}}\subseteq\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(2)}}/\mathbf{H}_{\underline{m}}\subseteq\ldots\subseteq\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(r-1)}}/\mathbf{H}_{\underline{m}}\subseteq\underline{\mathcal{E}}_{\underline{m}}[\mathfrak{p}^{\tilde{n}}]$

defines a $\Gamma_{0}(\mathfrak{p}^{\tilde{n}})$-level structure on the
Drinfeld shtuka $\underline{\mathcal{E}}_{\underline{m}}$, which we denote by
$(\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}})_{i}$.
In case that $(\mathbf{H}_{i})_{i}$ admits a
$(\mathfrak{p}^{-n}/A)^{r-1}$-generator $\iota$,
$(\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}})_{i}$
is generated by

$\displaystyle\iota_{\underline{m},\tilde{n},\tau}\colon(\mathfrak{p}^{-\tilde{n}}/A)^{r-1}$
$\displaystyle\cong\left({(\mathfrak{p}^{-m_{\tau(1)}-\tilde{n}}/A)\oplus\ldots\oplus(\mathfrak{p}^{-m_{\tau(r-1)}-\tilde{n}}/A)}\right)/\left({(\mathfrak{p}^{-m_{\tau(1)}}/A)\oplus\ldots\oplus(\mathfrak{p}^{-m_{\tau(r-1)}}/A)}\right)$
$\displaystyle\to\mathbf{H}_{\underline{m}+\underline{\tilde{n}}^{(r-1)}}/\mathbf{H}_{\underline{m}}(S).$

Moreover, the canonical subscheme from Corollary 5.5 for
$\underline{m^{\prime}}=(m^{\prime}_{1},\ldots,m^{\prime}_{r-1})$ with
$\tilde{n}\geq m^{\prime}_{1}\geq\ldots\geq m^{\prime}_{r-1}\geq 0$ is given
by

$(\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}})_{\underline{m}^{\prime}}\cong\mathbf{H}_{\underline{m}+\tau(\underline{m}^{\prime})}/\mathbf{H}_{\underline{m}}.$

###### Proof.

We follow the proof of [KM85, Theorem 6.7.4]. The question is fppf-local on
$S$, so we can assume that $\mathbf{H}$ has a generator. By reduction to the
universal case, we may further assume that $S$ is flat over $X^{\prime}$ and
noetherian by Theorem 4.9. Note that all assertions are clear away from $0$.
It thus suffices to show that the locus, where they are satisfied is closed.

The locus where each
$\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i-1)}}$
is ${\mathfrak{p}^{\tilde{n}}}$-cyclic is closed in $S$ by Theorem 5.1 (3).
This shows the first claim. For the second claim, the locus where
$\iota_{\underline{m},\tilde{n},\tau}$ is a generator of
$(\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}})_{i}$
is closed by [KM85, Proposition 1.9.1]. Moreover, the condition that
$(\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}})_{\underline{m}^{\prime}}\cong\mathbf{H}_{\underline{m}+\tau(\underline{m}^{\prime})}/\mathbf{H}_{\underline{m}}$
is closed by [KM85, Lemma 6.7.3]. This shows the last claim. ∎

Associating the $\Gamma_{0}(\mathfrak{p}^{\tilde{n}})$-level structure
$(\mathbf{H}_{\underline{m}+\underline{\tilde{n}}_{\tau}^{(i)}}/\mathbf{H}_{\underline{m}})_{i}$
on $\underline{\mathcal{E}}_{\underline{m}}$ to $(\mathbf{H}_{i})_{i}$ as in
the previous corollary defines a map of stacks

$F_{\underline{m},\tilde{n},\tau}\colon\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{\tilde{n}})}.$

######  Proposition 5.8.

The level map $F_{\underline{n},m,\tau}$ is schematic and finite locally free.

###### Proof.

Note that by Theorem 5.3 the map $F_{\underline{0},0}$ is schematic and finite
locally free. As a first step we show that
$F_{\underline{n},0}\colon\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
is schematic and finite locally free for all $\underline{m}$. In order to show
that the map is representable by a finite scheme, we consider the auxiliary
moduli problem
$\operatorname{Sht}_{r,\underline{m}-\text{isog},\Gamma_{0}(\mathfrak{p}^{n})}$
parametrising a Drinfeld shtuka $\underline{\mathcal{E}}$, a
$\mathfrak{p}^{n}$-isogeny
$f\colon\underline{\mathcal{E}}\hookrightarrow\underline{\mathcal{E}}^{\prime}$
such that $\operatorname{coker}(f)$ has rank $\sum_{i=1}^{r-1}m_{i}$ as
$\mathcal{O}_{S}$-module, and a $\Gamma_{0}(\mathfrak{p}^{n})$-level structure
$(\mathbf{H}_{i})_{i}$ on $\underline{\mathcal{E}}^{\prime}$. The projection
to $\underline{\mathcal{E}}$ defines then a map of stacks
$\operatorname{Sht}_{r,\underline{m}-\text{isog},\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
which is schematic and finite by Proposition 3.15 and Theorem 5.3. We also
have a map
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\underline{m}-\text{isog},\Gamma_{0}(\mathfrak{p}^{n})}$
sending $(\underline{\mathcal{E}},(\mathbf{H}_{i})_{i})$ to
$(\underline{\mathcal{E}}_{\underline{m}},\underline{\mathcal{E}},(\mathbf{H}_{i})_{i})$,
which identifies $\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ with
the substack of
$\operatorname{Sht}_{r,\underline{m}-\text{isog},\Gamma_{0}(\mathfrak{p}^{n})}$
where
$\underline{\mathcal{E}}=\underline{\mathcal{E}}^{\prime}_{\underline{m}}$. By
[KM85, Lemma 6.7.3], this is schematic and representable by a closed
immersion. The composition of the maps
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\underline{m}-\text{isog},\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r}$
is clearly given by $F_{\underline{m},0}$, which is thus schematic and finite.

Note that we have a commutative diagram

${\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}}$${\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{\tilde{n}})}}$${\operatorname{Sht}_{r}}$$\scriptstyle{F_{\underline{m},\tilde{n}}}$$\scriptstyle{F_{\underline{m},0}}$$\scriptstyle{F_{\underline{0},0}}$

with vertical arrows that are schematic and finite. In order to see that
$F_{\underline{m},\tilde{n},\tau}$ is schematic we argue as follows. We fix a
map $S^{\prime}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{\tilde{n}})}$
from some $S$-scheme $S^{\prime}$, in other words a Drinfeld shtuka
$(\underline{\mathcal{E}},(\mathbf{H}_{i})_{i})$ together with a
$\Gamma_{0}(\mathfrak{p}^{\tilde{n}})$-level structure. By composition with
$F_{\underline{0},0}$, we get a map $S^{\prime}\to\operatorname{Sht}_{r}$. By
the discussion above,
$S^{\prime\prime}=S^{\prime}\times_{\operatorname{Sht}_{r}}\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$
is representable by a finite $S$-scheme. Let
$(\underline{\mathcal{E}}^{\prime},(\mathbf{H}^{\prime}_{i})_{i})$ denote the
corresponding $\Gamma_{0}(\mathfrak{p}^{n})$-level structure. Then the fibre
product
$S^{\prime}\times_{\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{\tilde{n}})}}\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$
is the locus where the image of
$(\underline{\mathcal{E}}^{\prime},(\mathbf{H}^{\prime}_{i})_{i})$ under
$F_{\underline{m},\tilde{n},\tau}$ is given by
$(\underline{\mathcal{E}},(\mathbf{H}_{i})_{i})$. By [KM85, Lemma 6.7.3], this
is representable by a closed subscheme of $S^{\prime\prime}$.

As both $F_{\underline{m},0}$ and $F_{\underline{0},0}$ are finite, it is
immediate that $F_{\underline{m},\tilde{n},\tau}$ is finite as well. As both
$\operatorname{Sht}_{r.\Gamma_{0}(\mathfrak{p}^{n})}$ and
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{\tilde{n}})}$ are regular and
$(2r-1)$-dimensional, the level map is flat by miracle flatness. ∎

##  6\. Comparison with naive level structures and Bruhat-Tits theory

We compare the Drinfeld level structures defined above with naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures. The naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structures seem inadequate when $n>1$ as
the fibre above $0$ is missing points (compare Remark 2.20). We construct a
map from the stack of Drinfeld shtukas with naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure to our stack of Drinfeld
shtukas with Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level which is an open
immersion and an isomorphism away from $0$. Moreover, we show that the two
notions of level structures agree in the parahoric case. In this sense, the
Drinfeld level structures provide a compactification of the level maps.

Recall that we defined a naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structure
on a Drinfeld shtuka $\underline{\mathcal{E}}=(\mathcal{E},\varphi)$ of rank
$r$ as a flag of quotients as $\mathfrak{p}^{n}$-torsion finite shtukas

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

such that $\mathcal{L}_{i}$ is finite locally free of rank $i$ as
$\mathcal{O}_{D_{n,S}}$-module. Equivalently, using Proposition 2.14, a naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure is a chain of
$\mathfrak{p}^{n}$-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.

######  Lemma 6.1.

Let $\underline{\mathcal{E}}=(\mathcal{E},\varphi)$ be a Drinfeld shtuka of
rank $r$ over $S$ and let
$\mathcal{E}|_{D_{n,S}}\twoheadrightarrow\mathcal{L}$ be a quotient
$\mathfrak{p}^{n}$-torsion finite shtuka such that $\mathcal{L}$ is finite
locally free of rank 1 as $\mathcal{O}_{D_{n,S}}$-module. Then
$\operatorname{Dr}_{q}(\underline{\mathcal{L}})\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$
is an $\mathfrak{p}^{n}$-cyclic submodule scheme.

###### Proof.

We denote by
$\underline{\mathcal{L}}^{(i)}=\underline{\mathcal{L}}|_{D_{i,S}}$ for $1\leq
i\leq n$. Then $\underline{\mathcal{L}}^{(i)}$ is a locally free
$\mathcal{O}_{D_{i,S}}$-module of rank 1, and consequently a locally free
$\mathcal{O}_{S}$-module of rank $i$. Thus,

$\underline{\mathcal{L}}=\underline{\mathcal{L}}^{(n)}\twoheadrightarrow\underline{\mathcal{L}}^{(n-1)}\twoheadrightarrow\ldots\twoheadrightarrow\underline{\mathcal{L}}^{(2)}\twoheadrightarrow\underline{\mathcal{L}}^{(1)}\twoheadrightarrow
0$

corresponds via the finite shtuka equivalence to a flag of finite locally free
submodule schemes with strict ${\mathbb{F}_{q}}$-action

$0\subseteq\mathbf{H}^{(1)}\subseteq\ldots\subseteq\mathbf{H}^{(n-1)}\subseteq\mathbf{H}^{(n)}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}],$

where we denote by
$\mathbf{H}^{(i)}=\operatorname{Dr}_{q}(\underline{\mathcal{L}^{(i)}})$. It is
clear that $\mathbf{H}^{(i)}\subseteq\mathbf{E}[\mathfrak{p}^{i}]$ by
construction. As a next step, we inductively construct a generator of
$\mathbf{H}^{(i)}$ fppf-locally on $S$.

We may assume that $S=\operatorname{Spec}(R)$ is affine and that $\mathcal{L}$
is a free $\mathcal{O}_{D_{n,S}}=R[\varpi]/(\varpi^{n})$-module of rank 1.
Then, $\mathcal{L}^{(i)}\cong R[\varpi]/(\varpi^{i})$. We choose the standard
basis $1,\varpi,\ldots,\varpi^{i-1}$ of $\mathcal{L}^{(i)}$ as $R$-module. As
a map of finite free $R[\varpi]/(\varpi^{i})$-modules, $\varphi$ is given by
multplication by an element $\alpha=\sum_{j=0}^{i-1}\alpha_{j}\varpi^{j}\in
R[\varpi]/(\varpi^{i})$, and thus, its matrix as an $R$-linear map with
respect to the standard basis is given by

$\begin{pmatrix}\alpha_{0}&&&\\\ \alpha_{1}&\alpha_{0}&&\\\ \vdots&&\ddots\\\
\alpha_{i}&\alpha_{i-1}&\ldots&\alpha_{0}\end{pmatrix}.$

It follows that

$\mathbf{H}^{(i)}=\operatorname{Dr}_{q}(\underline{\mathcal{L}}^{(i)})=\operatorname{Spec}\left(R[t_{0},\ldots,t_{i-1}]/(t_{0}^{q}-\sum_{j=0}^{i-1}\alpha_{j}t_{j},t_{1}^{q}-\sum_{j=1}^{i-1}\alpha_{j-1}t_{j},\ldots,t_{i-1}^{q}-\alpha_{0}t_{i-1})\right).$

As the question is fppf-local on $R$, we may assume that $R$ contains a root
$\beta_{0}$ of the polynomial $t^{q-1}-\alpha_{0}$, a root $\beta_{1}$ of the
polynomial $t^{q}-\alpha_{0}t_{1}-\alpha_{1}\beta_{0}$ and inductively a root
$\beta_{j}$ of the polynomial
$t^{q}-\alpha_{0}t-\alpha_{1}\beta_{j-1}-\ldots-\alpha_{j}\beta_{0}$ for all
$0\leq j\leq i-1$. Then $(\beta_{i-1},\beta_{i-2},\ldots,\beta_{1},\beta_{0})$
is a section of $\mathbf{H}^{(i)}$ over $R$ by construction. We claim that the
map

$\displaystyle\iota^{(i)}\colon\mathfrak{p}^{-i}/\mathcal{O}_{0}$
$\displaystyle\to\mathbf{H}^{(i)}(R)$ $\displaystyle\varpi^{-i}$
$\displaystyle\mapsto(\beta_{i-1},\beta_{i-2},\ldots,\beta_{1},\beta_{0})$

is a $\mathfrak{p}^{-i}/\mathcal{O}_{0}$-generator of $\mathbf{H}^{(i)}$. We
proceed by induction on $i$.

Let $i=1$. In this case
$\mathbf{H}^{(1)}=\operatorname{Spec}(R[t]/(t_{0}^{q}-\alpha_{0}t_{0}))$. In
particular, $\mathbf{H}^{(1)}$ can be embedded in $\mathbb{A}^{1}_{R}$. Then
$\mathfrak{p}^{-1}/\mathcal{O}_{0}\to\mathbf{H}^{(1)}(R)$ given by
$\varpi^{-1}\mapsto\beta$ is a generator of $\mathbf{H}^{(1)}$, as
$\prod_{a\in{\mathbb{F}_{q}}}(t-a\beta)=t^{q}-\beta^{q-1}t=t^{q}-\alpha t$.

Let us now assume that the claim is true for $i\geq 1$. Note that the
subscheme $\mathbf{H}^{(i)}\subseteq\mathbf{H}^{(i+1)}$ is given by the locus
where $t_{i}=0$ by construction. Note that the map
$\iota^{(i+1)}|_{\mathfrak{p}^{-i}/\mathcal{O}_{0}}=\iota^{(i)}$ is given by
$\varpi^{-i}\mapsto(\beta_{i-1},\beta_{i-2},\ldots,\beta_{1},\beta_{0},0)$.
Thus, it factors through $\mathbf{H}^{(i)}$ and is a generator of
$\mathbf{H}^{(i)}$ by hypothesis.

Moreover, the quotient $\mathbf{H}^{(i+1)}/\mathbf{H}^{(i)}$ is then given by
the canonical inclusion

$R[t_{i}]/(t_{i}^{q}-\alpha_{0}t_{i})\to
R[t_{0},\ldots,t_{i-1},t_{i}]/(t_{0}^{q}-\sum_{j=0}^{i}\alpha_{j}t_{j},\ldots,t_{i}^{q}-\alpha_{0}t_{i}).$

Moreover, the image of the section $(\beta_{0},\ldots,\beta_{i-1})$ of
$\mathbf{H}^{(i+1)}$ in the quotient $\mathbf{H}^{(i+1)}/\mathbf{H}^{(i)}$ is
$\beta_{0}$. In particular, the map

$\iota^{(i+1)}\\!\mod\mathfrak{p}^{(i)}\colon\mathfrak{p}^{-1}/\mathcal{O}_{0}\cong(\mathfrak{p}^{-(i+1)}/\mathcal{O}_{0})/(\mathfrak{p}^{-i}/\mathcal{O}_{0})\to\left(\mathbf{H}^{(i+1)}/\mathbf{H}^{(i)}\right)(R)$

is well-defined and sends $\varpi^{-1}$ to $\beta_{0}$ and is thus a generator
of $\mathbf{H}^{(i+1)}/\mathbf{H}^{(i)}$ by the discussion of the case $i=1$
above. By [KM85, Lemma 1.11.3] it follows that $\iota^{{}^{(i+1)}}$ is a full
set of sections of $\mathbf{H}^{(i+1)}$.

Thus, $(\mathbf{H}^{(1)},\iota)$ is a generator of $\mathbf{H}^{(i+1)}$ in the
sense of Definition 4.2 and $\mathbf{H}^{(i+1)}$ is
$\mathfrak{p}^{i+1}$-cyclic. This shows the claim. ∎

######  Proposition 6.2.

Let $\underline{\mathcal{E}}=(\mathcal{E},\varphi)$ be a Drinfeld shtuka over
$S$ and let

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

be a naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structure on
$\underline{\mathcal{E}}$. Then

$0\subseteq\operatorname{Dr}_{q}(\underline{\mathcal{L}}_{1})\subseteq\ldots\operatorname{Dr}_{q}(\underline{\mathcal{L}}_{r-1})\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}]$

is a Drinfeld $\Gamma_{0}(\mathfrak{p}^{n})$-level structure on
$\underline{\mathcal{E}}$ in the sense of Definition 5.2.

###### Proof.

This follows directly from Lemma 6.1. ∎

Recall that a Drinfeld shtuka with a naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure is a bounded global
$\operatorname{GL}_{r,\Omega}$-shtuka for the Bruhat-Tits group scheme
$\operatorname{GL}_{r,\Omega}$ as defined in Remark 2.19. In particular, we
thus constructed a map of Deligne-Mumford stacks

(2)
$\operatorname{Sht}_{r,\Omega}\to\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}.$

As a next step, we show that the map (2) is an isomorphism in the case $n=1$.

######  Proposition 6.3.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$. Then every
$\Gamma_{0}(\mathfrak{p})$-level structure on $\underline{\mathcal{E}}$ comes
from a naive $\Gamma_{0}(\mathfrak{p})$-level structure.

###### Proof.

It suffices to show that for an $\mathfrak{p}$-cyclic submodule scheme
$\mathbf{H}\subseteq\underline{\mathcal{E}}[\mathfrak{p}]$, the corresponding
finite shtuka $\underline{M}_{q}(\mathbf{H})$ is finite locally free of rank 1
as $R\otimes\mathcal{O}_{0}/\mathfrak{p}\cong R$-module. But this is clear by
construction. ∎

######  Lemma 6.4.

Let $\underline{\mathcal{E}}\in\operatorname{Sht}_{r}(S)$ and assume its
characteristic is away from $0$. Then every
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure on $\underline{\mathcal{E}}$
comes from a naive $\Gamma_{0}(\mathfrak{p}^{n})$-level structure.

###### Proof.

Let $(\mathbf{H}_{i})_{1\leq i\leq r}$ be a
$\Gamma_{0}(\mathfrak{p}^{n})$-structure on $\underline{\mathcal{E}}$. As the
characteristic of $\underline{\mathcal{E}}$ is away from $0$, all the
$\mathbf{H}_{i}$ are finite étale over $S$. As the claim is fppf-local on the
base, we may choose a $(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}$-generator of
$(\mathbf{H}_{i})_{1\leq i\leq r}$. By Proposition 4.6, the $\mathbf{H}_{i}$
are then given by

$0\subseteq(\mathfrak{p}^{-n}/\mathcal{O}_{0})_{S}\subseteq(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{2}_{S}\subseteq\ldots\subseteq(\mathfrak{p}^{-n}/\mathcal{O}_{0})^{r-1}_{S}\subseteq\underline{\mathcal{E}}[\mathfrak{p}^{n}].$

By the finite shtuka equivalence, this corresponds to the flag of quotients

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

where the map $\mathcal{O}_{D_{n,S}}^{i+1}\to\mathcal{O}_{D_{n,S}}^{i}$ is
given by the projection to the first $i$ components and the Frobenius-linear
map on $\mathcal{O}_{D_{n,S}}^{i}$ is the trivial one. This is clearly a naive
$\Gamma_{0}(\mathfrak{p}^{n})$-level structure. ∎

######  Proposition 6.5.

The map (2) is schematic and a quasi-compact open immersion.

###### Proof.

By construction, $\operatorname{Sht}_{r,\Omega}$ is identified with the
substack of $\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ where all
the $\mathbf{H}_{i}$ correspond to finite locally free
$\mathcal{O}_{D_{n,S}}$-modules of rank $i$ via the finite shtuka equivalence,
or equivalently the locus, where all $\mathbf{H}_{i}/\mathbf{H}_{i-1}$
correspond to finite locally free $\mathcal{O}_{D_{n,S}}$-modules of rank $1$.

In order to show that this condition is representable by an open subscheme, we
work locally and assume that $S=\operatorname{Spec}(R)$ is affine. Let $M$ be
a $R[\varpi]/(\varpi^{n})$-module such that $M$ is free of rank $n$ as
$R$-module. Then $M$ is locally free of rank 1 as
$R[\varpi]/(\varpi^{n})$-module if and only if it is generated by a single
element.

Let $q\subseteq R$ be a prime ideal such that $M\otimes\kappa(q)$ is a one-
dimensional vector space over the residue field $\kappa(q)$ of
$R[\varpi]/(\varpi^{n})$ at $(q,\varpi)$ ($\kappa(q)$ is also the residue
field of $R$ at $q$). By Nakayama’s Lemma, there exists a
$a\in(R[\varpi]/(\varpi^{n}))\setminus(q,\varpi)$ such that $M[a^{-1}]$ is
free of rank 1. Let $a_{0}=a(0)$ be the constant term of $a$. Then
$M[a^{-1}]=M[a_{0}^{-1}]$. Hence, the principal open
$D(a_{0})\subseteq\operatorname{Spec}(R)$ is an open neighbourhood of $q$ such
that $M[a_{0}^{-1}]$ is locally free of rank 1 as
$R[a_{0}^{-1}][\varpi]/\varpi^{n}$-module over $D(a)$. Hence, the condition is
representable by an open immersion on the base scheme. ∎

By Propositions 5.8 and 6.3, we can interpret the level maps to
$\Gamma_{0}(\mathfrak{p})$-level structures as maps

$F_{\underline{m},1,\tau}\colon\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\operatorname{Sht}_{r,\mathfrak{f}_{\underline{m},\tau}},$

where $\mathfrak{f}_{\underline{m},\tau}$ is the alcove in the Bruhat-Tits
building corresponding to $\underline{m}$ and $\tau$. This system of maps is
compatible with level maps to parahoric levels given by smaller facets by
Corollary 5.5 and thus define a map

(3)
$F_{\Omega}\colon\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}.$

######  Proposition 6.6.

The map $F_{\Omega}$ is a closed immersion.

###### Proof.

As all the maps
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
are schematic and finite, so is their limit. Moreover, by the explicit moduli
description it is clear that the map $F_{\Omega}$ is a monomorphism. ∎

######  Theorem 6.7.

The map
$\operatorname{Sht}_{r,\Omega}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
is schematic and representable by a quasi-compact open immersion that is 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.

###### Proof.

The assertion for the parahoric case is Proposition 6.3. That the inclusion

$\operatorname{Sht}_{r,\Omega}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$

is schematic and representable by a quasi-compact locally closed immersion
follows from Propositions 6.5 and 6.6. That the image of
$\operatorname{Sht}_{r,\Omega}$ in
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ is dense follows from
the fact that the inclusion (2) is an isomorphism away from $0$ by Lemma 6.4
together with the flatness of
$\operatorname{Sht}_{r,\Gamma_{0}(\mathfrak{p}^{n})}$ over $X^{\prime}$ from
Theorem 5.3.

In order to see that the map
$\operatorname{Sht}_{r,\Omega}\to\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
is already open, we follow the proof of Proposition 6.5. One can again check
that a point
$(\mathcal{E}_{\underline{m}})_{\underline{m}}\in\varprojlim_{\mathfrak{f}\prec\Omega}\operatorname{Sht}_{r,\mathfrak{f}}$
comes from $\operatorname{Sht}_{\Omega}$ if and only if the cokernels of the
isogenies
$\underline{\mathcal{E}}_{(n,\ldots,n,0,0,\ldots,0)}\hookrightarrow\underline{\mathcal{E}}_{(n,\ldots,n,n,0,\ldots,0)}$
are locally free of rank 1 as $\mathcal{O}_{D_{n,S}}$-modules. By the argument
in the proof of Proposition 6.5, this condition is representable by an open
subscheme. ∎

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