$pn\cdot\left(\frac{|\pi_{F}|r}{\varpi}\right)^{n}\leqslant\frac{|\pi_{F}|^{n}r^{n}}{|n!|}|a_{n}(0)|\leqslant|a_{n}|_{X}r^{n}\to
0\quad\mbox{as}\quad n\to\infty$

which is impossible because $r>\varpi/|\pi_{F}|$ implies that
$\frac{|\pi_{F}|r}{\varpi}>1$. ∎

###  6.4. Estimates of $p$-adic valuations of binomial coefficients

The goal of this section is to estimate the $p$-adic growth rate of the
coefficient

(56)
$\sum_{r=0}^{n}\frac{\binom{\frac{k}{d}}{r}\binom{\frac{qk}{d}-(q-1)r}{(n-r)(q-1)}}{(n-r)(q-1)+1}$

that arises in the proof of Theorem 6.3.18 for $1\leqslant k\leqslant q$. In
particular we will see that for certain special values of $n$, the term
corresponding to the value of $r$ that maximises the $p$-adic valuation of the
denominator $(n-r)(q-1)+1$ will dominate this sum and moreover the $p$-adic
norm of this term will be unbounded as $n$ increases through these special
values. To achieve this we will appeal to a Theorem of Kummer to estimate the
$p$-adic valuation of the binomial coefficients that appear in the sum.

For the remainder of the section we fix $1\leqslant k\leqslant q$ and define
$f=v_{p}(q)$.

Let $\lambda\in\mathbb{Z}_{p}$ and $n\in\mathbb{N}$ be given, and consider
their $p$-adic expansions

$\lambda=\lambda_{0}+p\lambda_{1}+p^{2}\lambda_{2}+\cdots,\quad
n=n_{0}+pn_{1}+pn_{2}+\cdots$

where the $p$-adic digits $\lambda_{i}$ and $n_{i}$ all lie in
$\\{0,1,...,p-1\\}$. Since $n$ is assumed to be a natural number, we know that
$n_{i}=0$ for all sufficiently large $i$.

######  Definition 6.4.1.

Let $\lambda\in\mathbb{Z}_{p}$ and $n\in\mathbb{N}$ be as above.

  1. (a)

Let $i\geqslant 0$. The _$i$ -th carry function_ $\gamma_{i}(\lambda,n)$ is
defined as follows:

$\gamma_{0}(\lambda,n)=\left\\{\begin{array}[]{lll}1&\quad\mbox{if}&\lambda_{0}+n_{0}>p-1\\\
0&\quad\mbox{if}&\lambda_{0}+n_{0}\leqslant p-1\end{array}\right.$

and for $i\geqslant 1$, it is defined recursively by

$\gamma_{i}(\lambda,n)=\left\\{\begin{array}[]{lll}1&\quad\mbox{if}&\lambda_{i}+n_{i}+\gamma_{i-1}(\lambda,n)>p-1\\\
0&\quad\mbox{if}&\lambda_{i}+n_{i}+\gamma_{i-1}(\lambda,n)\leqslant
p-1.\end{array}\right.$

  2. (b)

We define

$L(\lambda,n):=\inf\\{i\geqslant 0:\gamma_{j}(\lambda,n)=0\quad\mbox{for
all}\quad j\geqslant i\\}.$

  3. (c)

For each $m\geqslant 0$, the _$m$ -th non-carry function_ is defined to be

$\mathcal{N}_{m}(\lambda,n):=|\\{0\leqslant i<m:\gamma_{i}(\lambda,n)=0\\}|.$

  4. (d)

We define $\langle\lambda|n\rangle$ to be the binomial coefficient
$\binom{\lambda+n}{n}$.

Thus $L(\lambda,n)$ records the position after which there are no further
carries when performing the addition of $\lambda$ and $n$ in $\mathbb{Z}_{p}$;
it is possible that $L(\lambda,n)=\infty$ but this happens only when
$\lambda_{i}=p-1$ for all sufficiently large $i$. More precisely we have the
following Lemma.

######  Lemma 6.4.2.

$L(\lambda,n)=\infty$ only when $\lambda$ is a negative integer and
$n\geqslant-\lambda$.

###### Proof.

Since $n\in\mathbb{N}$ there is $m\geqslant 1$ such that $n_{j}=0$ for all
$j\geqslant m$.

Suppose that $L(\lambda,n)=\infty$. Then by definition
$\gamma_{j}(\lambda,n)=1$ for infinitely many values of $j$. But, by 6.4.1(a),
if $\gamma_{j}(\lambda,n)=1$ for $j\geqslant m$ then
$\gamma_{j-1}(\lambda,n)=1$ and $\lambda_{j}=p-1$. It follows that
$\gamma_{j}(\lambda,n)=1$ for all $j\geqslant m-1$ and $\lambda_{j}=p-1$ for
all $j\geqslant m$ and $\lambda$ is a negative integer. Moreover
$(\lambda+n)_{j}=0$ for all $j\geqslant m$; that is $\lambda+n\in bN$ and so
$n\geqslant-\lambda$ as claimed. ∎

In our applications, $L(\lambda,n)$ will always be finite. The function
$\mathcal{N}_{m}(\lambda,n)$ records the number of non-carries that occurred
in the first $m$ digits when performing the addition of $\lambda$ and $n$ in
$\mathbb{Z}_{p}$.

We have now developed enough language to precisely state Kummer’s theorem on
the $p$-adic valuation of binomial coefficients.

######  Theorem 6.4.3 (Kummer, 1852).

Let $\lambda\in\mathbb{Z}_{p}$ and $n\in\mathbb{N}$ be given, and suppose that
$L(\lambda,n)<\infty$. Then

$v_{p}\langle\lambda|n\rangle=m-\mathcal{N}_{m}(\lambda,n)\quad\mbox{for
all}\quad m\geqslant L(\lambda,n).$

Under the assumption that $L(\lambda,n)<\infty$, the carrying will stop in
finite time in the sense that eventually all $\gamma_{i}(\lambda,n)$ are zero.
Then for sufficiently large $m$, the quantity on the right hand side is
independent of $m$ and returns the total number of carries that occurred
during the addition of $\lambda$ and $n$ in $\mathbb{Z}_{p}$ — this is perhaps
a more usual way of formulating Kummer’s theorem. We also note that if
$L(\lambda,n)=\infty$ then $\langle\lambda|n\rangle=0$ so
$v_{p}(\langle\lambda|n\rangle)=\infty$ in that case.

###### Proof of Theorem 6.4.3.

A proof in the case that $\lambda\in\mathbb{N}$ appeared in [31]. A modern
account of the proof in that case can be found in [26, Theorem 3.7].

Now let $\mu\in\mathbb{N}$ be such that $0\leqslant\mu<p^{L(\lambda,n)}$ and
$v_{p}(\lambda-\mu)\geqslant L(\lambda,n)$; that is the $p$-adic expansion of
$\mu$ is the truncation of the $p$-adic expansion of $\lambda$ at the
$L(\lambda,n)$th digit. Then it is easy to verify that $L(\lambda,n)=L(\mu,n)$
and $\mathcal{N}_{m}(\lambda,n)=\mathcal{N}_{m}(\mu,n)$ for all $m$. Moreover
$v_{p}\langle\lambda|n\rangle=v_{p}\langle\mu|n\rangle$ since
$v_{p}(\lambda+i)=v_{p}(\mu+i)\leqslant L(\lambda,n)$ for all $1\leqslant
i\leqslant n$. ∎

In most cases of interest to us, the possibility of carrying process _not_
stopping in finite time is ruled out by the following

######  Notation 6.4.4.

Given $\alpha_{0},\alpha_{1},\cdots,\alpha_{j-1}\in\\{0,1,\cdots,q-1\\}$,
write

$[\alpha_{0},\alpha_{1},\cdots,\alpha_{j-1}]_{j}:=\alpha_{0}q^{0}+\alpha_{1}q^{1}+\cdots+\alpha_{j-1}q^{j-1}.$

We call $\alpha_{i}$ the _$i$ th $q$-adic digit_ of this _$q$ -expansion_.

######  Lemma 6.4.5.

Suppose that $\lambda\in\mathbb{Z}_{p}$, $n,j\in\mathbb{N}$ with

$\lambda\equiv[\alpha_{0},\ldots,\alpha_{j-1}]_{j}\operatorname{mod}q^{j}\mbox{
and }n=[\beta_{0},\ldots,\beta_{j-1}]_{j}.$

  1. (a)

If $\beta_{i}+\alpha_{i}<q-1$ for some $0<i<j$ then
$\gamma_{(i+1)f-1}(\lambda,n)=0$.

  2. (b)

If $\beta_{j-1}+\alpha_{j-1}<q-1$ then $L(\lambda,n)<(j+1)f$.

###### Proof.

(a) Under the hypothesis $\beta_{i}+\alpha_{i}<q-1$

$\gamma_{(i+1)f-1}(\lambda,n)=\begin{cases}\gamma_{f-1}(\alpha_{i},\beta_{i})&\mbox{if
}\gamma_{if-1}(\lambda,n)=0\\\ \gamma_{f-1}(\alpha_{i}+1,\beta_{i})&\mbox{if
}\gamma_{if-1}(\lambda,n)=1\end{cases}$

and since $\alpha_{i}+1+\beta_{i}<q$ this is zero in either case.

(b) Since $n<q^{j}$, we have $n_{i}=0$ whenever $i\geqslant jf$. The result
now follows from part (a) and Definition 6.4.1(a). ∎

######  Definition 6.4.6.

For $n\geqslant 1$, let $M=M_{n}\in\mathbb{N}$ be largest possible such that
$1+q+\cdots+q^{M}\leqslant n$. We also define

$s:=s_{n}:=n-(1+q+\cdots+q^{M})\geqslant 0.$

It is easy to see that $s_{n}<q^{M_{n}+1}$ since otherwise we could make $M$
larger.

######  Lemma 6.4.7.

Let $0\leqslant r\leqslant n$ be an integer. Then

$v_{p}(r-s_{n})<(M_{n}+1)f$

whenever $r\neq s_{n}$.

###### Proof.

Since $s<q^{M+1}$ and $n-s=1+q+\cdots+q^{M}<q^{M+1}$,

$s-q^{M+1}<0\leqslant s\leqslant n<s+q^{M+1},$

which implies that

$\\{0,1,\cdots,n\\}\cap(s+q^{M+1}\mathbb{Z})=\\{s\\}.$

Suppose that $v_{p}(r-s)\geqslant(M+1)f$. Then $q^{M+1}$ divides $r-s$, so
$r\in s+q^{M+1}\mathbb{Z}$. Since $0\leqslant r\leqslant n$ by assumption, we
conclude that $r=s$.∎

######  Corollary 6.4.8.

Let $0\leqslant r\leqslant n$ be an integer. Then

$v_{p}((n-r)(q-1)+1)=\left\\{\begin{array}[]{lll}(M_{n}+1)f&\quad\mbox{if}&r=s_{n},\\\
v_{p}(r-s)&\quad\mbox{if}&r\neq s_{n}.\end{array}\right.$

###### Proof.

By definition of $s$, $v_{p}((n-s)(q-1)+1)=v_{p}(q^{M+1})=(M+1)f$. Now

$(n-r)(q-1)+1=(q-1)(s-r)\quad+\quad(n-s)(q-1)+1$

and $m:=v_{p}(r-s)$ is strictly less than $v_{p}(q^{M+1})=(M+1)f$ whenever
$r\neq s$ by Lemma 6.4.7. Since $v_{p}(q-1)=0$, the result now follows from
the non-Archimedean triangle inequality. ∎

Thus, combining the last two results, we see that $s_{n}$ has been chosen to
maximise $p$-adic value of the denominator of (56) for given $n$.

We are also interested in the following complicated-looking binomial
coefficient.

######  Definition 6.4.9.

For each $n,r\in\mathbb{N}$ with $0\leqslant r\leqslant n$, define

$S_{n,r}:=\langle\frac{qk}{q+1}-n(q-1)\hskip 2.84544pt|\hskip
2.84544pt(n-r)(q-1)\rangle=\binom{\frac{qk}{q+1}-r(q-1)}{(n-r)(q-1)}.$

In order to control this, until Corollary 6.4.17 below, we fix a possibly very
large positive integer $N\geqslant 0$, and define $n_{N}$ to be the smallest
positive integer such that

$n_{N}\equiv\frac{qk}{q^{2}-1}\operatorname{mod}q^{N}$

This expression obviously depends on $N$, but because of its frequent
appearance we will abbreviate it to $n:=n_{N}$ until Corollary 6.4.17. The
reason for this choice of the form of $n$ is the following result which will
enable us to ignore the $S_{n,r}$ in the numerator of (56) when estimating its
$p$-adic value.

######  Proposition 6.4.10.

$v_{p}(S_{n,r})\geqslant 0$ for all $0\leqslant r\leqslant n$ and
$v_{p}(S_{n,s_{n}})=0$.

###### Proof.

The first part is well-known since $qk/(q+1)$ is a $p$-adic integer.

We define $\alpha:=\frac{qk}{q+1}-n(q-1)\in\mathbb{Z}_{p}$ so that
$\alpha\equiv 0\operatorname{mod}q^{N}$. That is, the $p$-adic digits
$\alpha_{j}$ of $\alpha$ are zero whenever $j<Nf$. Similarly, since $M<N$,

$\beta:=(n-s)(q-1)\leqslant q^{M+1}-1<q^{N}$

so $\beta_{j}=0$ for all $j\geqslant Nf$. It follows that
$\gamma_{j}(\alpha,\beta)=0$ for all $j\geqslant 0$, and therefore
$v_{p}(S_{n,s})=v_{p}\langle\alpha|\beta\rangle=0$ by Theorem 6.4.3. ∎

For reasons that will become apparent later we now assume

$\begin{cases}N\mbox{ is even }&\mbox{if }1\leqslant k\leqslant q-1\mbox{ or
}k=2=q\\\ N\mbox{ is odd }&\mbox{if }k=q>2.\end{cases}$

It is also convenient to suppose that $N\geqslant 6$.

We can now compute the precise value of $M_{n}$ in terms of $N$ depending on
the values of $k$ and $q$.

######  Lemma 6.4.11.

$M_{n}=\begin{cases}N-1&\mbox{if }1\leqslant k\leqslant q-2\mbox{ or
}k=q>2,\\\ N-2&\mbox{if }\left(k=q-1\mbox{ and }q>2\right)\mbox{ or }k=q=2,\\\
N-3&\mbox{if }k=1,q=2.\end{cases}$

###### Proof.

Since $n\leqslant q^{N}$ it is immediate that $M\leqslant N-1$ in all cases.

We see that

$n\equiv
1+\frac{(q-k-1)q+(q-1)}{1-q^{2}}=2+\frac{(2q-k-1)q+(q-2)}{1-q^{2}}\operatorname{mod}q^{N}$

so, for $1\leqslant k\leqslant q-1$, since $N$ is even, the $q$-adic expansion
of $n$ is

$n=0+(q-k)q+(q-1)q^{2}+(q-1-k)q^{3}+\cdots+(q-1)q^{N-2}+(q-1-k)q^{N-1},$

For $k=q>2$, since $N$ is odd, the $q$-adic expansion of $n$ is

$n=0+0\cdot
q+(q-1)q^{2}+(q-1)q^{3}+(q-2)q^{4}+\cdots+(q-1)q^{N-2}+(q-2)q^{N-1},$

and for $k=q=2$, since $N$ is even,

$n=0+0\cdot
q+(q-1)q^{2}+(q-1)q^{3}+(q-2)q^{4}+\cdots+(q-2)q^{N-2}+(q-1)q^{N-1},$

where in all cases, the $q$-digits after the $3$rd digit repeat with period
$2$ until the last $q$-digit.

If $1\leqslant k\leqslant q-2$ then $q\geqslant 3$, and it is now easy to see
that

$1+q+q^{2}+\cdots+q^{N-1}<2q^{N-2}+q^{N-1}<n$

so $M=N-1$.

Similarly if $k=q-1$ and if $q\geqslant 3$, then

$1+q+q^{2}+\cdots+q^{N-2}<2q^{N-2}<n<q^{N-1}$

and so $M=N-2$, and if $k=q$ and $q\geqslant 3$, then

$1+q+q^{2}+\cdots+q^{N-1}<n<q^{N}$

so $M=N-2$.

The two cases where $q=2$ can be treated in a similar manner noting that when
$q=2$, $1+q+q^{2}+\cdots+q^{M}=q^{M+1}-1$. ∎

Our next job will be to compute the $q$-adic expansions of
$\frac{k}{q+1}-s_{n}$ and $s_{n}$ up to the $f(M+1)$st $p$-adic digit in all
cases.

######  Lemma 6.4.12.

Write $\lambda:=\frac{k}{q+1}\in\mathbb{Z}_{p}\backslash\mathbb{N}$.

  1. (a)

Suppose that $1\leqslant k\leqslant q-2$. Then $M$ is odd and

$\begin{array}[]{rclcccccr}s&=&[q-1,&q-k-2,&q-2,&q-k-2,&\cdots,&q-2,&q-k-2]_{M+1}\\\
\lambda-s&\equiv&[k+1,&1,&k+1,&1,&\cdots,&k+1,&1]_{M+1}\end{array}$

  2. (b)

Suppose that $k=q-1$ and $q>2$. Then $M$ is even and

$\begin{array}[]{rclccccr}s&=&[q-1,&q-1,&q-3,&\cdots,&q-1,&q-3]_{M+1}\\\
\lambda-s&\equiv&[0,&2,&0,&\cdots,&2,&0]_{M+1}\end{array}$

  3. (c)

Suppose that $k=1$ and $q=2$. Then $M$ is odd and

$\begin{array}[]{rclcccccr}s&=&[1,&1,&1,&0,&\cdots,&1,&0]_{M+1}\\\
\lambda-s&\equiv&[0,&0,&1,&0,&\cdots,&1,&0]_{M+1}\end{array}$

  4. (d)

Suppose that $k=2$ and $q=2$. Then $M$ is even and

$\begin{array}[]{rclccccccr}s&=&[1,&0,&1,&1,&0,&\cdots,&1,&0]_{M+1}\\\
\lambda-s&\equiv&[1,&0,&0,&1,&0&\cdots,&1,&0]_{M+1}\end{array}$

  5. (e)

Suppose that $k=q$ and $q>2$. Then $M$ is even and

$\begin{array}[]{rclccccr}s&=&[q-1,&q-2,&q-3,&\cdots,&q-2,&q-3]_{M+1}\\\
\lambda-s&\equiv&[1,&2,&1,&\cdots,&2,&1]_{M+1}\end{array}$

where $\equiv$ denotes congruence modulo $q^{M+1}$.

Note that all of the displayed expansions are recurrent with period $2f$ but
not purely periodic: the two $q$-digits just preceeding the $\cdots$ sign are
the repeating ones.

###### Proof.

The parity of $M$ can be deduced in each case from Lemma 6.4.11 and the parity
of $N$.

Since $q^{M+1}$ divides $q^{N}$ in all cases,

$\displaystyle n$ $\displaystyle\equiv$
$\displaystyle\frac{-qk}{1-q^{2}}\operatorname{mod}q^{M+1}\mbox{ so}$
$\displaystyle s$ $\displaystyle\equiv$
$\displaystyle\frac{-qk}{1-q^{2}}-\frac{1}{1-q}\operatorname{mod}q^{M+1}$
$\displaystyle\equiv$
$\displaystyle\frac{-qk-q-1}{1-q^{2}}\operatorname{mod}q^{M+1}$
$\displaystyle\equiv$ $\displaystyle 1+\frac{(q-2)+(q-k-2)q}{1-q^{2}}$
$\displaystyle\equiv$ $\displaystyle 2+\frac{(q-3)+(2q-k-2)q}{1-q^{2}},\mbox{
and }$ $\displaystyle\lambda-s$ $\displaystyle\equiv$
$\displaystyle\frac{k}{q+1}+\frac{qk+q+1}{1-q^{2}}=\frac{k+q+1}{1-q^{2}}\operatorname{mod}q^{M+1}.$

All the calculations can now be done in a straightforward manner. For the case
where $k=q$, it is helpful to note that $s\leqslant n<q^{2N+1}$ by definition
of $n$, so that the last two $q$-digits in the $q$-expansion of $s$ are $q-3$
and $q-2$. For the cases where $q=2$ it is helpful to remember that
$s+2^{M+1}=n+1$ and to use the $q$-expansions of $n$ found in the proof of
Lemma 6.4.11. ∎

######  Corollary 6.4.13.

In all five cases appearing in Lemma 6.4.12, the carrying in the sum
$s+(\lambda-s)$ stops before the $(M+1)f$ position. More precisely, we have

$L(\lambda-s,s)<(M+1)f.$

We note that the parity of $N$ in each case was chosen to make this true.

###### Proof.

In each case we can use the expressions for $s$ and $\lambda-s$ in Lemma
6.4.12 and Lemma 6.4.5(b). ∎

The remaining substantive result will be the following Proposition, whose
proof highlights the importance of _counting the_ non _-carry positions_ in
the sum $(\lambda-r)+r=\lambda$.

######  Proposition 6.4.14.

Let $\lambda\in\mathbb{Z}_{p}\backslash\mathbb{N}$, $s\in\mathbb{N}$ and
$\ell\geqslant 1$ be given, such that

$L(\lambda-s,s)<\ell.$

Then for any $r\in\mathbb{N}$ with $v_{p}(r-s)<\ell$, we have

$v_{p}\langle\lambda-r|r\rangle-v_{p}(r-s)\quad>\quad
v_{p}\langle\lambda-s|s\rangle-\ell.$

###### Proof.

Theorem 6.4.3 implies that

(57) $v_{p}\langle\lambda-s|s\rangle=\ell-\mathcal{N}_{\ell}(\lambda-s,s).$

Define the _$m$ -th truncation function_ $t_{m}:\mathbb{Z}_{p}\to\mathbb{N}$
for $m\geqslant 0$ by

$t_{m}\left(\sum_{i=0}^{\infty}\mu_{i}p^{i}\right)=\sum_{i=0}^{m-1}\mu_{i}p^{i}$

where $\mu_{i}\in\\{0,\ldots,p-1\\}$. Note that the sum on the left is
computed in $\mathbb{Z}_{p}$ but the sum on the right is computed in
$\mathbb{N}$. Because $L(\lambda-s,s)<\ell$ by assumption, Definition 6.4.1(b)
implies that $\gamma_{\ell-1}(\lambda-s,s)=0$. Therefore

(58)
$\mathcal{N}_{\ell}(\lambda-s,s)>\mathcal{N}_{\ell-1}(\lambda-s,s)\geqslant\mathcal{N}_{m}(\lambda-s,s)=\mathcal{N}_{m}(t_{m}(\lambda-s),t_{m}(s))$

whenever $0\leqslant m<\ell$. This inequality applies in particular when
$m:=v_{p}(r-s)$ which is strictly less than $\ell$ by hypothesis. Note also
that because $r\equiv s$ mod $p^{m}$, we know that $t_{m}(r)=t_{m}(s)$ and
$t_{m}(\lambda-r)=t_{m}(\lambda-s)$.

Next, we have the crude estimate

(59)
$\mathcal{N}_{j}(\lambda-r,r)\leqslant\mathcal{N}_{m}(t_{m}(\lambda-r),t_{m}(r))+(j-m)\quad\mbox{for
all}\quad j\geqslant m.$

Finally, let $j=\max\\{m,L(\lambda-r,r)\\}$ which is finite by Lemma 6.4.2
because $\lambda\notin\mathbb{N}$. Then Theorem 6.4.3 implies that

(60) $v_{p}\langle\lambda-r|r\rangle=j-\mathcal{N}_{j}(\lambda-r,r).$

Applying $(\ref{vpBin2})$, $(\ref{Crude})$, $(\ref{LastNoCarry})$ and
$(\ref{vpBin})$, we obtain

$\begin{array}[]{lll}v_{p}\langle\lambda-r|r\rangle-m&=&j-\mathcal{N}_{j}(\lambda-r,r)-m\\\
&\geqslant&-\mathcal{N}_{m}(t_{m}(\lambda-r),t_{m}(r))\\\
&=&-\mathcal{N}_{m}(t_{m}(\lambda-s),t_{m}(s))\\\
&>&-\mathcal{N}_{\ell}(\lambda-s,s)\\\
&=&v_{p}\langle\lambda-s|s\rangle-\ell\end{array}$

as required. ∎

We now put everything together to obtain our main estimate.

######  Theorem 6.4.15.

Let $0\leqslant r\leqslant n$ be given with $r\neq s$. Then

$v_{p}\left(\frac{\langle\lambda-r|r\rangle}{(n-r)(q-1)+1}\right)>v_{p}\left(\frac{\langle\lambda-s|s\rangle}{(n-s)(q-1)+1}\right).$

###### Proof.

By Corollary 6.4.8, it is enough to show that

$v_{p}(\langle\lambda-r|r\rangle)-v_{p}(r-s)>v_{p}(\langle\lambda-s|s\rangle)-(M+1)f.$

We set $\ell:=(M+1)f$. Then Corollary 6.4.13 tells us $L(\lambda-s,s)<\ell$,
and Lemma 6.4.7 tells us that $v_{p}(r-s)<\ell$. Now the required inequality
follows from Proposition 6.4.14. ∎

It remains to estimate the carries in the sum $s+(\lambda-s)$.

######  Proposition 6.4.16.

For each $N\geqslant 6$ of suitable parity depending on $k$ and $q$

$v_{p}\left(\frac{\langle\lambda-s|s\rangle}{s(q-1)+1}\right)\leqslant\frac{3-N}{2}.$

###### Proof.

We note that by Theorem 6.4.3 and Corollary 6.4.13,

$v_{p}(\langle\lambda-s|s\rangle)=(M+1)f-1-\mathcal{N}_{(M+1)f-1}(\lambda-s,s).$

Considering Lemma 6.4.5(a) and Lemma 6.4.12 we see that

$\mathcal{N}_{(M+1)f-1}(\lambda-s,s)\geqslant\left\lfloor\frac{M-2}{2}\right\rfloor.$

Moreover by Corollary 6.4.8 $v_{p}((n-s)(q-1)+1)=(M+1)f$.

However, $M\geqslant N-3$ by Lemma 6.4.11. Hence

$v_{p}\left(\frac{\langle\lambda-s|s\rangle}{s(q-1)+1}\right)\leqslant-\left\lfloor\frac{M}{2}\right\rfloor\leqslant-\frac{N-3}{2}.\qed$

Here is the main result of $\S$6.4.

######  Corollary 6.4.17.

The $p$-adic valuation of the rational number

$\sum_{r=0}^{n}\frac{\binom{\frac{k}{q+1}}{r}\binom{\frac{qk}{q+1}-(q-1)r}{(n-r)(q-1)}}{(n-r)(q-1)+1}$

is not bounded below as $n$ varies.

###### Proof.

Apply Proposition 6.4.10, Theorem 6.4.15 and Corollary 6.4.16. ∎

###  6.5. Proof of Theorem 5.3.12

We assume in this section that the hypotheses of Theorem 5.3.12 hold, namely:

  * •

$[\mathscr{L}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$
and $\omega[\mathscr{L}]\in\operatorname{PicCon}(\Upsilon)[p^{\prime}]$,

  * •

the order $d$ of $\omega[\mathscr{L}]$ in $\operatorname{PicCon}(\Upsilon)$
divides $q+1$,

  * •

$e\in d\mathbb{Z}$ is the order of $[\mathscr{L}]$ in
$\operatorname{PicCon}^{I}(\Upsilon)$,

  * •

$n\geqslant v_{\pi_{F}}(e)$ and $a\in\mathcal{O}_{F}$,

  * •

$N_{n+1}\leqslant J_{n+1}$.

The map $M_{n,d}$ from $\S\ref{UpperHP}$ sends
$\operatorname{PicCon}(\Upsilon)[d]^{I}$ into
$M_{0}(h(\Upsilon_{n}),\mathbb{Z}/d\mathbb{Z})^{I}$. Since
$\omega[\mathscr{L}]\in\operatorname{PicCon}(\Upsilon)[d]^{I}$, we can now use
Lemma 4.3.5(a) to make the following

######  Definition 6.5.1.

We let $k\in\\{1,2,\cdots,d\\}$ be the unique integer such that

$M_{n,d}(\omega[\mathscr{L}])=k\nu_{n}.$

Recall from Definition 5.3.9 and [8, Definition 4.1.1] the affinoid
$W:=W_{a,n}$:

$W=\operatorname{Sp}K\left\langle\tau,\frac{1}{\tau^{q-1}-1}\right\rangle\quad\mbox{where}\quad\tau:=\frac{x-a}{\pi_{F}^{n}}.$

We fix a primitive $(q-1)^{th}$ root $\zeta$ of $1$ in $K^{\times}$ using
Remark 6.2.11 and note the following basic properties of our affinoid domain
$W$.

######  Lemma 6.5.2.

  1. (a)

$W\cap V_{n}$ is obtained from $W$ by removing the open disc $\\{|\tau|<1\\}$.

  2. (b)

$W\cap V_{n+1}$ is obtained from $W$ by removing the $q$ open discs

$\\{|\tau-\pi_{F}|<|\pi_{F}|\\},\quad\\{|\tau-\pi_{F}\zeta|<|\pi_{F}|\\},\quad\cdots\quad\\{|\tau-\pi_{F}\zeta^{q-2}|<|\pi_{F}|\\},\quad\\{|\tau|<|\pi_{F}|\\}.$

###### Proof.

(a) $V_{n}$ is obtained from $\mathbb{D}$ by removing all open discs of radius
$|\pi_{F}|^{n}$ around all points in $\mathcal{O}_{F}$ with respect to the
$x$-coordinate. Now $W$ is contained in a closed disc
$\operatorname{Sp}K\langle\tau\rangle$ of radius $|\pi_{F}|^{n}$ which
contains exactly $q$ such open discs. We have removed $q-1$ of these when
forming $W$. Hence $W\cap V_{n}$ is obtained from $W$ by removing the last of
these open discs, namely $\\{|\tau|<1\\}$.

(b) This time $\operatorname{Sp}K\langle\tau\rangle$ contains $q^{2}$ open
discs of radius $|\pi_{F}|^{n+1}$ with respect to the $x$-coordinate. Of
these, $q(q-1)$ are contained in the $q-1$ open discs of radius
$|\pi_{F}|^{n}$ which are removed when we form $W$. This shows that $W$
contains exactly $q$ more such open discs that have to be removed when forming
$W\cap V_{n+1}$. ∎

When $n=0,q=5$, a picture of $W\supset W\cap V_{n+1}\supset W\cap V_{n}$ can
be found in $\S\ref{TheIntegral}$.

######  Theorem 6.5.3.

There exists $c\in K(\tau)\cap\mathcal{O}(W)^{\times}$ with
$\mathcal{S}(c)\subseteq F$, and an algebraic generator
$\dot{z}\in\mathscr{L}_{n+1}(W)$ with associated rational function

$\psi(\dot{z}^{\otimes d})=u\hskip 2.84544ptc\quad\mbox{where}\quad
u:=\frac{1}{(\tau^{q}-\pi_{F}^{q-1}\tau)^{k}}.$

###### Proof.

We define
$\mathcal{S}_{1}:=\\{0\\}\cup\\{1,\zeta,\zeta^{2},\cdots,\zeta^{q-2}\\}$ and
note that $\mathcal{S}_{1}$ forms a set of coset representatives for
$\pi_{F}\mathcal{O}_{F}$ in $\mathcal{O}_{F}$ containing $0$. Let
$\mathcal{S}$ be any set of coset representatives for
$\pi_{F}^{n+2}\mathcal{O}_{F}$ in $\mathcal{O}_{F}$ containing
$a+\pi_{F}^{n}\mathcal{S}_{1}=W(\overline{K})\cap\mathcal{O}_{F}$.

Using Definition 6.5.1 together with Lemma 4.3.5(b) we see that

$r_{n+1}(M_{n+1,d}(\omega[\mathscr{L}]))=M_{n,d}(\omega[\mathscr{L}])=k\nu_{n}=r_{n+1}(-k\nu_{n+1}).$

Lemma 4.3.5(c) then implies that $M_{n+1,d}(\omega[\mathscr{L}])=-k\nu_{n+1}$,
and hence

$M_{n+1,d}(\omega[\mathscr{L}])(D_{b,n+1})=-k+d\mathbb{Z}\quad\mbox{for
all}\quad b\in\mathcal{O}_{F}.$

Note that $-\frac{k}{d}\notin\mathbb{N}$ because $k\geqslant 1$. We can now
apply Theorem 4.3.7(a) with $n+1$ in place of $n$ with $k_{b}:=-k$ for all
$b\in\mathcal{S}$, to choose an algebraic generator
$\dot{y}\in\mathscr{L}_{n+1}(W)$ with associated rational function $v$ such
that

$v_{b}(v)=k\quad\mbox{for all}\quad b\in\mathcal{S}.$

Now, since
$u=\frac{1}{(\tau^{q}-\pi_{F}^{q-1}\tau)^{k}},\tau=\frac{x-a}{\pi^{n}_{F}}$
and $k\neq 0$, we see that $\mathcal{S}(u)=a+\pi^{n}_{F}\mathcal{S}_{1}$, and
that $v_{b}(u)=v_{b}(v)$ for all $b\in\mathcal{S}(u)$. Therefore the rational
function $c:=v/u$ satisfies
$\mathcal{S}(c)=\mathcal{S}(v)\backslash\mathcal{S}(u)=\mathcal{S}\backslash(a+\pi^{n}_{F}\mathcal{S}_{1})\subseteq
F$. This finite set does not meet $W(\overline{K})$ because $\mathcal{S}\cap
W(\overline{K})=a+\pi^{n}_{F}\mathcal{S}_{1}$. Hence
$c\in\mathcal{O}(W)^{\times}$ as claimed. ∎

######  Proposition 6.5.4.

Let $[\dot{z}]$ denote the image of $\dot{z}\in\mathscr{L}_{n+1}(W)$ in
$\mathscr{L}_{n}(W)$ under the natural restriction map
$\mathscr{L}_{n+1}(W)\to\mathscr{L}_{n}(W)$. Then the element

$\dot{y}:=(1-(\pi_{F}/\tau)^{q-1})^{k/d}[\dot{z}]\in\mathscr{L}_{n}(W)$

is an algebraic generator for $\mathscr{L}_{n}(W)$ with associated rational
function $c/\tau^{qk}$.

###### Proof.

Since $\dot{z}$ generates $\mathscr{L}_{n+1}(W)$ as an
$(\mathcal{O}_{W})_{\overline{W\cap V_{n+1}}}(W)$-module, we see that
$[\dot{z}]$ generates $\mathscr{L}_{n}(W)$ as an
$(\mathcal{O}_{W})_{\overline{W\cap V_{n}}}(W)$-module. Note that
$1-(\pi_{F}/\tau)^{q-1}\in
1+\mathcal{O}(W)\langle\pi_{F}/\tau^{2}\rangle^{\circ\circ}$, so by [8, Lemma
4.3.2(a)] it is a $d$th power in
$1+\mathcal{O}(W)\langle\pi_{F}/\tau^{2}\rangle^{\circ\circ}$. Since
$\mathcal{O}(W)\langle\pi_{F}/\tau^{2}\rangle\subseteq\mathcal{O}(W)_{\overline{W\cap
V_{n}}}$ in view of Lemma 6.5.2(a), we conclude that $\dot{y}$ also generates
$\mathscr{L}_{n}(W)$ as an $\mathcal{O}(W)_{\overline{W\cap V_{n}}}$-module as
required by Definition 4.3.6(a). Using Theorem 6.5.3, we see that the
associated rational function to $\dot{y}$ is

$\psi(\dot{y}^{\otimes
d})=\left(1-(\pi_{F}/\tau)^{q-1}\right)^{k}\frac{c}{(\tau^{q}-\pi_{F}^{q-1}\tau)^{k}}=\frac{c}{\tau^{qk}}.$

Next, $\mathcal{S}(c/\tau^{qk})\subseteq\\{a\\}\cup\mathcal{S}(c)\subseteq K$
by Theorem 6.5.3, so Definition 4.3.6(c) is satisfied. Since
$c\in\mathcal{O}(W)^{\times}$ and $k\neq 0$, we see that
$\mathcal{S}(c/\tau^{qk})\cap W=\\{a\\}$. Definition 4.3.6(d) is now satisfied
vacuously by $\dot{y}$, whereas Definition 4.3.6(e) holds because
$v_{a}(c/\tau^{qk})/d=-qk/d\notin\mathbb{N}$ as $k\geqslant 1$. ∎

After Theorem 6.5.3 and Proposition 6.5.4 and Corollary 4.3.12, we have
explicit presentations for the $\mathscr{D}_{n}(W)$-module
$\mathscr{L}_{n}(W)$ and $\mathscr{D}_{n+1}(W)$-module $\mathscr{L}_{n+1}(W)$.
Next, we will make the connection with the theory developed earlier on in
$\S\ref{PfOfCompOnW}$ by making a change of coordinates: recall from
$\S\ref{TheIntegral}$ the affinoid domain

$X:=\operatorname{Sp}K\left\langle x,\frac{1}{x^{q-1}-1}\right\rangle.$

######  Lemma 6.5.5.

Let $g_{a,n}:=\begin{pmatrix}1&-a\\\
0&\pi_{F}^{n}\end{pmatrix}\in\mathbb{B}(K)$. Then $g_{a,n}(W)=X$.

###### Proof.

We have $g_{a,n}\cdot z=\frac{z-a}{\pi_{F}^{n}}$ for all $z\in\overline{K}$.
Using the definitions of $W$ and $X$, we see that $z\in W(\overline{K})$ if
and only if $g_{a,n}\cdot z\in X(\overline{K})$. Hence $g_{a,n}(W)=X$. ∎

######  Corollary 6.5.6.

There is a $K$-algebra isomorphism

$\chi:\mathscr{D}_{n}(W)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{D}_{0}(X)$

such that $\chi(\tau)=x$ and $\chi(\pi_{F}^{n}\partial_{x})=\partial_{x}$, and
$\chi(\mathscr{D}_{n+m}(W))=\mathscr{D}_{m}(X)$ for all $m\geqslant 0$.

###### Proof.

Note that $\varrho(g_{a,n})=\pi_{F}^{-n}$ by Definition 2.5.1 and that
$g_{a,n}(W)=X$ by Lemma 6.5.5. The affinoid $W$ lies in
$\mathbb{A}(|\pi_{F}|^{n}\partial_{x}/\varpi)^{\dagger}$ by Lemma 5.3.10, so
we can apply Corollary 2.5.6 with $r=\varpi/|\pi_{F}|^{n}$ to get the required
$K$-algebra isomorphism

$\chi:=(g_{a,n})^{\dagger}_{\varpi/|\pi_{F}|^{n}}(W):\mathscr{D}_{n}(W)=\mathcal{D}^{\dagger}_{\varpi/|\pi_{F}|^{n}}(W)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{D}^{\dagger}_{\varpi}(X)=\mathscr{D}_{0}(X).$

Then $\chi(\tau)=g_{a,n}\cdot\tau=(g_{a,n}\cdot
x-a)/\pi_{F}^{n}=((a+\pi_{F}^{n}x)-a)/\pi_{F}^{n}=x$ and
$\chi(\partial_{x})=g_{a,n}\cdot\partial_{x}=\varrho(g_{a,n})\partial_{x}=\pi_{F}^{-n}\partial_{x}$
by Lemma 2.5.2(e).

Finally, we can apply Corollary 2.5.6 again with $r=\varpi/|\pi_{F}|^{n+m}$ to
see that $\chi$ maps $\mathscr{D}_{n+m}(W)$ onto $\mathscr{D}_{m}(X)$ for all
$m\geqslant 0$. ∎

Next, we record the presentations for $\mathscr{L}_{n}(W)$ and
$\mathscr{L}_{n+1}(W)$ after applying this change of coordinates as well as
untwisting via the untwisting automorphism $\theta_{\chi(c),d}^{-1}$ of
$\mathscr{D}_{0}(X)=\mathcal{D}^{\dagger}_{\varpi}(X)$ from Corollary 2.4.11.
More precisely:

######  Definition 6.5.7.

  1. (a)

Define $b:=\chi(c)\in\mathcal{O}(X)^{\times}$ and
$w:=(x^{q}-\pi_{F}^{q-1}x)^{-k}=\chi(u)$.

  2. (b)

Let
$M:=\mathscr{D}_{0}(X)\underset{\mathscr{D}_{n}(W)}{\otimes}{}\mathscr{L}_{n}(W)$
where $\mathscr{D}_{0}(X)$ is viewed as a $\mathscr{D}_{n}(W)$-module via the
isomorphism
$\theta_{b,d}^{-1}\circ\chi:\mathscr{D}_{n}(W)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{D}_{0}(X)$.

  3. (c)

Let
$M^{\prime}:=\mathscr{D}_{1}(X)\underset{\mathscr{D}_{n+1}(W)}{\otimes}{}\mathscr{L}_{n+1}(W)$
where $\mathscr{D}_{1}(X)$ is viewed as a $\mathscr{D}_{n+1}(W)$-module via
the isomorphism
$\theta_{b,d}^{-1}\circ\chi:\mathscr{D}_{n+1}(W)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{D}_{1}(X)$.

######  Theorem 6.5.8.

We have

$M^{\prime}\cong\frac{\mathscr{D}_{1}(X)}{\mathscr{D}_{1}(X)R_{\mathcal{S}(w)}(w,d)}\quad\mbox{and}\quad
M\cong\frac{\mathscr{D}_{0}(X)}{\mathscr{D}_{0}(X)R_{\mathcal{S}(x^{-qk})}(x^{-qk},d)}.$

###### Proof.

By Theorem 6.5.3 and Corollary 4.3.12, we have

$\mathscr{L}_{n+1}(W)=\mathscr{D}_{n+1}(W)\cdot{z}\cong\mathscr{D}_{n+1}(W)/\mathscr{D}_{n+1}(W)R_{\mathcal{S}(uc)}(uc,d).$

Now, applying Lemma 4.2.22(b) with $r=\varpi/|\pi_{F}|^{n}$ shows that

$\chi(R_{\mathcal{S}(uc)}(uc,d))\equiv
R_{\mathcal{S}(\chi(uc))}(\chi(uc),d)\quad\operatorname{mod}K^{\times}.$

Next, $\chi(uc)=wb$, so Lemma 4.2.17(c) gives

$R_{\mathcal{S}(\chi(uc))}(\chi(uc),d)=R_{\mathcal{S}(wb)}(wb,d)=\theta_{b,d}(R_{\mathcal{S}(wb)}(w,d)).$

Since $\mathcal{S}(w)\cap\mathcal{S}(b)=\emptyset$, $\mathcal{S}(wb)$ is the
disjoint union of $\mathcal{S}(w)$ and $\mathcal{S}(b)$, so
$\Delta_{\mathcal{S}(wb)}=\Delta_{\mathcal{S}(w)}\Delta_{\mathcal{S}(b)}$ and
$R_{\mathcal{S}(wb)}(w,d)=\Delta_{\mathcal{S}(b)}R_{\mathcal{S}(w)}(w,d)$.
Therefore

$\chi(R_{\mathcal{S}(uc)}(uc,d))\equiv\theta_{b,d}\left(\Delta_{\mathcal{S}(b)}R_{\mathcal{S}(w)}(w,d)\right)\quad\operatorname{mod}K^{\times}.$

However, $b$ and hence $\Delta_{\mathcal{S}(b)}$ are units in
$\mathcal{O}(X)$, so we have

(61)
$\theta_{b,d}^{-1}\chi\left(\mathscr{D}_{n+1}(W)R_{\mathcal{S}(uc)}(uc,d)\right)=\mathscr{D}_{1}(X)R_{\mathcal{S}(w)}(w,d).$

This implies that
$M^{\prime}\cong\frac{\mathscr{D}_{1}(X)}{\mathscr{D}_{1}(X)R_{\mathcal{S}(w)}(w,d)}$
as $\mathscr{D}_{1}(X)$-modules. The second isomorphism is deduced similarly
using Proposition 6.5.4 instead of Theorem 6.5.3. ∎

We assume until the end of $\S\ref{PfOfCOW}$ that the field $K$ is discretely
valued.

######  Corollary 6.5.9.

The $\mathscr{D}_{n}(W)$-module $\mathscr{L}_{n}(W)$ is simple if $k\neq d$.

###### Proof.

After Theorem 6.5.8, it is enough to check that
$\frac{\mathscr{D}_{0}(X)}{\mathscr{D}_{0}(X)R_{\mathcal{S}(x^{-qk})}(x^{-qk},d)}$
is a simple $\mathscr{D}_{0}(X)$-module. Now
$R_{\mathcal{S}(x^{-qk})}(x^{-qk},d)=x\partial_{x}+\frac{qk}{d}$ by Definition
4.2.10(b), and $-\frac{qk}{d}\in\mathbb{Z}_{p}\backslash\mathbb{Z}$ because
$q\equiv-1\operatorname{mod}d$ and $k\neq 0\operatorname{mod}d$ by our
assumption on $k$. We can now apply Theorem 4.4.1 to conclude. ∎

######  Proposition 6.5.10.

The natural action map

$\varphi:\mathscr{D}_{n}(W)\underset{\mathscr{D}_{n+1}(W)}{\otimes}{}\mathscr{L}_{n+1}(W)\longrightarrow\mathscr{L}_{n}(W)$

given by $\varphi(Q\otimes v):=Q\cdot\tau(v)$ for all $Q\in\mathscr{D}_{n}(W)$
and $v\in\mathscr{L}_{n+1}(W)$, is surjective.

###### Proof.

The restriction map $\tau:\mathscr{L}_{n+1}(W)\to\mathscr{L}_{n}(W)$ is non-
zero, so $\varphi$ is a non-zero $\mathscr{D}_{n}(W)$-linear map. If $k\neq d$
then $\mathscr{L}_{n}(W)$ is a simple $\mathscr{D}_{n}(W)$-module by Corollary
6.5.9, so $\varphi$ is surjective.

Suppose now that $k=d$. Inspecting the definition of the algebraic generator
$\dot{y}\in\mathscr{L}_{n}(W)$ from Proposition 6.5.4, we see that

$\tau^{q-1}\dot{y}=(\tau^{q-1}-\pi_{F}^{q-1})[\dot{z}]\in\operatorname{im}\varphi.$

Since $\mathscr{L}_{n}(W)=\mathscr{D}_{n}(W)\cdot\dot{y}$, it remains to show
that $\dot{y}\in\mathscr{D}_{n}(W)\cdot\tau^{q-1}\dot{y}$. Now,

$M=\mathscr{D}_{0}(X)\underset{\mathscr{D}_{n}(W)}{\otimes}{}\mathscr{L}_{n}(W)\cong\frac{\mathscr{D}_{0}(X)}{\mathscr{D}_{0}(X)R_{\mathcal{S}(x^{-qd})}(x^{-qd},d)}$

as $\mathscr{D}_{0}(X)$-modules by Theorem 6.5.8. However,
$\mathcal{S}(x^{-qd})=\\{0\\}$, and

$R_{\\{0\\}}(x^{-qd},d)=x\partial_{x}+q=R_{\\{0\\}}(x^{-q},1),$

so applying Theorem 4.2.16 with $u=x^{-q}$, $t=1$ and $d=1$, we see that

(62) $M\cong(j_{\ast}\mathcal{O}_{\mathbb{A}-\\{0\\}})_{\overline{U_{0}}}(X)$

as $\mathscr{D}_{0}(X)$-modules. Inside
$(j_{\ast}\mathcal{O}_{\mathbb{A}-\\{0\\}})_{\overline{U_{0}}}(X)$, we compute

$x^{-q}=(-1)^{q-1}\partial_{x}^{[q-1]}\cdot x^{-1}\in\mathscr{D}_{0}(X)\cdot
x^{q-1}x^{-q},$

and hence $\dot{y}\in\mathscr{D}_{n}(W)\cdot\tau^{q-1}\dot{y}$ because the
isomorphism (62) sends $1\otimes\dot{y}$ to $x^{-q}$. ∎

Next, we need to obtain the following upper bound.

######  Lemma 6.5.11.

We have $\ell(\ker\varphi)\leqslant q-1$.

###### Proof.

After applying the change of coordinates made precise by Theorem 6.5.8, the
map $\varphi$ induces a non-zero $\mathscr{D}_{0}(X)$-linear map

$\psi:\frac{\mathscr{D}_{0}(X)}{\mathscr{D}_{0}(X)R_{\mathcal{S}(w)}(w,d)}\cong\mathscr{D}_{0}(X)\underset{\mathscr{D}_{1}(X)}{\otimes}{}M^{\prime}\longrightarrow
M\cong\frac{\mathscr{D}_{0}(X)}{\mathscr{D}_{0}(X)R_{\mathcal{S}(x^{-qk})}(x^{-qk},d)}.$

Now, by applying Theorem 6.2.10 — see also Example 6.2.12 — we know that

$\begin{array}[]{lll}\operatorname{Cycl}\left(\frac{\mathscr{D}^{\dagger}_{\mathbb{Q}}(\mathscr{X})}{\mathscr{D}^{\dagger}_{\mathbb{Q}}(\mathscr{X})R_{\mathcal{S}(w)}(w,d)}\right)&=&[\mathscr{X}_{0}]+q[T^{\ast}_{0}\mathscr{X}_{0}],\quad\mbox{whereas}\\\
\operatorname{Cycl}\left(\frac{\mathscr{D}^{\dagger}_{\mathbb{Q}}(\mathscr{X})}{\mathscr{D}^{\dagger}_{\mathbb{Q}}(\mathscr{X})R(x^{-qk},d)}\right)&=&[\mathscr{X}_{0}]+[T^{\ast}_{0}\mathscr{X}_{0}]\end{array}.$

Since $\varphi$ is surjective by Proposition 6.5.10, so is $\psi$. Therefore

$\operatorname{Cycl}(\ker\psi)=(q-1)[T_{0}^{\ast}\mathscr{X}_{0}].$

Since $|\operatorname{Cycl}|$ is additive on short exact sequences and
$|\operatorname{Cycl}(V)|\geqslant 1$ for non-zero holonomic
$\mathscr{D}^{\dagger}_{\mathbb{Q}}(\mathscr{X})$-modules $V$, we see that
$\ell(V)\leqslant|\operatorname{Cycl}(V)|$ for every holonomic
$\mathscr{D}^{\dagger}_{\mathbb{Q}}(\mathscr{X})$-module $V$. Hence
$\ell(\ker\varphi)=\ell(\ker\psi)\leqslant|\operatorname{Cycl}(\ker\psi)|=q-1$.
∎

######  Definition 6.5.12.

Define $g=\begin{pmatrix}1&-\pi_{F}^{n+1}\\\ 0&1\end{pmatrix}\in N_{n+1}$ and
$h:=\begin{pmatrix}1&-\pi_{F}\\\ 0&1\end{pmatrix}\in N_{1}$.

It follows from Lemma 5.3.3 that
$\beta(g)\in\mathcal{D}_{r}(\mathbb{D})^{\times}$ whenever
$r<\varpi/|\pi_{F}|^{n+1}$. Since
$\varpi/|\pi_{F}|^{n}<\varpi/|\pi_{F}|^{n+1}$, $\beta(g)$ lies in
$\mathscr{D}_{n}(\mathbb{D})^{\times}$ and hence can also be viewed as an
element of $\mathscr{D}_{n}(W)^{\times}$.

######  Proposition 6.5.13.

$\beta(g)\notin\mathscr{D}_{n+1}(W)+\mathscr{D}_{n}(W)R_{\mathcal{S}(uc)}(uc,d).$

###### Proof.

Suppose for a contradiction that
$\beta(g)\in\mathscr{D}_{n+1}(W)+\mathscr{D}_{n}(W)R_{\mathcal{S}(uc)}(uc,d)$.
Then we can use equation $(\ref{ChiRwd})$ to deduce that

$\theta_{b,d}^{-1}\chi(\beta(g))\in\mathscr{D}_{1}(X)+\mathscr{D}_{0}(X)\cdot\mathscr{D}_{1}(X)R_{\mathcal{S}(w)}(w,d).$

Since $\chi(\pi_{F}^{n}\partial_{x})=\partial_{x}$ and $g\cdot
x-x=\pi_{F}^{n+1}$ by Lemma 5.2.2(a), we see that

$\chi(\beta(g))=\chi\left(\sum\limits_{m=0}^{\infty}\pi_{F}^{(n+1)m}\partial_{x}^{[m]}\right)=\sum\limits_{m=0}^{\infty}\pi_{F}^{m}\partial_{x}^{[m]}=\beta(h).$

By Proposition 5.2.13, we can find an element
$c_{b^{-1},d}(h)\in\mathcal{O}(X)^{\times}$ such that

$\theta_{b,d}^{-1}\chi(\beta(g))=\theta_{b^{-1},d}(\beta(h))=c_{b^{-1},d}(h)\beta(h).$

Since $\mathscr{D}_{1}(X)\subseteq\mathscr{D}_{0}(X)$, we conclude that
$c_{b^{-1},d}(h)\beta(h)\in\mathscr{D}_{1}(X)+\mathscr{D}_{0}(X)R_{\mathcal{S}(w)}(w,d)$.
Since $c_{b^{-1},d}(h)$ is a unit in $\mathcal{O}(X)$, this implies that
$\beta(h)\in\mathscr{D}_{1}(X)+\mathscr{D}_{0}(X)R_{\mathcal{S}(w)}(w,d)$.
This contradicts Theorem 6.3.21. ∎

Note that since $g\in N_{n+1}\leqslant J_{n+1}$ by our assumption on $J$, and
$W$ is $N_{n+1}$-stable by Lemma 5.3.10, the $\mathscr{D}_{n+1}(W)$-module
carries a compatible action of $g$.

######  Corollary 6.5.14.

The secret $g$-action from Lemma 5.1.6 on

$\mathscr{D}_{n}(W)\underset{\mathscr{D}_{n+1}(W)}{\otimes}\mathscr{L}_{n+1}(W)$

is non-trivial.

###### Proof.

Write $S:=\mathscr{D}_{n}(W)$ and $S^{\prime}:=\mathscr{D}_{n+1}(W)$. Choose
an algebraic generator $\dot{z}\in\mathscr{L}_{n+1}(W)$ as in Theorem 6.5.3
and write $r:=R_{\mathcal{S}(uc)}(uc,d)$. By Corollary 4.3.12, we can choose
$Q\in S^{\prime}$ such that $Q\cdot\dot{z}=g\cdot\dot{z}$. If the secret
action was trivial, then

$1\otimes\dot{z}=g\star(1\otimes\dot{z})=\beta(g)^{-1}\otimes
g\cdot\dot{z}=\beta(g)^{-1}\otimes
Q\cdot\dot{z}=\beta(g)^{-1}Q\otimes\dot{z}.$

Now $\operatorname{ann}_{S^{\prime}}(\dot{z})=S^{\prime}r$ by Corollary
4.3.12, so $\operatorname{ann}_{S}(1\otimes\dot{z})=Sr$ because $S$ is a flat
$S^{\prime}$-module by Lemma 4.3.10(a), Therefore $\beta(g)^{-1}Q-1\in Sr$.
Hence $Q-\beta(g)\in Sr$ so $\beta(g)\in S^{\prime}+Sr$, which contradicts
Proposition 6.5.13. ∎

Next, we need some elementary group theory. Let $k_{F}$ denote the residue
field of our $p$-adic field $F$ and let $k_{F}^{\times}=k_{F}-\\{0\\}$. We
view $k_{F}$ as a finite elementary abelian $p$-group of order $q$, and
$k_{F}^{\times}$ as a cyclic group of order $q-1$. As in $\S\ref{CPandSA}$,
let $\operatorname{Irr}_{\overline{K}}(k_{F})$ be the set of isomorphism
classes of simple $\overline{K}[k_{F}]$-modules, which is naturally in
bijection with the set $\widehat{k_{F}}$ of $\overline{K}^{\times}$-valued
characters of $k_{F}$. The group $k_{F}^{\times}$ acts on $k_{F}$ by
multiplication; since this action is by abelian group homomorphisms, we obtain
an induced action of $k_{F}^{\times}$ on $\widehat{k_{F}}$ given by

$(t\cdot\chi)(a)=\chi(t^{-1}a)\quad\mbox{for
all}\quad\chi\in\widehat{k_{F}},a\in k_{F},t\in k_{F}^{\times}.$

######  Lemma 6.5.15.

$k_{F}^{\times}$ acts transitively and freely on
$\widehat{k_{F}}\backslash\\{0\\}$.

###### Proof.

Suppose $t\in k_{F}^{\times}$ fixes $0\neq\chi\in\widehat{k_{F}}$ under the
above action. Then $(t^{-1}-1)a\in\ker\chi$ for all $a\in k_{F}$. Since
$k_{F}$ is a _field_ , $(t^{-1}-1)k_{F}=k_{F}$ unless $t=1$. But $\chi\neq 0$
forces $\ker\chi<k_{F}$, so indeed $t=1$. The result follows since
$|k_{F}^{\times}|=|\widehat{k_{F}}\backslash\\{0\\}|$. ∎

######  Corollary 6.5.16.

The action of $k_{F}^{\times}$ on $\operatorname{Irr}_{K}(k_{F})$ has exactly
two orbits.

###### Proof.

When $K=\overline{K}$, this follows directly from Lemma 6.5.15 because the
bijection $\operatorname{Irr}_{\overline{K}}(k_{F})\cong\widehat{k_{F}}$
respects the $k_{F}^{\times}$-action on both sides. In general, because our
ground field $K$ has characteristic zero, we may identify
$\operatorname{Irr}_{K}(k_{F})$ with the set of primitive idempotents of the
group ring $K[k_{F}]$, and every such idempotent is the sum of a
$\mathcal{G}_{K}$-orbit of primitive idempotents of $\overline{K}[k_{F}]$,
where $\mathcal{G}_{K}=\operatorname{Gal}(\overline{K}/K)$. Since the
$\mathcal{G}_{K}$-action on $\overline{K}[k_{F}]$ commutes with the
$k_{F}^{\times}$-action, we deduce from the case $K=\overline{K}$ that any two
non-principal idempotents in $K[k_{F}]$ lie in the same
$k_{F}^{\times}$-orbit. ∎

We now spell out the application of these ideas that we will need.

######  Corollary 6.5.17.

Let $B:=\begin{pmatrix}1&a\\\
0&1\end{pmatrix}\begin{pmatrix}\mathcal{O}_{F}^{\times}&0\\\
0&1\end{pmatrix}\begin{pmatrix}1&-a\\\ 0&1\end{pmatrix}$. Then

  1. (a)

$B$ stabilises $W=\operatorname{Sp}K\langle\tau,\frac{1}{\tau^{q-1}-1}\rangle$
where $\tau=\frac{x-a}{\pi_{F}^{n}}$.

  2. (b)

$B$ normalises $N_{n+1}=\begin{pmatrix}1&\pi_{F}^{n+1}\mathcal{O}_{F}\\\
0&1\end{pmatrix}$ and
$N_{n+2}=\begin{pmatrix}1&\pi_{F}^{n+2}\mathcal{O}_{F}\\\ 0&1\end{pmatrix}$.

  3. (c)

the $B$-action on $\operatorname{Irr}_{K}(N_{n+1}/N_{n+2})$ has exactly two
orbits.

###### Proof.

(a) This is an easy calculation using Lemma 5.2.2(a).

(b,c) We compute that $\begin{pmatrix}\alpha&0\\\ 0&1\end{pmatrix}$ conjugates
$\begin{pmatrix}1&\pi_{F}^{n+1}a\\\ 0&1\end{pmatrix}$ to
$\begin{pmatrix}1&\pi_{F}^{n+1}a\alpha^{-1}\\\ 0&1\end{pmatrix}$, so that the
conjugation $B$-action on $N_{n+1}/N_{n+2}$ is completely determined by the
multiplication action of $k_{F}^{\times}$ on $k_{F}$. Now apply Corollary
6.5.16. ∎

We can finally put everything together and prove Theorem 5.3.12.

###### Proof of Theorem 5.3.12.

Consider the commutative triangle

$\textstyle{\left(\mathscr{D}_{n}(W)\underset{\mathscr{D}_{n+1}(W)}{\otimes}{}\mathscr{L}_{n+1}(W)\right)_{N_{n+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{L}_{n}(W)}$$\textstyle{\left(\mathscr{D}_{n}(W)\underset{\mathscr{D}_{n+1}(W)}{\otimes}{}\mathscr{L}_{n+1}(W)\right)_{J_{n+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where we must show that the diagonal map is an isomorphism. Because $J_{n+1}$
contains $N_{n+1}$ by assumption, the vertical map is surjective. Proposition
6.5.10 implies that the diagonal map is surjective. Hence, to show that it is
an isomorphism it is enough to check that the horizontal map is injective. By
enlarging $K$ if necessary, we will assume that $K$ contains a primitive
$p^{th}$-root of unity $\zeta_{p}$.

We will apply Theorem 5.1.11 with the following parameters:
$S=\mathscr{D}_{n}(W)$, $S^{\prime}=\mathscr{D}_{n+1}(W)$,
$M=\mathscr{L}_{n}(W)$, $M^{\prime}=\mathscr{L}_{n+1}(W)$, $G$ is the subgroup
$B$ from Lemma 6.5.17, $H=N_{n+1}$ and $H^{\prime}=N_{n+2}$. Note that $M$ is
an $S\rtimes B$-module and $M^{\prime}$ is an $S^{\prime}\rtimes B$-module,
and these actions descend to $S\rtimes_{H}B$ and
$S^{\prime}\rtimes_{H^{\prime}}B$ respectively by Theorem 5.3.6. Note also
that the restriction map $M^{\prime}\to M$ is $B$-equivariant because our line
bundle $\mathscr{L}$ is assumed to be $I$-equivariant in Definition 5.3.2. All
points of Hypotheses 5.1.5 clearly hold, except possibly for points (e,f) of
Hypothesis 5.1.5 which, after decoding the notation, follow from Theorem
5.2.11 since $G=B\leqslant\mathcal{G}_{W}$.

Next, we verify the conditions (a-c) of Theorem 5.1.11.

(a) This is Corollary 6.5.14.

(b) This is Corollary 6.5.17(c).

(c) Since $K$ is assumed to contain $\zeta_{p}$ and since
$H/H^{\prime}=N_{n+1}/N_{n+2}\cong k_{F}$ is an elementary abelian $p$-group
of order $q$, we see that
$\left|\operatorname{Irr}_{K}(H/H^{\prime})\right|=q$. Now the required
inequality follows from Lemma 6.5.11.

Theorem 5.1.11 now tells us that the comparison map
$(S\otimes_{S^{\prime}}M^{\prime})_{H}\to M$ is injective as desired. ∎

##  7\. Coadmissibility and applications

###  7.1. Quasi-coherent modules over a tower of rings

We begin with some algebraic preliminaries. Recall that a _tower of rings_ is
a diagram

$S_{\bullet}=S_{0}\leftarrow S_{1}\leftarrow S_{2}\leftarrow\cdots$

of associative, unital rings, and unital ring homomorphisms. A _morphism_ of
towers of rings $f_{\bullet}:T_{\bullet}\to S_{\bullet}$ is a commutative
diagram of the form

$\textstyle{T_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{0}}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

An _$S_{\bullet}$ -module_ is a diagram $M_{\bullet}:=M_{0}\leftarrow
M_{1}\leftarrow M_{2}\leftarrow\cdots$ where each $M_{n}$ is an
$S_{n}$-module, and every connecting map $\mu^{M}_{n+1,n}:M_{n+1}\to M_{n}$ is
an $S_{n+1}$-linear map for $n\geqslant 1$, where $M_{n}$ is regarded as an
$S_{n+1}$-module via restriction of scalars along the ring homomorphism
$\mu^{S}_{n+1,n}:S_{n+1}\to S_{n}$. A _morphism of $S_{\bullet}$-modules_ is
defined in the obvious way. The space of _global sections_ of $M_{\bullet}$ is

$\Gamma(M_{\bullet}):=\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M_{n},$

this is naturally a module over
$\Gamma(S_{\bullet})=\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}S_{n}$.
We say that the $S_{\bullet}$-module $M_{\bullet}$ is _quasi-coherent_ if the
following map is an isomorphism for all $n\geqslant 0$:

$1\otimes\mu^{M}_{n+1,n}:S_{n}\underset{S_{n+1}}{\otimes}{}M_{n+1}\longrightarrow
M_{n}.$

Given a morphism $f_{\bullet}:T_{\bullet}\to S_{\bullet}$ and an
$S_{\bullet}$-module $M_{\bullet}$, we can form the _pullback_
$T_{\bullet}$-module $T_{\bullet}\otimes_{S_{\bullet}}M_{\bullet}$ defined by

$(T_{\bullet}\otimes_{S_{\bullet}}M_{\bullet})_{n}=T_{n}\otimes_{S_{n}}M_{n}\quad\mbox{for
all}\quad n\geqslant 0,$

whose connecting maps for $n\geqslant m$ are given by

$\mu^{T\otimes_{S}M}_{n,m}=\mu^{T}_{n,m}\otimes_{\mu^{S}_{n,m}}\mu^{M}_{n,m}:T_{n}\otimes_{S_{n}}M_{n}\longrightarrow
T_{m}\otimes_{S_{m}}M_{m}.$

######  Lemma 7.1.1.

If $M_{\bullet}$ quasi-coherent, then so is
$T_{\bullet}\otimes_{S_{\bullet}}M_{\bullet}$.

###### Proof.

Consider the following commutative square:

$\textstyle{T_{n}\underset{T_{n+1}}{\otimes}{}\left(T_{n+1}\underset{S_{n+1}}{\otimes}{}M_{n+1}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\mu^{T\otimes_{S}M}_{n+1,n}}$$\textstyle{T_{n}\underset{S_{n}}{\otimes}{}M_{n}}$$\textstyle{T_{n}\underset{S_{n+1}}{\otimes}{}M_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{T_{n}\underset{S_{n}}{\otimes}{}(S_{n}\underset{S_{n+1}}{\otimes}{}M_{n+1}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\mu^{M}_{n+1,n}}$

The vertical arrow on the left and the bottom horizontal arrow are
isomorphisms by the associativity of tensor product, whereas the vertical
arrow on the right is an isomorphism because $M_{\bullet}$ is quasi-coherent.
So the top horizontal arrow is also an isomorphism. ∎

For each integer $c\geqslant 0$, we can form the _shifted tower_
$S_{\bullet}[c]$:

$S_{\bullet}[c]=S_{c}\leftarrow S_{c+1}\leftarrow S_{c+2}\leftarrow\cdots$

so that $S_{\bullet}[c]_{n}=S_{n+c}$ for all $n\geqslant 0$. There is a _shift
functor_ $M_{\bullet}\mapsto M_{\bullet}[c]$ from $S_{\bullet}$-modules to
$S_{\bullet}[c]$-modules, given by $M_{\bullet}[c]_{n}=M_{n+c}$ for all
$n\geqslant 0$. The connecting maps $\mu^{S}_{n+c,c}:S_{n+c}\to S_{n}$ in
$S_{\bullet}$ induce a morphism of towers of rings

$\iota_{S}^{c}:S_{\bullet}[c]\longrightarrow S_{\bullet}$

and there is a similar morphism of towers of abelian groups
$\iota_{M}^{c}:M_{\bullet}[c]\to M_{\bullet}$.

######  Lemma 7.1.2.

Let $c\geqslant 0$ and let $M_{\bullet}$ be an $S_{\bullet}$-module.

  1. (a)

The map $\Gamma(\iota_{S}^{c}):\Gamma(S_{\bullet}[c])\to\Gamma(S_{\bullet})$
is a ring isomorphism.

  2. (b)

The map $\Gamma(\iota_{M}^{c}):\Gamma(M_{\bullet}[c])\to\Gamma(M_{\bullet})$
is an isomorphism of abelian groups.

###### Proof.

(a) Because $c\geqslant 0$, we have the projection map
$\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}S_{n}\to\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}S_{n+c}$,
and it is a two-sided inverse to
$\Gamma(\iota^{c}_{S})=\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\mu^{S}_{n+c,n}:\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}S_{n+c}\to\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}S_{n}$.

(b) This is entirely similar.∎

In fact, $\Gamma(\iota_{M}^{c})$ is a $\Gamma(S_{\bullet}[c])$-linear
isomorphism, when its codomain is regarded as a
$\Gamma(S_{\bullet}[c])$-module via the map $\Gamma(\iota^{c}_{S})$.

######  Proposition 7.1.3.

Let $c\geqslant 0$ and let $f_{\bullet}:T_{\bullet}\to S_{\bullet}$ and
$g_{\bullet}:S_{\bullet}[c]\to T_{\bullet}$ be morphisms of towers of rings
such that the diagram

$\textstyle{T_{\bullet}[c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\bullet}[c]}$$\scriptstyle{\iota^{c}_{T}}$$\textstyle{T_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\bullet}}$$\textstyle{S_{\bullet}[c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{c}_{S}}$$\scriptstyle{g_{\bullet}}$$\textstyle{S_{\bullet}}$

is commutative. Then:

  1. (a)

$\Gamma(f_{\bullet}):\Gamma(T_{\bullet})\to\Gamma(S_{\bullet})$ is a ring
isomorphism.

  2. (b)

Let $M_{\bullet}$ be a quasi-coherent $S_{\bullet}$-module and let
$L_{\bullet}:=T_{\bullet}\otimes_{S_{\bullet}[c]}M_{\bullet}[c]$. Then there
is a $\Gamma(T_{\bullet})$-linear isomorphism
$\Gamma(L_{\bullet})\to\Gamma(M_{\bullet})$, where the
$\Gamma(S_{\bullet})$-module $\Gamma(M_{\bullet})$ is regarded as a
$\Gamma(T_{\bullet})$-module via $\Gamma(f_{\bullet})$.

###### Proof.

(a) Applying $\Gamma$ gives the commutative diagram

$\textstyle{\Gamma(T_{\bullet}[c])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(f_{\bullet}[c])}$$\scriptstyle{\Gamma(\iota^{c}_{T})}$$\textstyle{\Gamma(T_{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(f_{\bullet})}$$\textstyle{\Gamma(S_{\bullet}[c])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(\iota^{c}_{S})}$$\scriptstyle{\Gamma(g_{\bullet})}$$\textstyle{\Gamma(S_{\bullet}).}$

Since the horizontal arrows are isomorphisms by Lemma 7.1.2(a), we conclude
that the diagonal arrow has both a right inverse and a left inverse. Hence it
is an isomorphism, and consequently so is $\Gamma(f_{\bullet})$.

(b) Noting that $L_{m}=T_{m}\otimes_{S_{m+c}}M_{m+c}$ for each $m\geqslant 0$,
we define additive maps

$M_{m+c}\quad\stackrel{{\scriptstyle\alpha_{m}}}{{\longrightarrow}}\quad
L_{m}\quad\stackrel{{\scriptstyle\beta_{m}}}{{\longrightarrow}}\quad M_{m}$

as follows: for $x,y\in M_{m+c}$ and $t\in T_{m}$ we set

(63) $\alpha_{m}(x)=1\otimes x\quad\mbox{and}\quad\beta_{m}(t\otimes
y)=f_{m}(t)\cdot\mu^{M}_{m+c,m}(y).$

Now let $n\geqslant m\geqslant 0$ and consider the following diagrams:

$\textstyle{M_{n+c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n}}$$\scriptstyle{\mu^{M}_{n+c,m+c}}$$\textstyle{L_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{n}}$$\scriptstyle{\mu^{L}_{n,m}}$$\textstyle{M_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{M}_{n,m}}$$\textstyle{M_{m+c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{m}}$$\textstyle{L_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{m}}$$\textstyle{M_{m}}$
$\textstyle{L_{m+c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{m+c}}$$\scriptstyle{\mu^{L}_{m+c,c}}$$\textstyle{L_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{m}}$$\textstyle{M_{m+c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{M}_{m+c,m}}$$\scriptstyle{\alpha_{m}}$$\textstyle{M_{m}.}$

The verification that these diagrams commute is straightforward; for example,
to show that $\alpha_{m}\circ\beta_{m+c}=\mu^{L}_{m+c,c}$, let $t\in T_{m+c}$
and $y\in M_{m+2c}$ and use the fact that $g_{m}\circ f_{m+c}=\mu^{T}_{m+c,m}$
to calculate as follows:

$\begin{array}[]{lll}\alpha_{m}(\beta_{m+c}(t\otimes
y))&=&\alpha_{m}(f_{m+c}(t)\cdot\mu^{M}_{m+2c,m+c}(y))\\\ &=&1\otimes
f_{m+c}(t)\cdot\mu^{M}_{m+2c,m+c}(y)\\\
&=&g_{m}(f_{m+c}(t))\otimes\mu^{M}_{m+2c,m+c}(y)\\\
&=&\mu^{T}_{m+c,m}(t)\otimes\mu^{M}_{m+2c,m+c}(y)\\\
&=&\mu^{L}_{m+c,m}(t\otimes y).\end{array}$

Thus we have defined morphisms of towers of abelian groups
$\alpha_{\bullet}:M_{\bullet}[c]\to L_{\bullet}$ and
$\beta_{\bullet}:L_{\bullet}\to M_{\bullet}$, and they fit into the following
commutative diagram:

$\textstyle{L_{\bullet}[c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{\bullet}[c]}$$\scriptstyle{\iota^{c}_{L}}$$\textstyle{L_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{\bullet}}$$\textstyle{M_{\bullet}[c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{c}_{M}}$$\scriptstyle{\alpha_{\bullet}}$$\textstyle{M_{\bullet}.}$

Considering the $T_{\bullet}$-module $L_{\bullet}$ as an
$S_{\bullet}[c]$-module via the morphism $g_{\bullet}:S_{\bullet}[c]\to
T_{\bullet}$, and the $S_{\bullet}$-module $M_{\bullet}$ as a
$T_{\bullet}$-module via the morphism $f_{\bullet}:T_{\bullet}\to
S_{\bullet}$, we can verify that $\alpha_{\bullet}$ becomes a morphism of
$S_{\bullet}[c]$-modules, whereas $\beta_{\bullet}$ becomes a morphism of
$T_{\bullet}$-modules. Apply the global sections functor $\Gamma$ to this
diagram. Using Lemma 7.1.2(b) we deduce as in part (a) that
$\Gamma(\beta_{\bullet}):\Gamma(L_{\bullet})\to\Gamma(M_{\bullet})$ is an
isomorphism of $\Gamma(T_{\bullet})$-modules. ∎

###  7.2. An alternative presentation for the algebra
$\wideparen{\mathcal{D}}(X,J)$

We now return to the setting of $\S\ref{CoadDrin}$ so that
$\mathbb{D}\subset\mathbb{P}^{1}$ is the closed unit disc, with local
coordinate $x$. Recall the $G$-topology $\mathbb{D}_{m}$ on $\mathbb{D}$ from
Definition 4.3.9.

######  Definition 7.2.1.

For each affinoid subdomain $X$ of $\mathbb{D}$, let $m_{X}$ be the least
integer $m$ such that $X\in\mathbb{D}_{m}$.

######  Lemma 7.2.2.

Let $X$ be an affinoid subdomain of $\mathbb{D}$ and let $m\geqslant 0$. The
derivation $\pi_{F}^{m}\partial_{x}$ of $\mathcal{O}(X)$ preserves
$\mathcal{O}(X)^{\circ}$ if and only if $m\geqslant m_{X}$.

###### Proof.

By Definitions 4.3.9(a) and 2.3.9(a), $m\geqslant m_{X}$ if and only if
$r(X)\leqslant\varpi/|{\pi_{F}}|^{m}$. By Corollary 2.3.19, this is equivalent
to $||\partial_{x}||_{\mathcal{O}(X)}\leqslant 1/|{\pi_{F}}|^{m}$. This, in
turn, is equivalent to $\pi_{F}^{m}\partial_{x}$ preserving
$\mathcal{O}(X)^{\circ}$ inside $\mathcal{O}(X)$. ∎

######  Notation 7.2.3.

We will write $\mathcal{A}:=\mathcal{O}(X)^{\circ}$ and
$\mathcal{L}:=\mathcal{A}\partial_{x}$ until the end of
$\S\ref{CompareSandT}$.

We refer the reader to [7, Definition 6.1] for the definition of Lie lattices.

######  Corollary 7.2.4.

Let $X$ be an affinoid subdomain of $\mathbb{D}$. For every $m\geqslant
m_{X}$,

  1. (a)

$\pi_{F}^{m}\mathcal{L}$ is an $\mathcal{A}$-Lie lattice in $\mathcal{T}(X)$,
and

  2. (b)

there is a natural isometric isomorphism of $K$-Banach algebras

$j:\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{D}_{|{\pi_{F}}|^{-m}}(X).$

###### Proof.

(a) Because $m\geqslant m_{X}$, Lemma 7.2.2 implies that
$\pi_{F}^{m}\mathcal{L}$ is a sub $(K^{\circ},\mathcal{A})$-Lie algebra of
$\mathcal{T}(X)$. Now apply [7, Definition 6.1].

(b) There is a unique $\mathcal{A}$-linear map
$j_{\pi_{F}^{m}\mathcal{L}}:\pi_{F}^{m}\mathcal{L}\to\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)=:D$
that sends $\pi_{F}^{m}\partial_{x}$ to $\pi_{F}^{m}\partial$. Let
$j_{\mathcal{A}}$ denote the inclusion of $\mathcal{A}$ into $D$. It is easy
to verify that $j_{\pi_{F}^{m}\mathcal{L}}$ respects Lie brackets and
satisfies
$[j_{\pi_{F}^{m}\mathcal{L}}(v),j_{\mathcal{A}}(a)]=j_{\pi_{F}^{m}\mathcal{L}}(v(a))$
for all $v\in\pi_{F}^{m}\mathcal{L}$ and $a\in\mathcal{A}$. The universal
property [7, §2.1] of the enveloping algebra $U(\pi_{F}^{m}\mathcal{L})$
implies that $j_{\pi_{F}^{m}\mathcal{L}}$ extends to an $K^{\circ}$-algebra
homomorphism $j:U(\pi_{F}^{m}\mathcal{L})\to D$. Because the image of this map
by construction lands in the unit ball of the $K$-Banach algebra $D$, it
extends further to a $K$-Banach algebra homomorphism
$j:\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}\to D$. Finally, Rinehart’s Theorem
[39, Theorem 3.1] implies that $\\{\pi_{F}^{mn}\partial_{x}^{n}:n\geqslant
0\\}$ is an orthonormal basis for the $\mathcal{O}(X)$-Banach module
$\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}$ in the sense of [22, §1.2, p.7],
whereas it follows from the construction of $D$ that
$\\{\pi_{F}^{mn}\partial^{n}:n\geqslant 0\\}$ is an orthonormal basis for $D$.
Since $j$ sends $\partial_{x}^{n}$ to $\partial^{n}$ for each $n\geqslant 0$,
we conclude that it is an isometric isomorphism. ∎

Recall from Notation 5.3.1 the _Iwahori subgroup_ $I$ of
$GL_{2}(\mathcal{O}_{F})$, given by

(64) $I:=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
GL_{2}(\mathcal{O}_{F}):c\equiv
0\operatorname{mod}\pi_{F}\mathcal{O}_{F}\right\\}.$

We fix a closed subgroup $J$ of $I$ and let $X$ be a $J$-stable affinoid
subdomain of $\mathbb{D}$. We now recall the construction of the algebra
$\wideparen{\mathcal{D}}(X,J)$ from [9, Definition 3.3.1].

Recall from [9, Definition 3.2.13] that $(\mathcal{J},H)$ is an _$\mathcal{A}$
-trivialising pair_ if $\mathcal{J}$ is a $J$-stable $\mathcal{A}$-Lie lattice
in $\operatorname{Der}_{K}(\mathcal{O}(X))$ and $H$ is an open normal subgroup
of $J$ such that $\rho_{X}(H)\leqslant\exp(p^{\epsilon}\mathcal{J})$ inside
$\operatorname{Aut}_{K}(\mathcal{O}(X))$, where
$\rho_{X}:J\to\operatorname{Aut}_{K}(\mathcal{O}(X))$ is the action of $J$ on
$\mathcal{O}(X)$. For each $\mathcal{A}$-trivialising pair $(\mathcal{J},H)$,
there is a crossed product $\widehat{U(\mathcal{J})_{K}}\rtimes_{H}J$, and
$\wideparen{\mathcal{D}}(X,J)$ is the inverse limit of these crossed products
taken over all possible $\mathcal{A}$-trivialising pairs $(\mathcal{J},H)$.

######  Lemma 7.2.5.

Let $X$ be a $J$-stable affinoid subdomain of $\mathbb{D}$. Then
$(\pi_{F}^{m}\mathcal{L},J_{m+\epsilon v_{\pi_{F}}(p)})$ is an
$\mathcal{A}$-trivialising pair whenever $m\geqslant m_{X}$.

###### Proof.

The affine formal model $\mathcal{A}=\mathcal{O}(X)^{\circ}$ in
$\mathcal{O}(X)$ is $J$-stable since $X$ is $J$-stable, and it is
$\pi_{F}^{m}\mathcal{L}$-stable by Lemma 7.2.2. Hence by [9, Corollary 4.3.7],
it will be enough to verify that
$(\pi_{F}^{m}\mathcal{O}(\mathbb{D})^{\circ}\partial_{x},J_{m+\epsilon
v_{\pi_{F}}(p)})$ is an $\mathcal{O}(\mathbb{D})^{\circ}$-trivialising pair.
Therefore we may assume that $X=\mathbb{D}$.

Recall congruence subgroup $J_{n}$ of $J$ from Definition 5.3.1 and note that
$\rho_{\mathbb{D}}(J_{n})$ acts trivially on $\mathcal{O}(\mathbb{D})^{\circ}$
modulo $\pi_{F}^{n}$. If $\mathcal{E}$ denotes the ring of $K^{\circ}$-linear
endomorphisms of $\mathcal{O}(\mathbb{D})^{\circ}$, then
$\rho_{\mathbb{D}}(J_{m+\epsilon v_{\pi_{F}}(p)})\leqslant
1+p^{\epsilon}\pi_{F}^{m}\mathcal{E}$. Since every element of
$\rho_{\mathbb{D}}(J)$ is an $K^{\circ}$-algebra automorphism of
$\mathcal{O}(\mathbb{D})^{\circ}$, it follows from the proof of [9, Lemma
3.2.5(a)] that $\log\rho_{\mathbb{D}}(J_{m+\epsilon v_{\pi_{F}}(p)})\leqslant
p^{\epsilon}\pi_{F}^{m}\operatorname{Der}_{K^{\circ}}\mathcal{O}(\mathbb{D})^{\circ}=p^{\epsilon}\pi_{F}^{m}\mathcal{O}(\mathbb{D})^{\circ}\partial_{x}$.
∎

After Lemma 7.2.5, [9, Theorem 3.2.12] and Corollary 7.2.4(b), we have at our
disposal the crossed product

$\hskip 2.84544pt\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)\underset{J_{m+\epsilon
v_{\pi_{F}}(p)}}{\rtimes}{}J$

for every $J$-stable affinoid subdomain $X$ of $\mathbb{D}$ and every
$m\geqslant m_{X}$. Note that the trivialisation of the $J_{m+\epsilon
v_{\pi_{F}}(p)}$-action on $\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)$ is
$j\circ\beta_{\pi_{F}^{m}\mathcal{L}}$, where $\beta_{\pi_{F}^{m}\mathcal{L}}$
is the trivialisation of the $J_{m+\epsilon v_{\pi_{F}}(p)}$-action on
$\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}$ obtained from [9, Theorem 3.2.12]
and Lemma 7.2.5, and $j$ comes from Corollary 7.2.4(b). On the other hand,
when $m\geqslant m_{X}$, by Proposition 5.3.4 we also have the crossed product

$\mathscr{D}_{m}(X)\underset{J_{m+1}}{\rtimes}{}J\hskip 2.84544pt=\hskip
2.84544pt\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{m}}(X)\underset{J_{m+1}}{\rtimes}{}J.$

######  Definition 7.2.6.

Let $X$ be a $J$-stable affinoid subdomain of $\mathbb{D}$ and let $m\geqslant
m_{X}$. Define

  1. (a)

$\mathcal{T}_{m}(X)\hskip 2.84544pt:=\hskip
2.84544pt\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)\underset{J_{m+\epsilon
v_{\pi_{F}}(p)}}{\rtimes}{}J$, and

  2. (b)

$\mathcal{S}_{m}(X):=\mathscr{D}_{m}(X)\underset{J_{m+1}}{\rtimes}{}J\hskip
2.84544pt=\hskip
2.84544pt\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{m}}(X)\underset{J_{m+1}}{\rtimes}{}J$.

Note that the algebra $\mathcal{T}_{m}(X)$ is functorial in $X$ and therefore
defines a presheaf $\mathcal{T}_{m}$ on the $G$-topology $\mathbb{D}_{m}/J$ —
see Definition 5.3.1(d).

######  Proposition 7.2.7.

For every $J$-stable affinoid subdomain $X$ of $\mathbb{D}$, the tower

$\mathcal{T}_{m_{X}}(X)\leftarrow\mathcal{T}_{m_{X}+1}(X)\leftarrow\mathcal{T}_{m_{X}+2}(X)\leftarrow\cdots$

gives a Fréchet-Stein presentation for the algebra
$\wideparen{\mathcal{D}}(X,J)$.

###### Proof.

Use Lemma 7.2.5, [9, Lemma 3.3.4] and the proof of [9, Theorem 3.4.8]. ∎

We will now make a precise comparison of the two constructions.

######  Theorem 7.2.8.

For every $m\geqslant 0$, there is a commutative diagram

$\textstyle{\mathcal{T}_{m+\epsilon
v_{\pi_{F}}(p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{m+\epsilon
v_{\pi_{F}}(p)}}$$\textstyle{\mathcal{T}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{m}}$$\textstyle{\mathcal{D}\rtimes
J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{m+\epsilon
v_{\pi_{F}}(p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{m}}$$\textstyle{\mathcal{S}_{m}.}$

of presheaves of $K$-algebras on $\mathbb{D}_{m}/J$.

###### Proof.

Fix $X\in\mathbb{D}_{m}/J$. We will first construct the $K$-algebra map

$f_{m}(X):\mathcal{T}_{m}(X)\to\mathcal{S}_{m}(X).$

Let $r$ be a real number satisfying
$\varpi/|{\pi_{F}}|^{m}<r<|{\pi_{F}}|^{-m}\min\\{\varpi/|{\pi_{F}}|,1\\}$.
Then in particular $r<|{\pi_{F}}|^{-m}$, so by Lemma 2.2.10 there is a
$K$-Banach algebra homomorphism
$f_{m,r}:\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)\to\mathcal{D}_{r}(X)$ which fixes
$\mathcal{O}(X)$ and sends $\partial\in\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)$ to
$\partial\in\mathcal{D}_{r}(X)$. This map is $J$-equivariant for the
$J$-actions on $\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)$ and $\mathcal{D}_{r}(X)$
that were constructed in Proposition 5.2.4. Because also
$\varpi/|{\pi_{F}}|^{m}<r<\varpi/|{\pi_{F}}|^{m+1}$, we can apply Lemma 5.3.3
together with Theorem 5.2.11 to construct the crossed product
$\mathcal{D}_{r}(X)\underset{J_{m+1}}{\rtimes}J$ as in the proof of
Proposition 5.3.4. Then by [9, Lemma 2.2.7], $f_{m,r}$ extends to a
$K$-algebra homomorphism

$f_{m,r}\rtimes 1_{J}:\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)\underset{J_{m+\epsilon
v_{\pi_{F}}(p)}}{\rtimes}{}J\quad\longrightarrow\quad\mathcal{D}_{r}(X)\underset{J_{m+1}}{\rtimes}J$

provided we can show that it respects the two trivialisations

$j\circ\beta_{\pi_{F}^{m}\mathcal{L}}:J_{m+\epsilon
v_{\pi_{F}}(p)}\longrightarrow\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)^{\times}\quad\mbox{and}\quad\beta:J_{m+1}\to\mathcal{D}_{r}(X)^{\times}.$

Recall from [9, Lemma 3.2.10(a)] the action map
$\psi_{\pi_{F}^{m}\mathcal{L}}:\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}\to\mathcal{B}(\mathcal{O}(X))$
so that $\psi_{\pi_{F}^{m}\mathcal{L}}\circ\beta_{\pi_{F}^{m}\mathcal{L}}$ is
the restriction of $\rho_{X}$ to $J_{m+\epsilon v_{\pi_{F}}(p)}$ by [9,
Definition 3.2.11]. Recall the action map
$\sigma_{r}:\mathcal{D}_{r}(X)\to\mathcal{B}(\mathcal{O}(X))$ from Lemma
2.3.12, and note that $\psi_{\pi_{F}^{m}\mathcal{L}}$ is just the restriction
of $\sigma_{r}$ to $\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}$; more precisely,
we have

(65) $\sigma_{r}\circ f_{m,r}\circ j=\psi_{\pi_{F}^{m}\mathcal{L}}$

where $j$ is the isomorphism from Corollary 7.2.4(b). Then

$\sigma_{r}\circ f\circ
j\circ\beta_{\pi_{F}^{m}\mathcal{L}}=\psi_{\pi_{F}^{m}\mathcal{L}}\circ\beta_{\pi_{F}^{m}\mathcal{L}}=\rho_{X|J_{m+\epsilon
v_{\pi_{F}}(p)}}=\sigma_{r}\circ\beta_{|J_{m+\epsilon v_{\pi_{F}}(p)}}$

by Proposition 5.2.10, and therefore $\beta_{|J_{m+\epsilon
v_{\pi_{F}}(p)}}=f\circ j\circ\beta_{\pi_{F}^{m}\mathcal{L}}$ because
$\sigma_{r}$ is injective by Lemma 2.3.12(b). This is precisely what is
required to apply [9, Lemma 2.2.7], and completes the construction of
$f_{m,r}\rtimes 1_{J}$. It is clear that if
$\varpi/|{\pi_{F}}|^{m}<r^{\prime}\leqslant r$ then $f_{m,r^{\prime}}\rtimes
1$ is compatible with $f_{m,r}\rtimes 1$ in the sense that
$f_{m,r^{\prime}}\rtimes 1_{J}$ is the composition of $f_{m,r}\rtimes 1_{J}$
with
$\mathcal{D}_{r}(X)\rtimes_{J_{m+1}}J\to\mathcal{D}_{r^{\prime}}(X)\rtimes_{J_{m+1}}J$.
We can now define

$f_{m}(X):\mathcal{T}_{m}(X)\to\mathcal{S}_{m}(X)$

to be the colimit of the maps $f_{m,r}\rtimes 1_{J}$ as $r$ approaches
$\varpi/|{\pi_{F}}|^{m}$ from above.

We will next similarly construct the $K$-algebra homomorphism

$g_{m}(X):\mathcal{S}_{m+\epsilon v_{\pi_{F}}(p)}(X)\to\mathcal{T}_{m}(X).$

Let $r$ be a real number such that $\varpi<r|{\pi_{F}}|^{m+\epsilon
v_{\pi_{F}}(p)}<\varpi/{\pi_{F}}$, so that the crossed product
$\mathcal{D}_{r}(X)\rtimes_{J_{m+\epsilon v_{\pi_{F}}(p)+1}}J$ is defined as
in the proof of Proposition 5.3.4. Note that then
$r|{\pi_{F}}|^{m}>\varpi/|{\pi_{F}}|^{\epsilon
v_{\pi_{F}}(p)}=|p|^{\frac{1}{p-1}-\epsilon}\geqslant 1$, so
$r\geqslant|{\pi_{F}}|^{-m}$. Hence by Lemma 2.2.10, there is a natural
$K$-Banach algebra homomorphism
$g_{m,r}:\mathcal{D}_{r}(X)\to\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)$ which fixes
$\mathcal{O}(X)$ and sends $\partial\in\mathcal{D}_{r}(X)$ to
$\partial\in\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)$. Now, the diagram

$\textstyle{\mathcal{D}_{r}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{m,r}}$$\scriptstyle{\sigma_{r}}$$\textstyle{\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{-1}}$$\scriptstyle{\cong}$$\textstyle{\mathcal{B}(\mathcal{O}(X))}$$\textstyle{\widehat{U(\pi_{F}^{m}\mathcal{L})_{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\pi_{F}^{m}\mathcal{L}}}$

is commutative, so that $\sigma_{r}=\psi_{\pi_{F}^{m}\mathcal{L}}\circ
j^{-1}\circ g_{m,r}$. Therefore

$\psi_{\pi_{F}^{m}\mathcal{L}}\circ j^{-1}\circ
g_{m,r}\circ\beta=\sigma_{r}\circ\beta=\rho_{X}=\psi_{\pi_{F}^{m}\mathcal{L}}\circ\beta_{\pi_{F}^{m}\mathcal{L}|J_{m+\epsilon
v_{\pi_{F}}(p)+1}}$

again by Proposition 5.2.10 and [9, Definition 3.2.11]. The factorisation (65)
together with Lemma 2.3.12(b) shows that $\psi_{\pi_{F}^{m}\mathcal{L}}$ is
injective, and we deduce that

$g_{m,r}\circ\beta=j\circ\beta_{\pi_{F}^{m}\mathcal{L}|J_{m+\epsilon
v_{\pi_{F}}(p)+1}}.$

Because the map $g_{m,r}$ is $J$-equivariant, this equation allows us to apply
[9, Lemma 2.2.7] and deduce that $g_{m,r}$ extends to a $K$-algebra
homomorphism

$g_{m,r}\rtimes 1_{J}:\mathcal{D}_{r}(X)\underset{J_{m+\epsilon
v_{\pi_{F}}(p)+1}}{\rtimes}{}J\quad\longrightarrow\quad\mathcal{D}_{|{\pi_{F}}|^{-m}}(X)\underset{J_{m+\epsilon
v_{\pi_{F}}(p)}}{\rtimes}{}J.$

Passing to the limit as $r$ approaches $\varpi/|{\pi_{F}}|^{m+\epsilon
v_{\pi_{F}}(p)}$ from above, we obtain the required $K$-algebra homomorphism

$g_{m}(X):\mathcal{S}_{m+\epsilon v_{\pi_{F}}(p)}(X)\to\mathcal{T}_{m}(X).$

Finally, we leave to the reader the straightforward verification that the
diagram in the statement of the Theorem is commutative, and that the morphisms
$f_{m}(X)$ and $g_{m}(X)$ are functorial in $X$. ∎

######  Corollary 7.2.9.

For each $n\geqslant 0$ there is an isomorphism

$\wideparen{\mathcal{D}}(-,J)\quad\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\quad\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{m\geqslant
n}}}\mathscr{D}_{m}\underset{J_{m+1}}{\rtimes}{}J$

of presheaves of $K$-algebras on $\mathbb{D}_{n}/J$.

###### Proof.

Use Proposition 7.2.7, Theorem 7.2.8 and Proposition 7.1.3(a). ∎

For future use, we establish here the following flatness result.

######  Theorem 7.2.10.

$\mathcal{S}_{n}(X)$ is a flat right $\mathcal{T}_{n}(X)$-module whenever
$X\in\mathbb{D}_{n}/J$.

###### Proof.

Write $c:=\epsilon v_{\pi_{F}}(p)$. Form the crossed product
$C:=\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{n}}(X)\underset{J_{n+c}}{\rtimes}{}J$,
and consider the following commutative diagram:

$\textstyle{\mathcal{T}_{n}(X)=\mathcal{D}_{|{\pi_{F}}|^{-n}}(X)\underset{J_{n+c}}{\rtimes}{}J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{n}(X)=\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{n}}(X)\underset{J_{n+1}}{\rtimes}{}J}$$\textstyle{C=\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{n}}(X)\underset{J_{n+c}}{\rtimes}{J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D:=\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{n}}(X).\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Note that $C$ is a flat right $D$-module being a crossed product of $D$ with
the group $J/J_{n+c}$, which allows us to deduce from the right-module version
of [40, Lemma 2.2] that $\mathcal{S}_{n}(X)$ is a flat right $C$-module. On
the other hand, we deduce from Theorem 3.4.1 and Lemma 3.2.2 that $C$ is a
flat right $\mathcal{T}_{n}(X)$-module. Hence $\mathcal{S}_{n}(X)$ is a flat
right $\mathcal{T}_{n}(X)$-module as claimed. ∎

###  7.3. Coadmissibility of $j_{\ast}\mathscr{L}$ on the local Drinfeld space

We begin with the following elementary result.

######  Lemma 7.3.1.

Let $D\hookrightarrow S$ be an extension of rings such that $S$ is finitely
generated and projective as a left $D$-module. Then every $S$-module $M$ which
is finitely presented as a $D$-module is also finitely presented as an
$S$-module.

###### Proof.

We can find a finite generating set $\\{v_{1},\ldots,v_{n}\\}$ for $M$ as a
$D$-module such that if $\theta:D^{n}\twoheadrightarrow M$ denotes the
corresponding surjection then $\ker\theta$ is also finitely generated. Clearly
$\\{v_{1},\ldots,v_{n}\\}$ also generates $M$ as an $S$-module; let
$\psi:S^{n}\twoheadrightarrow M$ be the corresponding surjection. Since $S$ is
a projective $D$-module, Schanuel’s Lemma implies that
$S^{n}\oplus\ker\theta\cong D^{n}\oplus\ker\psi$ as $D$-modules. Since $S$ is
finitely generated as a $D$-module, $\ker\psi$ is finitely generated as a
$D$-module being a homomorphic image of $S^{n}\oplus\ker\theta$. Hence
$\ker\psi$ is also finitely generated as an $S$-module. ∎

We now return to the setting of $\S\ref{CompSect}$. We assume
$[\mathscr{L}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$
satisfies Definition 5.3.2, and fix a closed subgroup $J$ of the Iwahori group
$I$.

######  Corollary 7.3.2.

Let $n\geqslant v_{\pi_{F}}(e)$ and let $X\in\mathbb{D}_{n}/J$. Then
$\mathscr{L}_{n}(X)$ is a finitely presented $\mathcal{S}_{n}(X)$-module.

###### Proof.

We know that $\mathscr{L}_{n}(X)$ is a finitely presented
$\mathscr{D}_{n}(X)$-module, by Theorem 4.3.7 and Corollary 4.3.12. The
$\mathscr{D}_{n}(X)$-action on $\mathscr{L}_{n}(X)$ extends to an
$\mathcal{S}_{n}(X)$-action by Theorem 5.3.6. By construction,
$\mathcal{S}_{n}(X)$ is a crossed product of $\mathscr{D}_{n}(X)$ with the
finite group $J/J_{n+1}$ and is therefore finitely generated and free as a
left $\mathscr{D}_{n}(X)$-module. Now apply Lemma 7.3.1. ∎

The following consequence will be useful later on.

######  Proposition 7.3.3.

Let $n\geqslant v_{\pi_{F}}(e)$ and let $X\in\mathbb{D}_{n}/J$. Then the map

$\mathcal{S}_{n}(X)\underset{\mathcal{S}_{n}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n}(\mathbb{D})\longrightarrow\mathscr{L}_{n}(X)$

induced by the $\mathcal{S}_{n}(X)$-action on $\mathscr{L}_{n}(X)$ is an
isomorphism.

###### Proof.

We abbreviate
$D:=\mathscr{D}_{n}(\mathbb{D}),D^{\prime}:=\mathscr{D}_{n}(X),S:=\mathcal{S}_{n}(\mathbb{D})$
and $S^{\prime}:=\mathcal{S}_{n}(X)$; these rings form a commutative square

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{\prime}.}$

There is a natural transformation $\eta:D^{\prime}\otimes_{D}-\to
S^{\prime}\otimes_{S}-$ between two functors from $S$-modules to
$D^{\prime}$-modules; since $S$ (respectively, $S^{\prime}$) is a crossed
product of $D$ (respectively, $D^{\prime}$) with the same finite group
$J/J_{n+1}$, we see that the map $\eta_{S}$ is an isomorphism. Since both
functors are right exact, we conclude using the Five Lemma that $\eta_{M}$ is
an isomorphism for every finitely presented $S$-module $M$.

Now let $M:=\mathscr{L}_{n}(\mathbb{D})$ and $M^{\prime}:=\mathscr{L}_{n}(X)$
and consider the commutative triangle

$\textstyle{D^{\prime}\underset{D}{\otimes}{}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{M}}$$\textstyle{S^{\prime}\underset{S}{\otimes}{}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\prime}.}$

Since $M$ is a finitely presented $S$-module by Corollary 7.3.2, the
horizontal arrow $\eta_{M}$ is an isomorphism by the above. Because the
diagonal arrow on the left is an isomorphism by Lemma 5.3.13(a), we conclude
that the diagonal arrow on the right is also an isomorphism. ∎

Recall from Definition 5.3.11 that $N=\begin{pmatrix}1&\mathcal{O}_{F}\\\
0&1\end{pmatrix}$.

######  Proposition 7.3.4.

Let $X$ be a $J$-stable affinoid subdomain of $\mathbb{D}$. Suppose that
$d\mid(q+1)$ and $N_{m_{0}+1}\leqslant J$ for some
$m_{0}\geqslant\max\\{m_{X},v_{\pi_{F}}(e)\\}$. Then the canonical action map

$\mathcal{S}_{n}(X)\underset{\mathcal{S}_{n+1}(X)}{\otimes}{}\mathscr{L}_{n+1}(X)\longrightarrow\mathscr{L}_{n}(X)$

is an isomorphism for all $n\geqslant m_{0}$.

###### Proof.

This map appears as the top horizontal arrow in the commutative diagram

$\textstyle{\mathcal{S}_{n}(X)\underset{\mathcal{S}_{n+1}(X)}{\otimes}{}\mathscr{L}_{n+1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{L}_{n}(X)}$$\textstyle{\mathcal{S}_{n}(X)\underset{\mathcal{S}_{n+1}(X)}{\otimes}{}\left(\mathcal{S}_{n+1}(X)\underset{\mathcal{S}_{n+1}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n+1}(\mathbb{D})\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{S}_{n}(X)\underset{\mathcal{S}_{n+1}(\mathbb{D})}{\otimes}\mathscr{L}_{n+1}(\mathbb{D}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Now $X\in\mathbb{D}_{n+1}/J$ because $n+1>n\geqslant m_{0}\geqslant m_{X}$, so
the vertical arrow on the left is an isomorphism by Proposition 7.3.3. Also,
because $n\geqslant m_{0}$ and because $N_{m_{0}+1}\leqslant J$ by assumption,
we have

$N_{n+1}=N_{m_{0}+1}\cap I_{n+1}\leqslant J\cap I_{n+1}=J_{n+1}.$

Therefore the vertical arrow on the right is an isomorphism by Corollary
5.3.17. The bottom horizontal arrow is an isomorphism by the associativity of
tensor products and the result follows. ∎

We can now invoke the formalism of $\S\ref{QcohTower}$ and prove our first
main result.

######  Theorem 7.3.5.

Let $X$ be a $J$-stable affinoid subdomain of $\mathbb{D}$. Suppose that
$d\mid(q+1)$ and $N_{m_{0}+1}\leqslant J$ for some
$m_{0}\geqslant\max\\{m_{X},v_{\pi_{F}}(e)\\}$. Then

  1. (a)

$(j_{\ast}\mathscr{L})(X)$ is a coadmissible
$\wideparen{\mathcal{D}}(X,J)$-module, and

  2. (b)

for each $m\geqslant m_{0}$, the canonical map

$\mathcal{S}_{m}(X)\underset{\wideparen{\mathcal{D}}(X,J)}{\otimes}{}(j_{\ast}\mathscr{L})(X)\longrightarrow\mathscr{L}_{m}(X)$

is an isomorphism.

###### Proof.

(a) For each $m\geqslant 0$, let $S_{m}:=\mathcal{S}_{m_{0}+m}(X)$,
$T_{m}:=\mathcal{T}_{m_{0}+m}(X)$ and $M_{m}:=\mathscr{L}_{m_{0}+m}(X)$. Then
$S_{\bullet}$ is a tower of rings, and $M_{\bullet}$ is a quasi-coherent
$S_{\bullet}$-module by Proposition 7.3.4. Theorem 7.2.8 gives us a diagram of
towers of rings

$\textstyle{T_{\bullet}[c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\bullet}[c]}$$\scriptstyle{\iota^{c}_{T}}$$\textstyle{T_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\bullet}}$$\textstyle{S_{\bullet}[c]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{c}_{S}}$$\scriptstyle{g_{\bullet}}$$\textstyle{S_{\bullet}}$

where $c:=\epsilon v_{\pi_{F}}(p)$. Note that
$\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}T_{m}\cong\wideparen{\mathcal{D}}(X,J)$
by Proposition 7.2.7. Now
$L_{\bullet}:=T_{\bullet}\otimes_{S_{\bullet}[c]}M_{\bullet}[c]$ is a quasi-
coherent $T_{\bullet}$-module by Lemma 7.1.1, and each

$L_{m}=T_{m}\otimes_{S_{m+c}}M_{m+c}$

is a finitely presented $T_{m}$-module because $M_{m+c}$ is a finitely
presented $S_{m+c}$-module by Corollary 7.3.2. In other words,
$\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}L_{m}$
is a coadmissible
$\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}T_{m}\cong\wideparen{\mathcal{D}}(X,J)$-module.
Now, Proposition 7.1.3(b) implies that there is a natural isomorphism

(66)
$\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M_{m}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}L_{m}.$

Since the restriction map
$(j_{\ast}\mathscr{L})(X)\to\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M_{m}$
is an isomorphism by Proposition 4.3.3, we conclude that
$(j_{\ast}\mathscr{L})(X)$ is a coadmissible
$\wideparen{\mathcal{D}}(X,J)$-module as claimed.

(b) Write $T_{\infty}:=\wideparen{\mathcal{D}}(X,J)$ and
$M_{\infty}:=\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M_{m}$.
By [44, Corollary 3.1], the canonical map
$\alpha_{m}:T_{m}\otimes_{T_{\infty}}M_{\infty}\to L_{m}$ is an isomorphism
for each $m\geqslant 0$. This map, together with the map
$\psi_{m}:S_{m}\otimes_{T_{\infty}}M_{\infty}\to M_{m}$ in question, appears
in the following commutative diagram:

$\textstyle{S_{m}\underset{T_{\infty}}{\otimes}{}M_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{m}}$$\scriptstyle{\cong}$$\textstyle{M_{m}}$$\textstyle{S_{m}\underset{T_{m}}{\otimes}{}(T_{m}\underset{T_{\infty}}{\otimes}{}M_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\alpha_{m}}$$\scriptstyle{\cong}$$\textstyle{S_{m}\underset{T_{m}}{\otimes}{}(T_{m}\underset{S_{m+c}}{\otimes}{}M_{m+c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{S_{m}\underset{S_{m+c}}{\otimes}{}M_{m+c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the arrow on the right is induced by the action of $S_{m}$ on $M_{m}$.
We see that this arrow is an isomorphism by a repeated application of
Proposition 7.3.4, and conclude that $\psi_{m}$ is also an isomorphism as
required. ∎

######  Lemma 7.3.6.

$j_{\ast}\mathscr{L}$ is a locally Fréchet $I$-equivariant
$\mathcal{D}$-module on $\mathbb{D}$.

###### Proof.

The local Drinfeld space $\Upsilon=\mathbb{D}\cap\Omega$ is an admissible
$I$-stable open subspace of $\mathbb{D}$. Since $\mathscr{L}$ is an
$I$-equivariant $\mathcal{D}$-module on $\Upsilon$, its pushforward
$j_{\ast}\mathscr{L}$ is an $I$-equivariant $\mathcal{D}$-module on
$\mathbb{D}$. According [9, Definition 3.6.1(a)] we must show that
$(j_{\ast}\mathscr{L})(U)$ carries a Fréchet topology for each affinoid
subdomain $U$ of $\mathbb{D}$, and that the action maps
$g^{j_{\ast}\mathscr{L}}(U):(j_{\ast}\mathscr{L})(U)\to(j_{\ast}\mathscr{L})(gU)$
are continuous for each $g\in I$. We observed in $\S\ref{UpperHP}$ that
$\Upsilon$ admits a quasi-Stein covering $(\Upsilon_{n})_{n=0}^{\infty}$.
Therefore $U\cap\Omega$ admits a quasi-Stein covering $(U\cap
V_{n})_{n=0}^{\infty}$ and there is a natural isomorphism

$\mathscr{L}(U\cap\Omega)=\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\mathscr{L}(U\cap
V_{n}).$

Since $\mathscr{L}$ is a coherent $\mathcal{O}$-module on $\Upsilon$ and since
$U\cap V_{n}$ is affinoid, each $\mathscr{L}(U\cap V_{n})$ is naturally a
$K$-Banach space. In this way, we see that
$(j_{\ast}\mathscr{L})(U)=\mathscr{L}(U\cap\Omega)$ carries a natural
$K$-Fréchet space topology.

The continuity of the maps
$g^{j_{\ast}\mathscr{L}}(U):(j_{\ast}\mathscr{L})(U)\to(j_{\ast}\mathscr{L})(gU)$
now follows from the continuity of the maps $g^{\mathscr{L}}(U\cap
V_{n}):\mathscr{L}(U\cap V_{n})\to\mathscr{L}(g(U\cap V_{n}))$, which holds by
viewing $\mathscr{L}(g(U\cap V_{n}))$ as an $\mathcal{O}(U\cap V_{n})$-modulle
via $g^{\mathcal{O}}(U\cap V_{n})$, automatically by [22, Corollary 1.2.4]. ∎

We can finally prove our first main local result.

######  Theorem 7.3.7.

Let
$[\mathscr{L}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$.
Suppose that the image $\omega[\mathscr{L}]$ of $[\mathscr{L}]$ in
$\operatorname{PicCon}(\Upsilon)$ is killed by $q+1$. Let
$j\colon\Upsilon\hookrightarrow\mathbb{D}$ be the open embedding, Then

$j_{\ast}\mathscr{L}\in\mathcal{C}_{\mathbb{D}/J}$

for any closed subgroup $J$ of $I$ which contains an open subgroup of
$N=\begin{pmatrix}1&\mathcal{O}_{F}\\\ 0&1\end{pmatrix}$.

###### Proof.

It is easy to see that the isomorphism in Lemma 4.3.3 is $J$-equivariant, and
so by that result and Theorem 7.3.5(a) we know that

$(j_{\ast}\mathscr{L})(\mathbb{D})=\mathscr{L}(\Upsilon)\cong\lim\limits_{\longleftarrow}\mathscr{L}_{n}(\mathbb{D})$

is a coadmissible $\wideparen{\mathcal{D}}(\mathbb{D},J)$-module. Because
$j_{\ast}\mathscr{L}$ is a locally Fréchet $I$-equivariant
$\mathcal{D}$-module on $\mathbb{D}$ by Lemma 7.3.6, and because $J\leqslant
I$, by [9, Definition 3.6.7] it remains to exhibit a continuous
$\mathcal{D}$-linear $J$-equivariant isomorphism

$\operatorname{Loc}^{\wideparen{\mathcal{D}}(\mathbb{D},J)}_{\mathbb{D}}\left((j_{\ast}\mathscr{L})(\mathbb{D})\right)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}j_{\ast}\mathscr{L}.$

Recall that by [9, Definition 3.5.12],
$\operatorname{Loc}^{\wideparen{\mathcal{D}}(\mathbb{D},J)}_{\mathbb{D}}((j_{\ast}\mathscr{L})(\mathbb{D}))$
is the unique extension of a sheaf
$\mathcal{P}^{\wideparen{\mathcal{D}}(\mathbb{D},J)}_{\mathbb{D}}((j_{\ast}\mathscr{L})(\mathbb{D}))$
on the basis $\mathbb{D}_{w}$ of $\mathbb{D}$, defined at [9, Definition
3.5.3]. Because
$\operatorname{Loc}^{\wideparen{\mathcal{D}}(\mathbb{D},J)}_{\mathbb{D}}((j_{\ast}\mathscr{L})(\mathbb{D}))$
and $j_{\ast}\mathscr{L}$ are sheaves, by [7, Theorem 9.1] it will suffice to
exhibit an isomorphism of $J$-equivariant $\mathcal{D}$-modules on
$\mathbb{D}_{w}$

$\varphi:\mathcal{P}^{\wideparen{\mathcal{D}}(\mathbb{D},J)}_{\mathbb{D}}((j_{\ast}\mathscr{L})(\mathbb{D}))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(j_{\ast}\mathscr{L})_{|\mathbb{D}_{w}}.$

Let $X$ be an affinoid subdomain of $\mathbb{D}$ and let $H$ be an $X$-small
open subgroup222Since $I$ preserves the free
$\mathcal{O}(\mathbb{D})^{\circ}$-Lie lattice
$\mathcal{O}(\mathbb{D})^{\circ}\partial_{x}$ in $\mathcal{T}(\mathbb{D})$, we
see that in fact, in this situation _every_ open subgroup $H$ of $J_{X}$ is
$X$-small. of $J_{X}=\operatorname{Stab}_{J}(X)$, in the sense of [9,
Definition 3.4.4]. Because $J$ contains an open subgroup of $N$ by assumption,
so does its open subgroup $H$. Hence we can find an integer
$m_{0}\geqslant\max\\{m_{X},v_{\pi_{F}}(e)\\}$ such that $N_{m_{0}+1}\leqslant
H$. For each $n\geqslant m_{0}$, let
$\mathcal{S}_{n}(X):=\mathscr{D}_{n}(X)\rtimes_{H_{n+1}}H$ and note that the
natural action map

$\mathcal{S}_{n}(X)\underset{\mathcal{S}_{n}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n}(\mathbb{D})\longrightarrow\mathscr{L}_{n}(X)$

is an isomorphism by Proposition 7.3.3 applied with $J$ replaced by $H$. Note
that

$\wideparen{\mathcal{D}}(X,H)\cong\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{n\geqslant
m_{0}}}}\mathcal{T}_{n}(X)\quad\mbox{and}\quad(j_{\ast}\mathscr{L})(\mathbb{D})\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\mathcal{N}_{n}(\mathbb{D})$

by Corollary 7.2.9 and equation (66), where we replace $J$ with $H$ so that

$\mathcal{T}_{n}(X):=\mathscr{D}_{n}(X)\underset{H_{n+1}}{\rtimes}{}H\quad\mbox{and}\quad\mathcal{N}_{n}(\mathbb{D}):=\mathcal{T}_{n}(\mathbb{D})\underset{\mathcal{S}_{n+c}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n+c}(\mathbb{D})$

and $c:=\epsilon v_{\pi_{F}}(p)$. Now, for each $n\geqslant m_{0}$ there is a
commutative diagram

$\textstyle{\mathcal{T}_{n}(X)\underset{\mathcal{T}_{n}(\mathbb{D})}{\otimes}{}\mathcal{N}_{n}(\mathbb{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{n}(X)\underset{\mathcal{S}_{n+c}(X)}{\otimes}{}\mathscr{L}_{n+c}(X)}$$\textstyle{\mathcal{T}_{n}(X)\underset{S_{n+c}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n+c}(\mathbb{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{\mathcal{T}_{n}(X)\underset{\mathcal{S}_{n+c}(X)}{\otimes}{}\left(\mathcal{S}_{n+c}(X)\underset{S_{n+c}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n+c}(\mathbb{D})\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the maps on the left and on the bottom are isomorphisms arising from the
associativity of tensor products. The vertical map on the right is an
isomorphism by Proposition 7.3.3, so the top horizontal map is an isomorphism.
Taking the inverse limit of these maps and using equation (66) together with
the definition of $\wideparen{\otimes}$ from [7, §7.3] and [9, Lemma 3.4.13],
we obtain a $\wideparen{\mathcal{D}}(X,H)$-linear isomorphism

$\varphi_{H}(X):\wideparen{\mathcal{D}}(X,H)\underset{\wideparen{\mathcal{D}}(\mathbb{D},H)}{\wideparen{\otimes}}{}(j_{\ast}\mathscr{L})(\mathbb{D})=\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{n\geqslant
m_{0}}}}\mathcal{T}_{n}(X)\underset{\mathcal{T}_{n}(\mathbb{D})}{\otimes}{}\mathcal{N}_{n}(\mathbb{D})\quad\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\quad(j_{\ast}\mathscr{L})(X).$

Write
$M(X,H):=\wideparen{\mathcal{D}}(X,H)\underset{\wideparen{\mathcal{D}}(\mathbb{D},H)}{\wideparen{\otimes}}{}(j_{\ast}\mathscr{L})(\mathbb{D})$,
let $X^{\prime}$ be an $H$-stable affinoid subdomain of $X$, let $H^{\prime}$
be another open subgroup of $J$ containing $H$ and let $g\in J$. We leave to
the reader the verification that the following diagram commutes:

$\textstyle{M(X^{\prime},H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{H}(X^{\prime})}$$\textstyle{(j_{\ast}\mathscr{L})(X^{\prime})}$$\textstyle{M(X,H^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{H^{\prime}}(X)}$$\textstyle{M(X,H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{H}(X)}$$\scriptstyle{g^{M}_{X,H}}$$\textstyle{(j_{\ast}\mathscr{L})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(gX,{}^{g}H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{{}^{g}H}(gX)}$$\textstyle{(j_{\ast}\mathscr{L})(gX)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{j_{\ast}\mathscr{L}}(X)}$

It follows that the isomorphisms $\varphi_{H}(X)$ are compatible as $H$
shrinks to $1$, and commute with the restriction maps on both sides. Passing
to the limit over all $X$-small $H$ we obtain the required $J$-equivariant
$\mathcal{D}$-linear isomorphism

$\varphi:\mathcal{P}^{\wideparen{\mathcal{D}}(\mathbb{D},J)}_{\mathbb{D}}((j_{\ast}\mathscr{L})(\mathbb{D}))=\lim\limits_{\stackrel{{\scriptstyle\longleftarrow}}{{H}}}M(-,H)\quad\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\quad
j_{\ast}\mathscr{L}.\qed$

###  7.4. Coadmissibility and irreducibility

Let $j:\Omega\hookrightarrow\mathbb{P}^{1}$ and
$j_{0}:\Upsilon\hookrightarrow\mathbb{D}$ be the open inclusions and let
$G_{0}:=\mathbb{GL}_{2}(\mathcal{O}_{F})$.

######  Theorem 7.4.1.

Let
$[\mathscr{L}]\in\operatorname{PicCon}^{G_{0}}(\Omega)_{\operatorname{tors}}$.
Then

$j_{\ast}\mathscr{L}\in\mathcal{C}_{\mathbb{P}^{1}/H}$

for every closed subgroup $H$ of $G_{0}$ which contains an open subgroup of
$\mathbb{SL}_{2}(\mathcal{O}_{F})$.

###### Proof.

Using the argument given in the proof of Lemma 7.3.6, we see that
$j_{\ast}\mathscr{L}$ is a $G_{0}$-equivariant locally Fréchet
$\mathcal{D}$-module on $\mathbb{P}^{1}$. Let $w:=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\in G_{0}$ so that $\\{\mathbb{D},w\mathbb{D}\\}$ forms an
admissible affinoid covering of $\mathbb{P}^{1}$ by two copies of the closed
unit disc $\mathbb{D}$. Note that the Iwahori subgroup $I$ from
$(\ref{IwahoriEq})$ stabilises $\mathbb{D}$, so the opposite Iwahori subgroup
${}^{w}I:=wIw^{-1}$ stabilises $w\mathbb{D}$; we will show that
$j_{\ast}\mathscr{L}$ is $\\{\mathbb{D},w\mathbb{D}\\}$-coadmissible in the
sense of [9, Definition 3.6.7(a)].

Next, consider the restriction $\mathscr{L}_{|\Upsilon}$ so that
$[\mathscr{L}_{|\Upsilon}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$.
Because $\omega[\mathscr{L}]\in\operatorname{PicCon}(\Omega)^{G_{0}}$ is
killed by $q+1$ by [8, Corollary 4.3.9], we see that
$\omega[\mathscr{L}_{0}]\in\operatorname{PicCon}(\Upsilon)^{I}$ is also killed
by $q+1$. Since $H$ contains an open subgroup of
$\mathbb{SL}_{2}(\mathcal{O}_{F})$, $J:=H\cap I$ contains an open subgroup of
$N$. Therefore
$(j_{\ast}\mathscr{L})_{|\mathbb{D}}=j_{0,\ast}(\mathscr{L}_{|\Upsilon})\in\mathcal{C}_{\mathbb{D}/J}$
by Theorem 7.3.7. Entirely similarly we deduce that
$(j_{\ast}\mathscr{L})_{|w\mathbb{D}}\in\mathcal{C}_{w\mathbb{D}/(H\cap{}^{w}I)}$.
Hence $j_{\ast}\mathscr{L}\in\mathcal{C}_{\mathbb{P}^{1}/H}$ by [9, Definition
3.6.7]. ∎

We will now work towards the irreducibility of $j_{\ast}\mathscr{L}$ viewed as
an object in the abelian category $\mathcal{C}_{\mathbb{P}^{1}/G_{0}}$. We
start in the following abstract setting. Let $X$ be a set equipped with a
$G$-topology, let $\mathcal{S}$ be a sheaf of rings on $X$, let $\mathcal{U}$
be a covering of $X$ and let $\mathcal{M}$ be a sheaf of $\mathcal{S}$-modules
on $X$. We say $\mathcal{M}$ is _$\mathcal{U}$ -quasi-coherent_ if the
canonical map

$\mathcal{S}(V)\underset{\mathcal{S}(U)}{\otimes}{}\mathcal{M}(U)\longrightarrow\mathcal{M}(V)$

is an isomorphism whenever $U\in\mathcal{U}$ and $V$ is an admissible open
subset of $U$. For $U,V\in\mathcal{U}$, write $U\sim_{\mathcal{M}}V$ if
$\mathcal{M}(U\cap V)\neq 0$ and let $\cong_{\mathcal{M}}$ be the equivalence
relation on $\mathcal{U}$ generated by $\sim_{\mathcal{M}}$. We say that
$\mathcal{U}$ is _$\mathcal{M}$ -connected_ if there is only one
$\cong_{\mathcal{M}}$ equivalence class in $\mathcal{U}$.

######  Proposition 7.4.2.

Let $\mathcal{M}$ be an $\mathcal{S}$-module on $X$ and let $\mathcal{U}$ be
an admissible covering of $X$. Suppose that

  1. (a)

$\mathcal{M}$ is $\mathcal{U}$-quasi-coherent,

  2. (b)

$\mathcal{U}$ is $\mathcal{M}$-connected, and

  3. (c)

$\mathcal{M}(U)$ is a simple $\mathcal{S}(U)$-module for all
$U\in\mathcal{U}$.

Then $\mathcal{M}$ contains no non-zero proper $\mathcal{U}$-quasi-coherent
submodules.

###### Proof.

Let $\mathcal{N}$ be a $\mathcal{U}$-quasi-coherent $\mathcal{S}$-submodule of
$\mathcal{M}$. We first show that for any $U,V\in\mathcal{U}$ with
$U\sim_{\mathcal{M}}V$, $\mathcal{N}(U)=\mathcal{M}(U)$ if and only
$\mathcal{N}(V)=\mathcal{M}(V)$. If not, then because $\mathcal{M}(U)$ and
$\mathcal{M}(V)$ are simple, we may assume that without loss of generality
that $\mathcal{N}(U)=0$ and $\mathcal{N}(V)=\mathcal{M}(V)$. But then because
both $\mathcal{N}$ and $\mathcal{M}$ are $\mathcal{U}$-quasi-coherent,

$\begin{array}[]{lllll}\mathcal{M}(U\cap V)&=&\mathcal{S}(U\cap
V)\underset{\mathcal{S}(V)}{\otimes}{}\mathcal{M}(V)&=&\mathcal{S}(U\cap
V)\underset{\mathcal{S}(V)}{\otimes}{}\mathcal{N}(V)\\\ &=&\mathcal{N}(U\cap
V)&=&\mathcal{S}(U\cap
V)\underset{\mathcal{S}(U)}{\otimes}{}\mathcal{N}(U)=0\end{array}$

which contradicts the hypothesis that $U\sim_{\mathcal{M}}V$.

Suppose now that the $\mathcal{S}$-submodule $\mathcal{N}$ is non-zero. Then
we can find at least one $U\in\mathcal{U}$ such that $\mathcal{N}(U)\neq 0$,
and hence $\mathcal{N}(U)=\mathcal{M}(U)$ because $\mathcal{M}(U)$ is simple.
Since $\mathcal{U}$ is $\mathcal{M}$-connected, we conclude that in fact
$\mathcal{N}(U)=\mathcal{M}(U)$ for all $U\in\mathcal{U}$. Since $\mathcal{N}$
and $\mathcal{M}$ are both $\mathcal{U}$-quasi-coherent, we see that

$\mathcal{N}(W)=\mathcal{S}(W)\underset{\mathcal{S}(U)}{\otimes}{}\mathcal{N}(U)=\mathcal{S}(W)\underset{\mathcal{S}(U)}{\otimes}{}\mathcal{M}(U)=\mathcal{M}(W)$

for any $U\in\mathcal{U}$ and any admissible open $W\subseteq U$; thus
$\mathcal{N}_{|U}=\mathcal{M}_{|U}$ for all $U\in\mathcal{U}$. Since
$\mathcal{U}$ is an admissible covering of $X$ and since $\mathcal{N}$ and
$\mathcal{M}$ are sheaves, we conclude that $\mathcal{N}=\mathcal{M}$.∎

Here is how we apply this general result. Recall the sheaf of rings
$\mathscr{D}_{n}=\mathcal{D}^{\dagger}_{\varpi/|{\pi_{F}}|^{n}}$ from
Definition 4.3.9 on the $G$-topology $\mathbb{D}_{n}$ on $\mathbb{D}$, and
recall that the truncated line bundle
$\mathscr{L}_{n}:=j_{0,\ast}(\mathscr{L}_{\overline{\Upsilon_{n}}})$ is a
$\mathscr{D}_{n}$-module by Corollary 4.3.11.

######  Theorem 7.4.3.

Let
$[\mathscr{L}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$.
Suppose that the image $\omega[\mathscr{L}]$ of $[\mathscr{L}]$ in
$\operatorname{PicCon}(\Upsilon)_{\operatorname{tors}}$ is non-zero and of
order $d$ dividing $q+1$. Let $j_{0}:\Upsilon\hookrightarrow\mathbb{D}$ be the
inclusion, let $n\geqslant 0$ and let
$\mathscr{L}_{n}:=j_{0,\ast}(\mathscr{L}_{\overline{\Upsilon_{n}}})$. Then
$\mathscr{L}_{n}(\mathbb{D})$ is a simple
$\mathscr{D}_{n}(\mathbb{D})$-module.

###### Proof.

Let $\mathcal{U}_{n}$ be the covering
$\\{W_{1,n},\ldots,W_{h_{n},n},\Upsilon_{n}\\}$ from Definition 5.3.9. We can
use Lemma 5.3.13(a) to see that $\mathscr{L}_{n}$ is a
$\mathcal{U}_{n}$-quasi-coherent $\mathscr{D}_{n}$-module on $\mathbb{D}_{n}$.
Then by construction, $\mathscr{L}_{n}(W_{i,n}\cap\Upsilon_{n})$ is non-zero
for any $i=1,\ldots,h_{n}$, so the covering $\mathcal{U}_{n}$ is
$\mathscr{L}_{n}$-connected.

Using Lemma 4.3.5(d), we see that
$\omega[\mathscr{L}|_{\Upsilon_{n}}]\in\operatorname{Con}(\Upsilon_{n})^{I}[d]$
is non-zero. Hence
$k\nu_{n}=M_{n,d}(\omega[\mathscr{L}])=\mu_{\Upsilon_{n},d}(\theta_{d}(\omega[\mathscr{L}|_{\Upsilon_{n}}]))$
is non-zero as well by [8, Corollary 4.3.4]. Since $k\in\\{1,\cdots,d\\}$ by
Definition 6.5.1, Lemma 4.3.5(a) now implies that $k\neq d$. Hence
$\mathscr{L}_{n}(W_{i,n})$ is a simple $\mathscr{D}_{n}(W_{i,n})$-module for
each $i$ by Corollary 6.5.9.

On the other hand, $\mathscr{L}_{n}(\Upsilon_{n})$ is a simple
$\mathscr{D}_{n}(\Upsilon_{n})$-module by Corollary 4.2.13, because
$\Upsilon_{n}$ is a connected affinoid subdomain of $\mathbb{D}$. Hence
$\mathscr{L}_{n}$ contains no non-zero proper $\mathcal{U}_{n}$-quasi-coherent
$\mathscr{D}_{n}$-submodules by Proposition 7.4.2.

Let $L$ be a $\mathscr{D}_{n}(\mathbb{D})$-submodule of
$\mathscr{L}_{n}(\mathbb{D})$, and consider the presheaf of
$\mathscr{D}_{n}$-modules
$\mathcal{L}:=\mathscr{D}_{n}\underset{\mathscr{D}_{n}(\mathbb{D})}{\otimes}{}L$.
For any $\mathbb{D}_{n}$-admissible covering $\mathcal{V}$ of $\mathbb{D}$,
the augmented Čech complex
$C^{\bullet}_{\operatorname{aug}}(\mathcal{V},\mathscr{D}_{n})$ is exact by
Theorem 2.3.10. Each of its terms is flat as a right
$\mathscr{D}_{n}(\mathbb{D})$-module by Lemma 4.3.10(b). Because the covering
$\mathcal{V}$ is finite by Definition 2.3.9(b), the complex is bounded above,
and therefore, by induction on its length, each of its syzygies is flat as a
right $\mathscr{D}_{n}(\mathbb{D})$-module. Tensoring this complex on the
right by $L$ over $\mathscr{D}_{n}(\mathbb{D})$ and splicing, we conclude that
the augmented Čech complex
$C^{\bullet}_{\operatorname{aug}}(\mathcal{V},\mathcal{L})$ is also exact. In
particular, this means that $\mathcal{L}$ is a sheaf on $\mathbb{D}_{n}$ and
$\mathcal{L}(\mathbb{D})=L$. Applying Lemma 4.3.10(b) again, we see that
$\mathcal{L}$ is a subsheaf of
$\mathscr{D}_{n}\underset{\mathscr{D}_{n}(\mathbb{D})}{\otimes}{}\mathscr{L}_{n}(\mathbb{D})\cong\mathscr{L}_{n}$.
Since $\mathcal{L}$ is $\mathcal{U}_{n}$-quasi-coherent by construction, we
conclude from the first paragraph that either $\mathcal{L}=0$ or
$\mathcal{L}=\mathscr{L}_{n}$. Because $L=\mathcal{L}(\mathbb{D})$, we
conclude that either $L=0$ or $L=\mathscr{L}_{n}(\mathbb{D})$ as required. ∎

Recall the sheaves $\mathcal{T}_{n}$ and $\mathcal{S}_{n}$ on
$\mathbb{D}_{n}/J$ from Definition 7.2.6.

######  Corollary 7.4.4.

With the notation and assumptions of Theorem 7.4.3,
$\mathscr{L}_{n}(\mathbb{D})$ is a simple
$\mathcal{S}_{n}(\mathbb{D})$-module, whenever $n\geqslant v_{\pi_{F}}(e)$.

###### Proof.

We know that the $\mathscr{D}_{n}(\mathbb{D})$-action on
$\mathscr{L}_{n}(\mathbb{D})$ extends to the crossed product
$\mathcal{S}_{n}(\mathbb{D})=\mathscr{D}_{n}(\mathbb{D})\rtimes_{J_{n+1}}J$ by
Theorem 5.3.6, and that $\mathscr{L}_{n}(\mathbb{D})$ is a simple
$\mathscr{D}_{n}(\mathbb{D})$-module by Theorem 7.4.3. So it is per force
simple as an $\mathcal{S}_{n}(\mathbb{D})$-module. ∎

######  Theorem 7.4.5.

Let
$[\mathscr{L}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$.
Suppose that the image $\omega[\mathscr{L}]$ of $[\mathscr{L}]$ in
$\operatorname{PicCon}(\Upsilon)$ is non-zero and its order divides $q+1$.
Then $\mathscr{L}(\Upsilon)$ is a topologically irreducible
$\wideparen{\mathcal{D}}(\mathbb{D},J)$-module for any closed subgroup $J$ of
$I$ which contains an open subgroup of $N$.

###### Proof.

We know that $M_{\infty}:=\mathscr{L}(\Upsilon)$ is a coadmissible module over
the Fréchet-Stein algebra $\wideparen{\mathcal{D}}(\mathbb{D},J)$ by Lemma
4.3.3 and Theorem 7.3.5. Let $S_{n}:=\mathcal{S}_{n}(\mathbb{D})$ and
$T_{n}:=\mathcal{T}_{n}(\mathbb{D})$ for each $n\geqslant 0$; it follows from
Lemma 7.2.2 that the integer $m_{\mathbb{D}}$ from Definition 7.2.1 is zero,
so the tower of rings $T_{\bullet}$ gives a Fréchet-Stein presentation of
$T_{\infty}:=\wideparen{\mathcal{D}}(\mathbb{D},J)$ by Proposition 7.2.7.
Hence each $T_{n}$ is a flat right $T_{\infty}$-module by [44, Remark 3.2]. On
the other hand, $S_{n}$ is a flat right $T_{n}$-module by Theorem 7.2.10 and
hence it is also a flat right $T_{\infty}$-module.

Now suppose that $L$ is a non-zero, closed, proper $T_{\infty}$-submodule of
$M_{\infty}$. Then by [44, Corollary 3.3], $T_{n}\otimes_{T_{\infty}}L$ and
$T_{n}\otimes_{T_{\infty}}(M_{\infty}/L)$ are both non-zero for all
sufficiently large integers $n$. Let $c:=\epsilon v_{\pi_{F}}(p)$; because the
map $T_{\infty}\to T_{n}$ factors through $S_{n+c}$ by Theorem 7.2.8, it
follows that $S_{n+c}\otimes_{T_{\infty}}L$ and
$S_{n+c}\otimes_{T_{\infty}}(M_{\infty}/L)$ are also both non-zero. Since
$S_{n+c}$ is a flat $T_{\infty}$-module by the first paragraph, we conclude
that $S_{n+c}\otimes_{T_{\infty}}M_{\infty}$ is not a simple $S_{n+c}$-module.
However this module is isomorphic to $M_{n+c}$ by Theorem 7.3.5(b), which
contradicts Corollary 7.4.4. ∎

######  Corollary 7.4.6.

Let
$[\mathscr{L}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$.
Suppose that the image $\omega[\mathscr{L}]$ of $[\mathscr{L}]$ in
$\operatorname{PicCon}(\Upsilon)$ is non-zero and its order divides $q+1$. Let
$J$ be a closed subgroup of $I$ which contains an open subgroup of $N$. Then
$j_{0,\ast}\mathscr{L}$ is a simple object in $\mathcal{C}_{\mathbb{D}/J}$.

###### Proof.

This follows from Theorem 7.4.5 and [9, Theorem 3.6.11]. ∎

######  Theorem 7.4.7.

Let
$[\mathscr{L}]\in\operatorname{PicCon}^{G_{0}}(\Omega)_{\operatorname{tors}}$
whose image $\omega[\mathscr{L}]$ in
$\operatorname{PicCon}(\Omega)_{\operatorname{tors}}$ is non-zero. Then
$j_{\ast}\mathscr{L}$ is a simple object in $\mathcal{C}_{\mathbb{P}^{1}/H}$
for _every_ closed subgroup $H$ of $G_{0}$ which contains an open subgroup of
$\mathbb{SL}_{2}(\mathcal{O}_{F})$.

###### Proof.

Note first that $j_{\ast}\mathscr{L}\in\mathcal{C}_{\mathbb{P}^{1}/H}$ by
Theorem 7.4.1. Let $w:=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\in G_{0}$ and
recall the covering $\\{\mathbb{D},w\mathbb{D}\\}$ of $\mathbb{P}^{1}$ from
the proof of Theorem 7.4.1.

We will apply Corollary 7.4.6 to
$[\mathscr{L}|_{\Upsilon}]\in\operatorname{PicCon}^{I}(\Upsilon)_{\operatorname{tors}}$,
but first we have to check the required conditions on
$\omega[\mathscr{L}|_{\Upsilon}]$ hold. Since
$(q+1)\cdot\omega[\mathscr{L}]=0$ by [8, Corollary 4.3.9], the order of
$\omega[\mathscr{L}|_{\Upsilon}]$ in $\operatorname{PicCon}(\Upsilon)$ divides
$q+1$. Consider the following commutative diagram:

$\textstyle{\operatorname{PicCon}(\Omega)^{G_{0}}[q+1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{PicCon}(\Upsilon)^{I}[q+1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Con}(\Upsilon_{0})^{G_{0}}[q+1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Con}(\Upsilon_{0})^{I}[q+1].}$

Note that $\Upsilon_{0}=\Omega_{0}$ in view of Definition 4.3.2(b) and [8,
Definition 4.2.12(b)]. Then it follows from [8, Proposition 3.1.9(b) and
Proposition 4.3.8(a)] that the vertical restriction map on the left is an
isomorphism. Since the bottom horizontal map is an inclusion, it follows from
Lemma 4.3.5(d) that the top horizontal map is injective. Since
$\omega[\mathscr{L}]\neq 0$ in $\operatorname{PicCon}(\Omega)^{G_{0}}[q+1]$ by
assumption, it follows that $\omega[\mathscr{L}|_{\Upsilon}]\neq 0$ in
$\operatorname{PicCon}(\Upsilon)^{I}[q+1]$ as required.

The group $J:=H\cap I$ contains an open subgroup of $N$ because $H$ contains
an open subgroup of $\mathbb{SL}_{2}(\mathcal{O}_{F})$ by assumption. We can
now apply Corollary 7.4.6 to see that
$(j_{\ast}\mathscr{L})_{|\mathbb{D}}=j_{0,\ast}(\mathscr{L}_{|\Upsilon})$ is a
simple object in $\mathcal{C}_{\mathbb{D}/H\cap I}$.

Similarly, ${}^{w}H\cap I$ contains an open subgroup of $N$ because $H$
contains an open subgroup of $\mathbb{SL}_{2}(\mathcal{O}_{F})$. A similar
argument to the above shows that we can apply Corollary 7.4.6 to
$w^{\ast}(\mathscr{L}_{|w\Upsilon})\in\operatorname{PicCon}^{I}(\Upsilon)[e]$
and $J:={}^{w}H\cap I$ to deduce that
$j_{0,\ast}w^{\ast}\mathscr{L}_{|w\Upsilon}$ is a simple object in
$\mathcal{C}_{\mathbb{D}/{}^{w}H\cap I}$. Hence
$(j_{\ast}\mathscr{L})_{|w\mathbb{D}}=w_{\ast}(j_{0,\ast}w^{\ast}\mathscr{L}_{|w\Upsilon})$
is a simple object in $\mathcal{C}_{w\mathbb{D}/H\cap{}^{w}I}$.

Now, consider a short exact sequence $0\to\mathcal{M}\to
j_{\ast}\mathscr{L}\to\mathcal{N}\to 0$ in $\mathcal{C}_{\mathbb{P}^{1}/H}$
with $\mathcal{M}\neq 0$. By [9, Proposition 3.6.10(b)] it induces exact
sequences
$0\to\mathcal{M}_{|\mathbb{D}}\to(j_{\ast}\mathscr{L})_{|\mathbb{D}}\to\mathcal{N}_{|\mathbb{D}}\to
0$ in $\mathcal{C}_{\mathbb{D}/H\cap I}$ and
$0\to\mathcal{M}_{|w\mathbb{D}}\to(j_{\ast}\mathscr{L})_{|w\mathbb{D}}\to\mathcal{N}_{|w\mathbb{D}}\to
0$ in $\mathcal{C}_{w\mathbb{D}/H\cap{}^{w}I}$. Since $\mathcal{M}\neq 0$, we
may assume without loss of generality that $\mathcal{M}_{|\mathbb{D}}\neq 0$.
Then the map $\mathcal{M}_{|\mathbb{D}}\to(j_{\ast}\mathscr{L})_{|\mathbb{D}}$
is an isomorphism, and $\mathcal{N}_{|\mathbb{D}}=0$. Next,
$\mathcal{M}_{|w\mathbb{D}}$ cannot be zero, as otherwise
$(j_{\ast}\mathscr{L})_{|\mathbb{D}\cap
w\mathbb{D}}\cong(\mathcal{M}_{|\mathbb{D}})_{|\mathbb{D}\cap
w\mathbb{D}}=(\mathcal{M}_{|w\mathbb{D}})_{|\mathbb{D}\cap w\mathbb{D}}=0$
which is not the case. So in fact $\mathcal{M}_{|w\mathbb{D}}\neq 0$, which
forces $\mathcal{N}_{|w\mathbb{D}}=0$. Hence $\mathcal{N}=0$ and
$\mathcal{M}\to j_{\ast}\mathscr{L}$ is an isomorphism. ∎

Let $D(H,K)$ denote the locally $F$-analytic distribution algebra of a locally
$F$-analytic group $H$ with coefficients in $K$. Let
$\mathfrak{g}=\operatorname{Lie}(H)\otimes_{F}K$ denote the $K$-Lie algebra of
$H$ and let $\mathfrak{{m}}_{0}=D(H,K)\cdot(\mathfrak{g}U(\mathfrak{g})\cap
Z(U(\mathfrak{g})))$ be the closed ideal of $D(H,K)$ generated by the kernel
of the trivial infinitesimal character $Z(U(\mathfrak{g}))\to F$ that sends
$\mathfrak{g}$ to zero. We have the following general ‘rigid-analytic
equivariant Beilinson-Bernstein localisation’ result.

######  Theorem 7.4.8.

Let $\mathbb{G}$ be a connected, split semisimple algebraic group over $F$,
let $\mathbb{X}$ be its flag variety and let $\mathbf{X}$ be the $K$-rigid
analytification of $\mathbb{X}\times_{F}K$. For every open subroup $H$ of
$\mathbb{G}(F)$, the functor of global sections

$\Gamma(\mathbf{X},-):\mathcal{C}_{\mathbf{X}/H}\longrightarrow\\{M\in\mathcal{C}_{D(H,K)}:\mathfrak{{m}}_{0}\cdot
M=0\\}.$

is a well-defined equivalence of categories.

###### Proof.

First, note that by [9, Theorem 6.5.1], $D(H,K)$ is isomorphic as a Fréchet-
Stein algebra to a certain completed skew-group algebra denoted
$\wideparen{U}(\mathfrak{g},H)$ that was introduced at [9, Definition 6.2.7].
We wish to apply [9, Theorem 6.4.9] to the $K$-algebraic group
$\mathbb{G}\times_{F}K$ and the continuous inclusion
$H\hookrightarrow\mathbb{G}(K)$. In [9] it was assumed that the algebraic
group $\mathbb{G}$ is simply connected; however this assumption is only used
in the proof of [9, Theorem 5.3.5], where it can be easily avoided altogether
(by choosing an $K^{\circ}$-module basis for $\mathfrak{h}$) or instead by
working (as we are currently doing) with a discretely valued ground field $K$.
With this in mind, Theorem 7.4.8 then follows from [9, Theorem 6.4.9], once we
observe that the functor of global sections $\Gamma(\mathbf{X},-)$ is quasi-
inverse to the localisation functor
$\operatorname{Loc}_{\mathbf{X}}^{\wideparen{U}(\mathfrak{g},G)}$, in view of
[9, Theorem 6.4.8]. ∎

Recall that $G^{0}=\\{g\in\mathbb{GL}_{2}(F):v_{\pi_{F}}(\det g)=0\\}$.

######  Corollary 7.4.9.

Suppose
$[\mathscr{L}]\in\operatorname{PicCon}^{G^{0}}(\Omega)_{\operatorname{tors}}$
whose image $\omega[\mathscr{L}]$ in $\operatorname{PicCon}(\Omega)$ is non-
zero. Then $\mathscr{L}(\Omega)$ is a coadmissible and topologically
irreducible $D(H,K)$-module for _any_ closed subgroup $H$ of $G^{0}$
containing an open subgroup of $\mathbb{SL}_{2}(F)$.

###### Proof.

Let $Z$ denote the centre of $\operatorname{GL}_{2}(F)$; note that $Z$ is
contained in $G^{0}$ and that it acts trivially on the $K$-analytic projective
line $\mathbb{P}^{1}$ via Möbius transformations. Since $[\mathscr{L}]$ is a
torsion element of $\operatorname{PicCon}^{G^{0}}(\Omega)$,
$\mathscr{L}^{\otimes e}$ is $G^{0}$-equivariantly isomorphic to
$\mathcal{O}_{\Omega}$ for some $e\geqslant 1$. Since $Z$ acts trivially on
$\mathcal{O}_{\Omega}$, some finite-index open subgroup $Z^{\prime}$ of $Z$
acts trivially on $\mathscr{L}$. We can then choose a finite-index open
subgroup $H^{\prime}$ of $H$ such that $H^{\prime}\cap Z\leqslant Z^{\prime}$.
Then the $H^{\prime}$-action on $j_{\ast}\mathscr{L}$ factors through
$\overline{H^{\prime}}:=H^{\prime}Z/Z$ and Theorem 7.4.7 tells us that
$j_{\ast}\mathscr{L}\in\mathcal{C}_{\mathbb{P}^{1}/\overline{H^{\prime}}}$ is
a simple object.

Our condition on $H$ guarantees that its image $\overline{H}:=HZ/Z$ is open in
$\mathbb{PGL}_{2}(F)=\operatorname{GL}_{2}(F)/Z$. Hence
$\overline{H^{\prime}}$ is open in $\mathbb{PGL}_{2}(F)$ as well. We now apply
Theorem 7.4.8 to $\mathbb{G}:=\mathbb{PGL}_{2,F}$ and
$\overline{H^{\prime}}\hookrightarrow\mathbb{PGL}_{2}(F)$ to see that
$\mathscr{L}(\Omega)$ is a coadmissible and topologically irreducible
$D(\overline{H^{\prime}},K)$-module, killed by $\mathfrak{m}_{0}$. Since
$H\cap Z^{\prime}$ acts trivially on $\mathscr{L}(\Omega)$, it follows by
inflation that the same statements hold for $\mathscr{L}(\Omega)$ viewed as a
$D(H^{\prime},K)$-module. Hence they also hold _a fortiori_ for
$\mathscr{L}(\Omega)$ viewed as a $D(H,K)$-module.∎

###  7.5. $j_{\ast}\mathcal{O}_{\Omega}$ and proof of Corollary B

We begin by recording a very general and very important example of a
coadmissible $G$-equivariant $\mathcal{D}$-module.

######  Proposition 7.5.1.

Let $G$ be a $p$-adic Lie group acting continuously on a smooth rigid
$K$-analytic space $X$. Then

  1. (1)

$\mathcal{O}_{X}$ is a coadmissible $G$-equivariant $\mathcal{D}$-module on
$X$: $\mathcal{O}_{X}\in\mathcal{C}_{X/G}$.

  2. (2)

$\mathcal{O}_{X}$ is a simple object in $\mathcal{C}_{X/G}$ whenever $X$ is
connected.

###### Proof.

(1) Certainly $\mathcal{O}_{X}$ is a $G$-equivariant locally Fréchet
$\mathcal{D}$-module on $X$. Then in view of [9, Definition 3.6.7], it is
enough to consider the case where $(X,G)$ is small in the sense of [9,
Definition 3.4.4]. We will show that in this case

  1. (a)

$\mathcal{O}(X)$ is a coadmissible $\wideparen{\mathcal{D}}(X,G)$-module, and

  2. (b)

there is a natural continuous $G-\mathcal{D}$-linear isomorphism

$\operatorname{Loc}^{\wideparen{\mathcal{D}}(X,G)}_{X}(\mathcal{O}(X))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}_{X}.$

(a) Because $(X,G)$ is small, we can choose a $G$-stable affine formal model
$\mathcal{A}\subset\mathcal{O}(X)$ and a $G$-stable free $\mathcal{A}$-Lie
lattice $\mathcal{L}\subseteq\mathcal{T}(X)$. By replacing $\mathcal{L}$ by
$\pi\mathcal{L}$ if necessary, we may assume that
$[\mathcal{L},\mathcal{L}]\subseteq\pi\mathcal{L}$ and
$\mathcal{L}\cdot\mathcal{A}\subseteq\pi\mathcal{A}$. Since
$\pi^{n}\mathcal{L}\subseteq\mathcal{T}(X)$ is an $\mathcal{A}$-Lie lattice,
$\mathcal{A}$ is naturally a $U(\pi^{n}\mathcal{L})$-module. Since
$\mathcal{A}$ is $\pi$-adically complete, the $U(\pi^{n}\mathcal{L})$-action
extends to $\mathcal{U}_{n}:=\widehat{U(\pi^{n}\mathcal{L})}$ and clearly
$1\in\mathcal{A}$ generates $\mathcal{A}$ as a $\mathcal{U}_{n}$-module. Since
$\mathcal{T}(X)$ is a free $\mathcal{A}$-Lie lattice, we can choose an
$\mathcal{A}$-module basis $\\{\partial_{1},\cdots,\partial_{d}\\}$ for
$\mathcal{L}$. Then the left ideal $\mathcal{I}$ in $\mathcal{U}_{n}$
generated by this finite set annihilates $1\in\mathcal{A}$:
$\mathcal{I}\subseteq\operatorname{ann}_{\mathcal{U}_{n}}(1)$.

For the reverse inclusion, let $Q\in\operatorname{ann}_{\mathcal{U}_{n}}(1)$
and write
$Q=\sum\limits_{\alpha\in\mathbb{N}^{d}}f_{\alpha}(\pi^{n}\partial)^{\alpha}$
for some family $(f_{\alpha})_{\alpha\in\mathbb{N}^{d}}\subset\mathcal{A}$
converging to zero as $|\alpha|\to\infty$. Then $Q\cdot 1=0$ implies that
$f_{0}=0$, and since $f_{\alpha}\to 0$ as $|\alpha|\to\infty$ we see that for
every $m\geqslant 0$, $Q$ lies in $\mathcal{I}+\pi^{m}\mathcal{U}_{n}$. Since
$[\mathcal{L},\mathcal{L}]\subseteq\pi\mathcal{L}$, [9, Proposition 4.1.6(b)]
tells us that the cyclic $\mathcal{U}_{n}$-module
$\mathcal{U}_{n}/\mathcal{I}$ is $\pi$-adically separated. Hence
$Q\in\mathcal{I}$ and $\operatorname{ann}_{\mathcal{U}_{n}}(1)=\mathcal{I}$.
This implies that $\mathcal{A}$ is a finitely presented
$\mathcal{U}_{n}$-module, and hence that
$\mathcal{O}(X)=\mathcal{A}\otimes_{K^{\circ}}K$ is a finitely presented
$\mathcal{U}_{n}\otimes_{K^{\circ}}K$-module for every $n\geqslant 0$.

Using [9, Corollary 3.3.7], choose a good chain $G_{\bullet}$ for
$\mathcal{L}$. In view of [9, Definitions 3.3.3 and 3.2.13] this means that
$G_{0}\geqslant G_{1}\geqslant\cdots$ is a chain of open normal subgroups of
$G$ with trivial intersection, such that $G_{n}\leqslant
G_{\pi^{n}\mathcal{L}}$ for all $n\geqslant 0$. Define

$\mathscr{T}_{n}(X):=\widehat{U(\pi^{n}\mathcal{L})_{K}}=\mathcal{U}_{n}\otimes_{K^{\circ}}K\quad\mbox{and}\quad\mathscr{S}_{n}(X):=\mathscr{T}_{n}(X)\rtimes_{G_{n}}G$

for each $n\geqslant 0$; then by [9, Lemma 3.3.4] we have the Fréchet-Stein
presentation

$\wideparen{\mathcal{D}}(X,G)\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\mathscr{S}_{n}(X).$

Since $G_{n}\leqslant G_{\pi^{n}\mathcal{L}}$, it follows from [9, Definition
3.2.11] that for every $g\in G_{n}$, the action of
$\beta_{\pi^{n}\mathcal{L}}(g)\in\mathcal{U}_{n}$ on $\mathcal{A}$ agrees with
the action of $g\in G_{n}$. Therefore the natural $\mathcal{U}_{n}\rtimes
G$-action on $\mathcal{A}$ factors through the crossed product
$\mathcal{U}_{n}\rtimes_{G_{n}}G$, and hence the natural
$\mathscr{T}_{n}(X)$-action on $\mathcal{O}(X)$ extends to
$\mathscr{S}_{n}(X)=\mathscr{T}_{n}(X)\rtimes_{G_{n}}G$. Since
$\mathcal{O}(X)$ is a finitely presented $\mathscr{T}_{n}(X)$-module, Lemma
7.3.1 implies that it is also a finitely presented
$\mathscr{S}_{n}(X)$-module. To see that $\mathcal{O}(X)$ is a coadmissible
$\wideparen{\mathcal{D}}(X,G)$-module, it remains to see that for every
$n\geqslant 0$, the following natural maps are isomorphisms:

(67)
$\mathscr{S}_{n}(X)\underset{\mathscr{S}_{n+1}(X)}{\otimes}{}\mathcal{O}(X)\longrightarrow\mathcal{O}(X).$

Fix $n\geqslant 0$. We saw above that
$\operatorname{ann}_{\mathscr{T}_{n}(X)}(1)=\sum\limits_{i=1}^{n}\mathscr{T}_{n}(X)\partial_{i}$.
Therefore

(68)
$\mathscr{T}_{n}(X)^{d}\longrightarrow\mathscr{T}_{n}(X)\to\mathcal{O}(X)\to
0$

is a presentation of $\mathcal{O}(X)$ as a $\mathscr{T}_{n}(X)$-module, where
the first map sends $(Q_{i})_{i=1}^{d}\in\mathscr{T}_{n}(X)^{d}$ to
$\sum\limits_{i=1}^{d}Q_{i}\partial_{i}$ and the second map sends
$Q\in\mathscr{T}_{n}(X)$ to $Q\cdot 1$. This presentation implies that the
canonical $\mathscr{T}_{n}(X)$-linear action map

$\mathscr{T}_{n}(X)\underset{\mathscr{T}_{n+1}(X)}{\otimes}{}\mathcal{O}(X)\longrightarrow\mathcal{O}(X)$

is an isomorphism. Since this map factors through
$\mathscr{S}_{n}(X)\underset{\mathscr{S}_{n+1}(X)}{\otimes}{}\mathcal{O}(X)$
by Theorem 5.1.8, we conclude that the map (67) is an isomorphism as well.

(b) After possibly replacing $G$ by an open subgroup and inspecting [9,
3.5.12, 3.5.1 and 3.5.6], it suffices to prove that the natural action map

$\wideparen{\mathcal{D}}(Y,G)\underset{\wideparen{\mathcal{D}}(X,G)}{\wideparen{\otimes}}{}\mathcal{O}(X)\longrightarrow\mathcal{O}(Y)$

is an isomorphism, whenever $Y$ is a $G$-stable affinoid subdomain of $X$. For
this, in view of [9, Lemma 3.3.4] and [7, Lemma 7.3], it is enough to prove
that

$\mathscr{S}_{n}(Y)\underset{\mathscr{S}_{n}(X)}{\otimes}{}\mathcal{O}(X)\longrightarrow\mathcal{O}(Y)$

is an isomorphism for all sufficiently large $n$. We can use Theorem 5.1.8
again to reduce to showing that
$\mathscr{T}_{n}(Y)\underset{\mathscr{T}_{n}(X)}{\otimes}{}\mathcal{O}(X)\to\mathcal{O}(Y)$
is an isomorphism for all sufficiently large $n$. Since
$\\{\partial_{1},\cdots,\partial_{d}\\}$ is still a basis for $\mathcal{T}(Y)$
as an $\mathcal{O}(Y)$-module, this follows immediately from the explicit
presentations
$\mathcal{O}(X)\cong\mathscr{T}_{n}(X)/\sum\limits_{i=1}^{d}\mathscr{T}_{n}(X)\partial_{i}$
and
$\mathcal{O}(Y)\cong\mathscr{T}_{n}(Y)/\sum\limits_{i=1}^{d}\mathscr{T}_{n}(Y)\partial_{i}$
coming from (68).

(2) Suppose first that $X$ is a small affinoid, in the sense that
$\mathcal{T}(X)$ is a free $\mathcal{O}(X)$-module. Then because $X$ is
connected, $\mathcal{O}(X)$ is a simple $\mathcal{D}(X)$-module and therefore
a simple coadmissible $\wideparen{\mathcal{D}}(X,H)$-module for any open
subgroup $H$ of $G$ such that $(X,H)$ is small. Hence
$\mathcal{O}_{X}\in\mathcal{C}_{X/H}$ is a simple object by [9, Theorem
3.6.11] in this case, and it is therefore _a fortiori_ simple in
$\mathcal{C}_{X/G}$.

Returning to the general case, we can choose an admissible open covering
$\mathcal{U}$ of $X$ consisting of small, connected, affinoids. Suppose that
$\mathcal{M}$ is a subobject of $\mathcal{O}_{X}$ and consider
$\mathcal{V}_{1}:=\\{U\in\mathcal{U}:\mathcal{M}|_{U}=\mathcal{O}_{U}\\}$ and
$\mathcal{V}_{2}=\\{U\in\mathcal{U}:\mathcal{M}|_{U}=0\\}$. Then $\mathcal{U}$
is the disjoint union of $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ by the first
paragraph. Now if $U\in\mathcal{V}_{1}$ and $V\in\mathcal{V}_{2}$ with $U\cap
V\neq\emptyset$, then

$(\mathcal{M}|_{V})|_{U\cap V}=\mathcal{M}|_{U\cap
V}=(\mathcal{M}|_{U})|_{U\cap V}=\mathcal{O}_{U\cap V}$

since $U\in\mathcal{V}_{1}$. Thus $\mathcal{M}|_{V}\neq 0$ a contradiction.
Since $X$ is connected if follows that $\mathcal{U}=\mathcal{V}_{1}$ or
$\mathcal{U}=\mathcal{V}_{2}$ i.e. $\mathcal{M}=\mathcal{O}_{X}$ or
$\mathcal{M}=0$ as required. ∎

Recall that $j:\Omega\hookrightarrow\mathbb{P}^{1}$ is the open embedding. We
write $G:=\mathbb{GL}_{2}(F)$ and observe that there is a natural injective
morphism of $G$-equivariant locally Fréchet $\mathcal{D}$-modules on
$\mathbb{P}^{1}$

$\alpha:\mathcal{O}_{\mathbb{P}^{1}}\to j_{\ast}\mathcal{O}_{\Omega}.$

######  Proposition 7.5.2.

$\operatorname{coker}\alpha$ lies in $\mathcal{C}_{\mathbb{P}^{1}/G}$.

###### Proof.

By Proposition 7.5.1(1), $\mathcal{O}_{\mathbb{P}^{1}}$ lies in
$\mathcal{C}_{\mathbb{P}^{1}/G}$. The trivial line bundle with flat connection
$\mathcal{O}_{\Omega}$ is lies in $\operatorname{Con}^{G}(\Omega)$ by [8,
Lemma 3.2.4]. Applying Theorem 7.4.1 to $[\mathcal{O}_{\mathbb{P}^{1}}]$ with
$H=G_{0}$, we see that $j_{\ast}\mathcal{O}_{\Omega}$ lies in
$\mathcal{C}_{\mathbb{P}^{1}/G_{0}}$ as well. Since it is naturally a
$G$-equivariant locally Fréchet $\mathcal{D}$-module on $\mathbb{P}^{1}$, [9,
Definition 3.6.7] implies that $j_{\ast}\mathcal{O}_{\Omega}$ lies in
$\mathcal{C}_{\mathbb{P}^{1}/G}$. Finally, the morphism $\alpha$ lies in
$\mathcal{C}_{\mathbb{P}^{1}/G}$, and we can conclude by applying [9, Lemma
3.7.5]. ∎

######  Lemma 7.5.3.

$\operatorname{coker}\alpha$ is supported on $\mathbb{P}^{1}(F)$ in the sense
of [5, Definition 1.2.1(a)].

###### Proof.

The restriction of $j_{\ast}\mathcal{O}_{\Omega}$ back to $\Omega$ is equal to
$\mathcal{O}_{\Omega}$. Therefore $(\operatorname{coker}\alpha)_{|\Omega}=0$.
Since $\mathbb{P}^{1}=\Omega\cup\mathbb{P}^{1}(F)$, this means that
$\operatorname{coker}\alpha$ is supported on $\mathbb{P}^{1}(F)$. ∎

Recall from $\S\ref{AffTrans}$ that
$\mathbb{B}=\\{g\in\mathbb{GL}_{2}:g_{21}=0\\}$ denotes the subgroup scheme of
upper-triangular matrices in $\mathbb{GL}_{2}$, and let $B:=\mathbb{B}(F)$.
Let $\iota:\\{0\\}\hookrightarrow\mathbb{P}^{1}$ be the closed embedding.
Recall the algebra of smooth distributions $D^{\infty}(B,K)$ from [42].

In what follows, we will silently identify the category
$\mathcal{C}_{D^{\infty}(B,K)}$ with $\mathcal{C}_{\\{0\\}/B}$, using [9,
Theorem B(c)]. Recall also the functors
$\mathcal{H}^{0}_{\\{0\\}},\operatorname{ind}_{B}^{G},\iota_{+}$ and
$\iota^{\natural}$ from [5].

######  Corollary 7.5.4.

Let $\mathcal{N}:=\mathcal{H}^{0}_{\\{0\\}}(\operatorname{coker}\alpha)$. Then
there is an isomorphism in $\mathcal{C}_{\mathbb{P}^{1}/G}$

$\operatorname{coker}\alpha\cong\operatorname{ind}_{B}^{G}\mathcal{N}.$

###### Proof.

The $G$-orbit of the point $0$ is equal to $\mathbb{P}^{1}(F)$, and the
stabiliser of $0$ under the Möbius action of $G$ on $\mathbb{P}^{1}(F)$ is
equal to $B$. By Proposition 7.5.2 and Lemma 7.5.3, we see that
$\operatorname{coker}\alpha$ is an object of $\mathcal{C}^{G\cdot
0}_{\mathbb{P}^{1}/G}$. Now use [5, Theorem A]. ∎

For the following calculations with local cohomology, recall that by [48, §5]
for any rigid analytic space $X$, there is a natural equivalence of categories
between the abelian sheaves on $X$ and the abelian sheaves on the Huber space
$\tilde{X}$ associated with $X$. See also $\S\ref{HuberSpace}$ and [5, §2.1]
for further details.

We now begin to study the object
$\mathcal{N}=\mathcal{H}^{0}_{\\{0\\}}(\operatorname{coker}\alpha)$ in
$\mathcal{C}^{\\{0\\}}_{\mathbb{P}^{1}/B}$. This sheaf is supported at
$\\{0\\}$ only, and therefore is completely determined by its restriction
$\mathcal{N}_{|\mathbb{D}}$ to $\mathbb{D}$. On the other hand,
$\mathcal{N}_{|\mathbb{D}}$ lies in $\mathcal{C}^{\\{0\\}}_{\mathbb{D}/B_{0}}$
where $B_{0}:=\mathbb{B}(\mathcal{O}_{F})$, and it is therefore completely
determined by its global sections $\mathcal{N}(\mathbb{D})$ as a
$\wideparen{\mathcal{D}}(\mathbb{D},B_{0})$-module, in view of [9, Theorem
B(c)]. We first compute the $\mathcal{D}(\mathbb{D})$-action on
$\mathcal{N}(\mathbb{D})$ as follows.

######  Lemma 7.5.5.

$\mathcal{N}(\mathbb{D})$ is isomorphic to
$H^{1}_{\\{0\\}}(\mathbb{D},\mathcal{O}_{\mathbb{D}})$ as a
$\mathcal{D}(\mathbb{D})$-module.

###### Proof.

Recall the notation $\Upsilon=\mathbb{D}\cap\Omega$. Let
$j_{0}:\Upsilon\hookrightarrow\mathbb{D}$ be the open inclusion and let
$\alpha_{0}$ be the restriction of $\alpha$ to $\mathbb{D}$. Then we have the
short exact sequence

$0\to\mathcal{O}_{\mathbb{D}}\stackrel{{\scriptstyle\alpha_{0}}}{{\longrightarrow}}j_{0,\ast}\mathcal{O}_{\Upsilon}\to\operatorname{coker}(\alpha_{0})\to
0$

of locally Fréchet $\mathcal{D}$-modules on $\mathbb{D}$. By the definition of
local cohomology that can be found at [27, p.2], we have

$\mathcal{N}(\mathbb{D})=H^{0}_{\\{0\\}}(\mathbb{D},(\operatorname{coker}\alpha)_{|\mathbb{D}})=H^{0}_{\\{0\\}}(\mathbb{D},\operatorname{coker}\alpha_{0}).$

Now, using [27, Proposition 1.1(b)], we get the long exact sequence

(69)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\\{0\\}}(\mathbb{D},\mathcal{O}_{\mathbb{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\\{0\\}}(\mathbb{D},\operatorname{coker}\alpha_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{0\\}}(\mathbb{D},\mathcal{O}_{\mathbb{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{0\\}}(\mathbb{D},\operatorname{coker}\alpha_{0}).}$

Note that this is a long exact sequence of $\mathcal{D}(\mathbb{D})$-modules.
We claim that the middle terms in this long exact sequence vanish:

(70)
$H^{0}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})=H^{1}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})=0.$

By [27, Corollary 1.9], these terms appear in the long exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathbb{D}\backslash\\{0\\},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{D}\backslash\\{0\\},j_{0,\ast}\mathcal{O}_{\Upsilon}).}$

The third arrow in the first row is
$\mathcal{O}(\Upsilon)\to\mathcal{O}(\Upsilon\backslash\\{0\\})$, which is an
isomorphism since $0\notin\Upsilon$. This already implies that
$H^{0}_{\\{0\\}}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})=0$, and that the
second arrow in the second row is injective. The five-term exact sequence of
low degree terms associated the Leray spectral sequence applied to the map
$j_{0}:\Upsilon\to\mathbb{D}$ and the sheaf $\mathcal{O}_{\Upsilon}$ begins
with $0\to H^{1}(\mathbb{D},j_{0,\ast}\mathcal{O}_{\Upsilon})\to
H^{1}(\Upsilon,\mathcal{O}_{\Upsilon})$. But $\Upsilon$ is a quasi-Stein rigid
analytic space, so $H^{1}(\Upsilon,\mathcal{O}_{\Upsilon})=0$ by Kiehl’s
Theorem. This completes the proof of (70). The connecting map in (69) now
gives a $\mathcal{D}(\mathbb{D})$-linear isomorphism

$\mathcal{N}(\mathbb{D})=H^{0}_{\\{0\\}}(\mathbb{D},\operatorname{coker}\alpha_{0})\cong
H^{1}_{\\{0\\}}(\mathbb{D},\mathcal{O}_{\mathbb{D}}).\qed$

######  Lemma 7.5.6.

We have $\dim_{K}\iota^{\natural}\mathcal{N}=1$.

###### Proof.

The definition of $\iota^{\natural}$ can be found in the statement of [5,
Theorem 3.4.17]:

$\iota^{\natural}(\mathcal{M})=\mathpzc{Hom}_{\mathcal{O}_{\\{0\\}}}\left(\Omega_{\\{0\\}},\hskip
1.42271pt\iota^{\natural}\hskip
1.42271pt(\Omega_{\mathbb{P}^{1}}\underset{\mathcal{O}_{\mathbb{P}}^{1}}{\otimes}{}\mathcal{M})\right)$

for any object $\mathcal{M}$ of
$\mathcal{C}^{\\{0\\}}_{\mathbb{P}^{1}/B_{0}}$: first one side-switches
$\mathcal{M}$ to make a $B_{0}$-equivariant _right_ $\mathcal{D}$-module on
$\mathbb{P}^{1}$, then one applies the pullback functor $\iota^{\natural}$ for
equivariant right $\mathcal{D}$-modules, and then one side-switches back
again. The pullback functor for right $\mathcal{D}$-modules $\iota^{\natural}$
is defined at [5, Definition 3.4.13]. We see that only the
$\mathcal{O}_{\mathbb{P}^{1}}$-module structure on
$\Omega_{\mathbb{P}^{1}}\underset{\mathcal{O}_{\mathbb{P}}^{1}}{\otimes}{}\mathcal{M}$
is used in the definition of this functor. Since $\mathcal{M}$ is supported at
$\\{0\\}$, and since
$\Omega_{\mathbb{P}^{1}}(\mathbb{D})=\mathcal{O}(\mathbb{D})dx$ is a free
$\mathcal{O}(\mathbb{D})$-module of rank $1$, we see that for any
$\mathcal{M}\in\mathcal{C}^{\\{0\\}}_{\mathbb{P}^{1}/B_{0}}$ we have a
$K$-vector space isomorphism