# Rigid Analytic Vectors in Locally Analytic Representations

<EMAIL_ADDRESS>and  Aranya Lahiri  Department of Mathematics, Indiana
University, Bloomington

###### Abstract.

Let $H$ be a uniform pro-$p$ group. Associated to $H$ are rigid analytic
affinoid groups ${\mathbb{H}}_{n}$, and their “wide open” subgroups
${\mathbb{H}}_{n}^{\circ}$. Denote by $D^{\rm la}(H)=C^{\rm
la}(H)^{\prime}_{b}$ the locally analytic distribution algebra of $H$ and by
$D({\mathbb{H}}_{n}^{\circ},H)$ Emerton’s ring of
${\mathbb{H}}_{n}^{\circ}$-rigid analytic distributions on $H$. If $V$ is an
admissible locally analytic representation of $H$, and if
$V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$ denotes the subspace of
${\mathbb{H}}_{n}^{\circ}$-rigid analytic vectors (with its intrinsic
topology), then we show that the continuous dual of
$V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$ is canonically isomorphic to
$D({\mathbb{H}}_{n}^{\circ},H)\otimes_{D^{\rm la}(H)}V^{\prime}$. From this we
deduce the exactness of the functor $V\rightsquigarrow
V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$ on the category of admissible locally
analytic representations of $H$.

###### 2010 Mathematics Subject Classification:

11Sxx

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Uniform pro-$p$ groups, $F$-uniform groups, and rigid analytic groups
  4. 4 Distribution Algebras (summary of results)
  5. 5 Application to rigid analytic vectors
  6. 6 Example: a weak Fréchet-Stein structure on $D^{\rm la}({\mathbb{Z}_{p}},E)$

##  1\. Introduction

Let ${\mathbb{Q}}_{p}\subseteq F\subseteq E$ be finite extensions of
${\mathbb{Q}}_{p}$. Locally analytic representations of locally $F$-analytic
groups as introduced by Schneider-Teitelbaum [9] have occupied a central stage
in the $p$-adic local Langlands program in recent years. For a compact
$F$-analytic group $G$ the locally analytic distribution algebra $D^{\rm
la}(G,E)$ associated to $G$ carries the structure of a Fréchet-Stein algebra,
and the category of coadmissible modules over $D^{\rm la}(G,E)$ is, by
definition, anti-equivalent to the category of admissible locally $F$-analytic
representations of $G$ over $E$, cf. [10].

To describe the main results of this paper, we assume in the introduction that
the base field $F$ is ${\mathbb{Q}}_{p}$. Throughout this paper we denote by
$H$ a uniform pro-$p$ group. There is canonically associated to $H$ a rigid
analytic affinoid group ${\mathbb{H}}$ and a sequence of affinoid subgroups
${\mathbb{H}}_{n}\subset{\mathbb{H}}$, for $n\in{\mathbb{N}}$. Each
${\mathbb{H}}_{n}$ in turn contains a canonically defined “wide open” subgroup
${\mathbb{H}}_{n}^{\circ}$ (which is in general not affinoid). Let ${D^{\rm
la}(H,E)}:=C^{\rm la}(H,E)^{\prime}_{b}$ be the locally analytic distribution
algebra, which is the strong dual of the space of $E$-valued locally analytic
functions on $H$. The multiplication on ${D^{\rm la}(H,E)}$ is a convolution
product. Furthermore, denote by $C^{\rm
la}(H,E)_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}\subset C^{\rm la}(H,E)$ the
subspace of functions which are rigid analytic for the translation action of
${\mathbb{H}}_{n}^{\circ}$, and let $D({\mathbb{H}}_{n}^{\circ},H):=(C^{\rm
la}(H,E)_{{\mathbb{H}}_{n}^{\circ}-{\rm an}})^{\prime}_{b}$ be its strong dual
space, which is again equipped with the structure of an $E$-algebra. The
inclusion $C^{\rm la}(H,E)_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}\hookrightarrow
C^{\rm la}(H,E)$ induces by duality a morphism of $E$-algebras

(1.1) ${D^{\rm la}(H,E)}\longrightarrow D({\mathbb{H}}_{n}^{\circ},H)\;.$

If $V$ is an admissible locally analytic representation of $H$ over $E$, then,
by definition, the continuous dual space $V^{\prime}_{b}$ (equipped with the
topology of bounded convergence) is a coadmissible ${D^{\rm la}(H,E)}$-module.
The space of ${\mathbb{H}}_{n}^{\circ}$-rigid analytic vectors
$V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$ of $V$ (which carries an intrinsic
topology finer than the subspace topology) plays a crucial role in M.
Emerton’s treatment of locally analytic representation theory, cf. [5]. In
that [5, Thm. 1.2.11, Thm. 6.1.19] Emerton has shown that the canonical
homomorphism of topological $D({\mathbb{H}}_{n}^{\circ},H)$-modules

(1.2) $D({\mathbb{H}}_{n}^{\circ},H)\otimes_{D^{\rm
la}(H,E)}V^{\prime}_{b}\longrightarrow(V_{{\mathbb{H}}_{n}^{\circ}-{\rm
an}})^{\prime}_{b}$

gives rise to an isomorphism
$D({\mathbb{H}}_{n}^{\circ},H){\widehat{\otimes}}_{D^{\rm
la}(H,E)}V^{\prime}\longrightarrow V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$. The
key result of this note is that one does not need to use the completed tensor
product here. In other words:

######  Theorem 1.3.

(cf. 5.2) The map 1.2 is bijective.

Then we note that results of Emerton and Schneider-Teitelbaum imply that

######  Theorem 1.4.

(cf. 4.4) The homomorphism 1.1 is flat.

Combining these two results gives

######  Theorem 1.5.

(cf. 5.3) The functor $V\rightsquigarrow V_{{\mathbb{H}}_{n}^{\circ}-{\rm
an}}$ from the category of locally analytic admissible $H$-representations to
the category of $E$-vector spaces is exact.

Most probably the results in this note are well known to the experts. But the
author has not seen 1.3 or 1.5 stated like this in literature.111After a first
version of this paper was written and posted on arXiv, it was pointed out to
us that the results of this paper are contained in the appendix of the
preprint [4]. While the arguments given here and in the appendix of [4] are
similar, we make heavier use of the results of [10] and place more emphasis on
the interaction of Fréchet-Stein and weak Fréchet-Stein structures. Moreover,
our treatment of good analytic subgroups is based on the concept of uniform
pro-$p$ groups and thus offers an alternative and self-contained point of
view. It will be clear to the reader how much this paper owes to the profound
work done in [10] and [5].

Notation and conventions. As above, let ${{\mathbb{Q}}_{p}}\subseteq
F\subseteq E$ be finite field extensions. We denote by ${\mathcal{O}}_{F}$ the
ring of integers of $F$. Further, we denote by $|.|$ the unique extension to
$E$ (resp $F$) of the $p$-adic absolute value $|.|_{p}$ on ${\mathbb{Q}}_{p}$
normalized by $|p|_{p}=p^{-1}$.

##  2\. Preliminaries

######  2.1.

Weak Fréchet-Stein algebras. For general background on locally convex vector
spaces over non-archimidean fields we refer the reader to [7] and [5]. For a
locally convex $E$-algebra $A$ the notion of a weak Fréchet-Stein structure on
$A$ has been defined in [5, Def 1.2.6]. We recall the definition here.

Let $A$ be a locally convex $E$-algebra, a weak Fréchet-Stein structure on $A$
consists of the following data:

  1. (1)

A sequence $\\{A_{n}\\}_{n}$ of hereditarily complete, locally convex
topological $E$-algebras $A_{n}$, for each $n\geq 1$.

  2. (2)

A BH map222A continuous linear map between Hausdorff topological $E$-vector
spaces $U\rightarrow V$ is called BH if it admits a factorization of the form
$U\rightarrow T\rightarrow V$, where $T$ is a Banach $E$-space [5, Def.
1.1.13] $\tau^{A}_{n+1,n}:A_{n+1}\rightarrow A_{n}$, that is a continuous
homomorphism of locally convex topological $E$ vectors spaces, for each $n\geq
1$.

  3. (3)

An isomorphism of topological $A$-modules $A\cong\varprojlim_{n}A_{n}$, where
each of the maps $A\rightarrow A_{n}$ has dense image. The right hand side is
given a projective limit topology from the transition maps induced from part
(2).

Such a locally convex topological algebra $A$ with a choice of weak Fréchet-
Stein structure is called a weak Fréchet-Stein algebra.

######  Example 2.2.

The locally analytic distribution algebra of ${\mathbb{Z}_{p}}$. Let $D^{\rm
la}({\mathbb{Z}_{p}},E)$ be the strong dual of the space of locally analytic
functions ${\mathbb{Z}_{p}}\rightarrow E$. We observe that

$C^{\rm la}({\mathbb{Z}_{p}},E)_{p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}-{\rm
an}}=\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\;,$

where

${\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}):=\left\\{\sum_{n}c_{n}(x-a)^{n}|\;c_{n}\in
E,\lim_{n\rightarrow\infty}c_{n}r^{n}=0\;\forall\;r<|p|^{n}\right\\}\;.$

By compactness of ${\mathbb{Z}_{p}}$ it follows that

$C^{\rm la}({\mathbb{Z}_{p}},E)=\varinjlim_{n\geq
0}\Big{(}\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\Big{)}\;,$

and hence by [7, Prop. 16.10 (iii)]

$D^{\rm la}({\mathbb{Z}_{p}},E)=\varprojlim_{n\geq
0}\Big{(}\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\Big{)}^{\prime}_{b}\;.$

Where the spaces are topologized by the locally convex inductive and
projective limit topology respectively with the natural restrictions (and
their duals) as transition maps. Let us set

$A_{n}:=\Big{(}\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\Big{)}^{\prime}_{b}$

We show in Cor. 6.7 that $A_{n}\cong{\mathcal{O}}_{r_{n}}(X)^{\dagger}$, where

${\mathcal{O}}_{r_{n}}(X)^{\dagger}:=\Big{\\{}\sum_{n=0}^{\infty}a_{n}T^{n}\;\Big{|}\;\lim_{n\rightarrow\infty}a_{n}R^{n}\\\
=0\text{\;for \;some\;}R>r_{n}\Big{\\}}$

is the set of overconvergent functions on a closed disk of radius
$r_{n}:=|p|^{\frac{1}{p^{n}(p-1)}}$.

Let us denote by $\partial$ the generator of the Lie algebra of
${\mathbb{Z}_{p}}$, given by $\partial(f)=f^{\prime}(0)$. Put
$q^{(m)}_{i}=\left\lfloor\frac{i}{p^{m}}\right\rfloor$, and set

$D^{(m)}_{n}:=\left\\{\sum_{i=0}^{\infty}b_{i}\frac{q_{i}^{(m)}!}{i!}(p^{n}\partial)^{i}\;\Big{|}\;b_{i}\in
E,\,\lim_{i\rightarrow\infty}b_{i}\rightarrow 0\right\\}\;.$

Then, the map $\partial\rightarrow\log(1+T)$, induces an isomorphism
$\varinjlim_{m}D^{(m)}_{n}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A_{n}$.
Each of the $D^{(m)}_{n}$ are Noetherian Banach algebras and the locally
convex inductive limit topology coincides under this identification with the
natural topology of the space of overconvergent functions. Each of the natural
restriction maps $A_{n+1}\rightarrow A_{n}$ factors through
$A_{n+1}\rightarrow D_{n}^{(0)}\rightarrow A_{n}$. It can then be shown that
the isomorphism $D^{\rm la}({\mathbb{Z}_{p}},E)\cong\varprojlim_{n}A_{n}$
defines a weak Fréchet-Stein structure on $D^{\rm la}({\mathbb{Z}_{p}},E)$,
and the isomorphism $D^{\rm
la}({\mathbb{Z}_{p}},E)\cong\varprojlim_{n}D_{n}^{(0)}$ defines a Fréchet-
Stein structure (whose definition is recalled below) on $D^{\rm
la}({\mathbb{Z}_{p}},E)$. $\blacksquare$

Next we recall the notion of coadmissible modules over weak Fréchet-Stein
algebras and the stronger notion of Fréchet-Stein algebras.

######  2.3.

Coadmissible modules. Let $A$ be a weak Fréchet-Stein algebra with a choice of
weak Fréchet-Stein structure $A\cong\varprojlim_{n}A_{n}$. A locally convex
topological $A$-module $M$ is called coadmissible [5, Def. 1.2.8] (with
respect to the chosen weak Fréchet-Stein structure of $A$) if there exist the
following data:

  1. (1)

A sequence $\\{M_{n}\\}_{n\geq 1}$ of finitely generated topological
$A_{n}$-modules, for each $n\geq 1$.

  2. (2)

An isomorphism $A_{n}{\widehat{\otimes}}_{A_{n+1}}M_{n+1}\cong M_{n}$, for
each $n\geq 1$.

  3. (3)

An isomorphism of topological $A$-modules $M\cong\varprojlim_{n}M_{n}$, where
the right hand side is given a projective limit structure and topology from
the transition maps induced from part (2).

######  Remark 2.4.

The completed tensor product in 2.3 (2) is defined as follows, cf. [5, after
1.2.3]. If $V$ and $W$ are two locally convex $E$-vector spaces then the
locally convex projective tensor product topology on $V\otimes_{E}W$ is the
universal topology for jointly continuous bilinear maps $V\times W\rightarrow
U$ of locally convex $E$-vector spaces. We let $V\otimes_{E,\pi}W$ denote
$V\otimes_{E}W$ equipped with this projective tensor product topology. Let
$A\rightarrow B$ be a continuous homomorphism of locally convex topological
$E$-algebras and let $M$ be a locally convex topological $A$-module. If
$B\otimes_{A}M$ is endowed by the quotient topology obtained by regarding it
as a quotient of $B\otimes_{E,\pi}M$, then $B\otimes_{A}M$ is a locally convex
topological $B$-module, cf. [5, 1.2.3]. We let $B{\widehat{\otimes}}_{A}M$ be
the completion of the locally convex $E$-vector space $B\otimes_{A}M$, cf. [7,
7.5]. By [5, 1.2.2] the completion $B{\widehat{\otimes}}_{A}M$ carries a
canonical structure of a locally convex topological $B$-module. $\blacksquare$

######  2.5.

Given a weak Fréchet-Stein algebra $A$, with a weak Fréchet-Stein structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}A_{n}$, and
a coadmissible $A$-module $M$, a sequence $\\{M_{n}\\}_{n}$ as in 2.3 (1) is
referred to as an $\\{A_{n}\\}_{n}$-sequence for $M$. We let ${\rm
pr}^{M_{\bullet}}_{n}:M\rightarrow M_{n}$ be the map with is the composition
of the chosen isomorphism
$M\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}M_{n}$
followed by the canonical projection $\varprojlim_{n}M_{n}\rightarrow M_{n}$.
We also recall that if $M$ is a coadmissible module with respect to a certain
weak Fréchet-Stein structure then it is coadmissible with respect to any weak
Fréchet-Stein structure on the same algebra [5, Prop. 1.2.9].

######  2.6.

Fréchet-Stein algebras. A locally convex $E$-algebra $A$ is said to have a
Fréchet-Stein structure [5, Def. 1.2.10] if it is endowed with a weak Fréchet-
Stein structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}B_{n}$ with
the additional requirements that each $B_{n}$ is left Noetherian and each map
$B_{n+1}\rightarrow B_{n}$ is right flat. Such an algebra equipped with a
Fréchet-Stein structure is called a Fréchet-Stein algebra. We begin with
recalling the following useful theorems.

######  Theorem 2.7.

[10, Cor. 3.1] Let $A$ be a Fréchet-Stein algebra, equipped with a Fréchet-
Stein structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}B_{n}$. Let
$M\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}M_{n}$ be a
coadmissible $A$-module, where $\\{M_{n}\\}_{n}$ is a
$\\{B_{n}\\}_{n}$-sequence. Then the map

${\rm id}_{B_{n}}\otimes{\rm
pr}^{M_{\bullet}}_{n}:B_{n}\otimes_{A}M\longrightarrow
B_{n}\otimes_{B_{n}}M_{n}=M_{n}$

is an isomorphism for all $n\geq 1$.

######  Remark 2.8.

The argument in [10, Cor. 3.1] crucially uses that the $B_{n}$ are (left)
Noetherian. We will later adapt the argument and generalize the preceding
theorem in Thm. 2.11. $\blacksquare$

When $A$ is a Fréchet-Stein algebra, equipped with a weak Fréchet-Stein
structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}A_{n}$, we
have the following

######  Theorem 2.9.

[5, Thm. 1.2.11 (i)] Let $A$ be a Fréchet-Stein algebra, equipped with a weak
Fréchet-Stein structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}A_{n}$. Let
$M\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}M_{n}$ be a
coadmissible $A$-module, where $\\{M_{n}\\}_{n}$ is an
$\\{A_{n}\\}_{n}$-sequence. Then the map

${\rm id}_{A_{n}}\otimes{\rm
pr}^{M_{\bullet}}_{n}:A_{n}\otimes_{A}M\longrightarrow
A_{n}\otimes_{A_{n}}M_{n}=M_{n}$

induces an isomorphism $A_{n}{\widehat{\otimes}}_{A}M\rightarrow M_{n}$ for
all $n\geq 1$.

We will also need the following properties of algebras of compact type.

######  Lemma 2.10.

[6, Prop. 5.1.1] Let
$C\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varinjlim_{m}C^{(m)}$ be
an $E$-algebra of compact type with Noetherian Banach algebras $C^{(m)}$ over
$E$.

(i) A finitely generated module $P$ over $C$ has a a unique compact type
topology
$P\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varinjlim_{m}P_{m}$,
called the canonical topology, such that $C\times P\rightarrow P$ is a
continuous map. We can choose each $P_{m}$ to be a finitely generated module
over $C^{(m)}$.

(ii) A finitely generated $C$-submodule of $P$ is closed in the canonical
topology.

(iii) Any $C$-linear map between finitely generated $C$-modules is continuous
and strict with respect to the canonical topology of part (i).

Our main goal in this section is to prove the

######  Theorem 2.11.

Let $A$ be a weak Fréchet-Stein algebra with a weak Fréchet-Stein structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}A_{n}$,
where each $A_{n}$ is assumed to be an $E$-algebra of compact type. Suppose
$\\{B_{n},\tau^{B}_{n+1,n}\\}_{n\geq 1}$ is a projective system of left
Noetherian Banach $E$-algebras with the following properties

  1. (1)

There is a morphism
$(i_{n})_{n}:\\{B_{n},\tau^{B}_{n+1,n}\\}_{n}\rightarrow\\{A_{n},\tau^{A}_{n+1,n}\\}_{n}$
of projective systems, such that all $B_{n}\xrightarrow{i_{n}}A_{n}$ are
continuous morphisms of $E$-algebras. In particular,
$i_{n}\circ\tau^{B}_{n+1,n}=\tau^{A}_{n+1,n}\circ i_{n+1}$.

  2. (2)

The map

$i:=\varprojlim_{n}i_{n}:\varprojlim_{n}B_{n}\longrightarrow\varprojlim_{n}A_{n}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}A$

is an isomorphism of topological $E$-algebras, with the additional property
that the sequence $\\{B_{n},\tau^{B}_{n+1,n}\\}_{n\geq 1}$ is a Fréchet-Stein
structure on $A$.333cf. the following remark

Let $M\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}M_{n}$
be a coadmissible module with respect to the weak Fréchet-Stein structure
given by $\\{A_{n}\\}_{n}$, with each $M_{n}$ a finitely generated
$A_{n}$-module equipped with the canonical compact type topology. Then the
canonical map

$\phi_{n}:={\rm id}_{A_{n}}\otimes{\rm
pr}^{M_{\bullet}}_{n}:A_{n}\otimes_{A}M\longrightarrow M_{n}$

is a bijection.

######  Remark 2.12.

When we assume in the statement of the preceding theorem that the sequence
$\\{B_{n},\tau^{B}_{n+1,n}\\}_{n\geq 1}$ is a Fréchet-Stein structure on $A$,
then this means that the following conditions are supposed to be satisfied:

  1. (1)

For every $n\in{\mathbb{N}}$ the transition map
$\tau^{B}_{n+1,n}:B_{n+1}\rightarrow B_{n}$ makes $B_{n}$ a right flat
$B_{n+1}$-module.

  2. (2)

Each map ${\rm pr}^{B_{\bullet}}_{n}:A\rightarrow B_{n}$, obtained from
composing $i^{-1}:A\rightarrow\varprojlim_{n}B_{n}$ with the canonical
projection map $\varprojlim_{n}B_{n}\rightarrow B_{n}$, has dense image.

As each $B_{n}$ is assumed to be a Banach algebra, it is hereditarily
complete, cf. [5, after Def. 1.1.39]. $\blacksquare$

###### Proof.

As mentioned in 2.8, this is an adaptation of the argument in [10, Cor. 3.1].
Instead of assuming that the $A_{n}$ are Noetherian we generalize it for
algebras of compact type using lemma 2.10. We also mention that the proof of
the injectivity of $\phi_{n}$ given below is essentially the same as the proof
in [10, Cor. 3.1] with additional details furnished.

(i) Surjectivity. We consider $B_{n}$ as an $A$-module via the map ${\rm
pr}^{B_{\bullet}}_{n}:A\rightarrow B_{n}$ introduced in 2.12. By 2.5 $M$ is
also a coadmissible module for the Fréchet-Stein structure
$A\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}B_{n}$ on
$A$. Let $\\{M_{n}^{B}\\}_{n}$ be a $\\{B_{n}\\}_{n}$-sequence for $M$. By 2.7
the map

${\rm id}_{B_{n}}\otimes{\rm
pr}^{M^{B}_{\bullet}}_{n}:B_{n}\otimes_{A}M\longrightarrow
B_{n}\otimes_{B_{n}}M^{B}_{n}=M^{B}_{n}$

is an isomorphism, and $B_{n}\otimes_{A}M$ is hence a finitely generated
$B_{n}$-module (because $M^{B}_{n}$ is a finitely generated $B_{n}$-module).
Furthermore, $A_{n}\otimes_{A}M\cong A_{n}\otimes_{B_{n}}B_{n}\otimes_{A}M$.
Consequently ${\rm im}(\phi_{n})$ is a finitely generated $A_{n}$-submodule of
$M_{n}$. Therefore, ${\rm im}(\phi_{n})$ is closed in the canonical topology
of $M_{n}$ (Lemma 2.10 (ii), here $C=A_{n},P=M_{n}$). By 2.9, the map
$\phi_{n}={\rm id}_{A_{n}}\otimes{\rm
pr}^{M_{\bullet}}_{n}:A_{n}\otimes_{A}M\rightarrow M_{n}$ induces an
isomorphism
$A_{n}{\widehat{\otimes}}_{A}M\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}M_{n}$,
and the image of $\phi_{n}$ is thus dense in $M_{n}$. Density and closedness
of the image shows that $\phi_{n}$ is surjective.

(ii) Injectivity. We will use the surjectivity result from above to prove
injectivity of $\phi_{n}$. Suppose $x:=b_{1}\otimes x_{1}+\ldots+b_{r}\otimes
x_{r}\in A_{n}\otimes_{A}M$ is such that

$({\rm id}_{A_{n}}\otimes{\rm pr}^{M_{\bullet}})(x)=b_{1}.{\rm
pr}^{M_{\bullet}}_{n}(x_{1})+\ldots+b_{r}.{\rm pr}^{M_{\bullet}}_{n}(x_{r})=0$

in $M_{n}$. For $\ell\geq n$ we consider the map

$\psi_{\ell}:A_{\ell}^{r}\longrightarrow
M_{\ell}\;,\;\;(a_{1},\ldots,a_{r})\longmapsto a_{1}.{\rm
pr}^{M_{\bullet}}_{\ell}(x_{1})+\ldots+a_{r}.{\rm
pr}^{M_{\bullet}}_{\ell}(x_{r})\;.$

Let us denote the kernel of this map by $N_{\ell}$. Thus, for each $\ell\geq
n$ we have an exact sequence

$0\rightarrow N_{\ell}\hookrightarrow
A^{r}_{\ell}\stackrel{{\scriptstyle\psi_{\ell}}}{{\longrightarrow}}M_{\ell}\;.$

Setting $N=\varprojlim_{\ell}N_{\ell}$ and passing to the projective limit
(which is left exact), we get an exact sequence

$0\rightarrow N\hookrightarrow
A^{r}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}M\;,$

where the rightmost map is given by $(a_{1},\ldots,a_{r})\mapsto
a_{1}.x_{1}+\ldots+a_{r}.x_{r}$. The category of coadmissible modules is
closed under taking kernels [5, Thm. 1.2.11 (ii)], [10, Cor. 3.4 (ii)]. In
particular $N$ is a coadmissible module with the corresponding
$\\{A_{\ell}\\}_{\ell}$-sequence $\\{N_{\ell}\\}_{\ell}$. Further, $N_{\ell}$
is a finitely generated module over $A_{\ell}$ (by definition of “coadmissible
module”) and it then follows from the above argument that the map from ${\rm
id}_{A_{n}}\otimes{\rm pr}^{N_{\bullet}}_{n}:A_{n}\otimes_{A}N\rightarrow
N_{n}$ is surjective. We can then find elements
$(c_{1}^{(1)},\ldots,c_{r}^{(1)}),\ldots,(c_{1}^{(s)},\ldots,c_{r}^{(s)})$ of
$N\subseteq A^{r}$ whose images under ${\rm pr}_{n}^{N_{\bullet}}:N\rightarrow
N_{n}$ generate $N_{n}$ as an $A_{n}$-module. Recall the element
$x=b_{1}\otimes x_{1}+\cdots+b_{r}\otimes x_{r}$ from above. By definition of
$x$ we see that $(b_{1},\ldots,b_{r})\in\ker(\psi_{n})=N_{n}$. Thus there
exists $f_{1},\ldots,f_{s}\in A_{n}$ such that

$(b_{1},\ldots,b_{r})=f_{1}.({\rm
pr}_{n}^{A_{\bullet}}(c_{1}^{(1)}),\ldots,{\rm
pr}^{A_{\bullet}}_{n}(c_{r}^{(1)}))+\ldots+f_{s}.({\rm
pr}^{A_{\bullet}}_{n}(c_{1}^{(s)}),\ldots,{\rm
pr}^{A_{\bullet}}_{n}(c_{r}^{(s)}))$

It follows that

$\displaystyle b_{1}\otimes x_{1}+\ldots+b_{r}\otimes x_{r}$
$\displaystyle=\sum_{j=1}^{r}(b_{j}\otimes
x_{j})=\sum_{j=1}^{r}\Big{(}\sum_{i}f_{i}.{\rm
pr}^{A_{\bullet}}_{n}(c_{j}^{(i)})\Big{)}\otimes x_{j}$
$\displaystyle=\sum_{j=1}^{r}\Big{(}\sum_{i}f_{i}\otimes
c_{j}^{(i)}x_{j}\Big{)}=\sum_{i}f_{i}\otimes\Big{(}\sum_{j}c_{j}^{(i)}x_{j}\Big{)}$
$\displaystyle=\sum f_{i}\otimes 0=0$

The third equality follows from $a_{n}.{\rm pr}^{A_{\bullet}}_{n}(b)\otimes
m=a_{n}\otimes b.m\in A_{n}\otimes_{A}M$. So,
$\phi_{n}:A_{n}\otimes_{A}M\rightarrow M_{n}$ is also injective. ∎

##  3\. Uniform pro-$p$ groups, $F$-uniform groups, and rigid analytic groups

######  3.1.

Uniform pro-$p$ groups. In this section, $H$ will always denote a pro-$p$
group, whose group law we always write multiplicatively. $H$ is called
powerful [3, Def. 3.1. (i)] if $p$ is odd and $H/\overline{H^{p}}$ is abelian
or $p=2$ and $H/\overline{H^{4}}$ is abelian. The lower $p$-series
$(P_{i}(H))_{i\geq 1}$ of $H$ is defined by

$P_{1}(H)=H\;,\;\;P_{i+1}(H):=\overline{P_{i}(H)^{p}[P_{i}(H),H]}$

where $[P_{i}(H),H]$ is the commutator of these two groups within $H$.

A powerful pro-$p$ group $H$ is called uniformly powerful pro-$p$ or uniform
pro-$p$ [3, Def. 4.1] if

  1. (1)

$H$ is finitely generated as a pro-$p$ group.

  2. (2)

For all $i$ the $p$-th power map $x\mapsto x^{p}$ induces an isomorphism of
abelian groups

$P_{i}(H)/P_{i+1}(H)\longrightarrow P_{i+1}(H)/P_{i+2}(H)$

If $H$ is a topologically finitely generated powerful pro-$p$ group, then
$P_{i+1}(H)=\\{x^{p^{i}}{\;|\;}x\in H\\}$ for every $i\geq 0$, cf. [3, 3.6
(iii)]. If $H$ is uniform pro-$p$, then the map $x\mapsto x^{p^{i}}$ is a
homeomorphism $H\rightarrow P_{i+1}(H)$, in particular, each element $z\in
P_{i+1}(H)$ has a unique $p^{i}$-th root in $H$ which we denote by
$z^{p^{-i}}$.

######  3.2.

The ${\mathbb{Z}_{p}}$-Lie algebra associated to a uniform pro-$p$ group [3,
sec. 4.3, 4.5]. In this paragraph, let $H$ be a uniform pro-$p$ group, and set
$H_{n}=P_{n+1}(H)=\\{x^{p^{n}}{\;|\;}x\in H\\}$ for $n\geq 0$. For $x,y\in H$,
the element $x^{p^{n}}y^{p^{n}}$ lies in $H_{n}$ and hence has a unique
$p^{n}$-th root $(x^{p^{n}}y^{p^{n}})^{p^{-n}}$, as recalled above. By the
discussion in [3, sec. 4.3], the limit

$x+y=\lim_{n\rightarrow\infty}(x^{p^{n}}y^{p^{n}})^{p^{-n}}$

exists in $H$ and gives $H$ the additional structure of an abelian group. The
multiplication ${\mathbb{Z}}\times H\rightarrow H$ extends continuously to
${\mathbb{Z}_{p}}\times H\rightarrow H$, and provides $(H,+)$ with the
structure of a finitely generated free ${\mathbb{Z}_{p}}$-module, whose rank
is equal to the minimal number of topological generators of $H$.

As usual, we denote by $[x,y]_{H}=xyx^{-1}y^{-1}$ the commutator of $x,y\in
H$. Note that $[x^{p^{n}},y^{p^{n}}]_{H}\in[H_{n},H_{n}]\subseteq H_{2n+1}$
for $x,y\in H$. We then endow $H$ with a Lie bracket by setting

$[x,y]_{\rm
Lie}:=\lim_{n\rightarrow\infty}[x^{p^{n}},y^{p^{n}}]_{H}^{p^{-2n}}\;,$

where the limit exists by [3, 4.28]. By [3, Thm. 4.30] the triple
$L_{H}:=(H,\,+\,,\,[\,,\,]_{\rm Lie})$ is a ${\mathbb{Z}_{p}}$-Lie algebra
which is powerful in the sense of [3, sec. 9.4], i.e.,

$[L_{H},L_{H}]\subseteq p^{\epsilon}L_{H}\;,$

where $\epsilon=1$ if $p$ is odd and is equal to $2$ if $p=2$. We denote by
$\exp_{H}$ the map $L_{H}\rightarrow H$ which is the identity map on the
underlying sets. The ${{\mathbb{Q}}_{p}}$-Lie algebra
$L_{H}\otimes_{\mathbb{Z}_{p}}{{\mathbb{Q}}_{p}}$ is canonically isomorphic to
the Lie algebra ${\rm Lie}(H)$ of $H$ as a $p$-adic Lie group,444as defined
in, for instance, [8, ch. VII] and the exponential map ${\rm
Lie}(H)\dashrightarrow H$, which is only defined on a sufficiently small
lattice in ${\rm Lie}(H)$, is equal to $\exp_{H}$ (as defined here), when
restricted to $L_{H}\subset{\rm Lie}(H)$.

By [3, 9.10] the functor $H\rightsquigarrow L_{H}$ defines an equivalence of
between the category of uniform pro-$p$ groups and the category of powerful
${\mathbb{Z}_{p}}$-Lie algebras.

######  3.3.

From powerful ${\mathbb{Z}_{p}}$-Lie algebras to uniform pro-$p$ groups. The
Baker-Campbell-Hausdorff (BCH) formula [3, 6.28] is a formal series

$\Phi(X,Y):=\sum_{n=1}^{\infty}u_{n}(X,Y)\;,$

whose terms $u_{n}(X,Y)$ are in the free associative ${\mathbb{Q}}$-algebra
generated by two variables $X$ and $Y$. If we denote by $[f,g]=fg-gf$ the
commutator of two elements in this ring, then we have the following formulas

$u_{1}(X,Y)=X+Y\;,\;\;u_{2}(X,Y)=\frac{1}{2}[X,Y]\;,$

$u_{n}(x,y)=\sum_{\mbox{{\tiny$\begin{array}[]{c}e=(e_{1},\ldots,e_{n})\in{\mathbb{N}}^{n}\\\
e_{1}+\ldots+e_{n}=n-1\end{array}$}}}q_{e}[X,Y]_{e}\;.$

Here

$[X,Y]_{e}=\Big{[}\ldots[X,[\underbrace{Y,\ldots,Y}_{e_{1}},\underbrace{X,\ldots,X}_{e_{2}},\ldots]\ldots\Big{]}$

denotes an iterated commutator, and the $q_{e}$ are certain constants in
${\mathbb{Q}}$. If $L$ is a powerful ${\mathbb{Z}_{p}}$-Lie algebra, then for
any $x,y\in L$ each term $u_{n}(x,y)$, which a priori is in
$L\otimes{\mathbb{Q}}$, lies actually in $L$, and the series $\Phi(x,y)$
converges in $L$. Moreover, the map

$L\times L\longrightarrow L\;,\;\;(x,y)\mapsto x\cdot_{\Phi}y:=\Phi(xy)$

defines a group structure on $L$. By [3, 9.8] the pair $(L,\,\cdot_{\Phi}\,)$
is a uniform pro-$p$ group. In fact the functor from the category of powerful
${\mathbb{Z}_{p}}$-Lie algebras to the category of uniform pro-$p$ groups,
$(L,[\,,\,])\rightsquigarrow(L,\,\cdot_{\Phi}\,)$ is a quasi-inverse to the
functor $H\rightsquigarrow L_{H}$, cf. [3, 9.10].

######  Definition 3.4.

A locally $F$-analytic group555cf. [8, sec. 13], where those groups are called
”Lie groups over $F$” $H$ is called $F$-uniform pro-$p$ if the following
conditions hold:

  1. (1)

$H$ is a uniform pro-$p$ group.

  2. (2)

The $\mathbb{Z}_{p}$-Lie algebra $L_{H}$, which embeds canonically into ${\rm
Lie}(H)=L_{H}\otimes_{\mathbb{Z}_{p}}{{\mathbb{Q}}_{p}}$ (cf. end of sec.
3.2), is stable under multiplication by ${\mathcal{O}}_{F}$ on ${\rm Lie}(H)$.
(In particular, $L_{H}$ has the structure of an ${\mathcal{O}}_{F}$-module.)

######  3.5.

Rigid analytic groups associated to $F$-uniform groups. Let $H$ be an
$F$-uniform pro-$p$ group. The ${\mathcal{O}}_{F}$-module $L_{H}$ is free of
finite rank $d$ over ${\mathcal{O}}_{F}$. We choose an
${\mathcal{O}}_{F}$-basis $\partial_{1},\ldots,\partial_{d}$ of $L_{H}$. For
$x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}\in{\mathcal{O}}_{F}$ we can write

$\Phi(x_{1}\partial_{1}+\ldots+x_{d}\partial_{d},y_{1}\partial_{1}+\ldots+y_{d}\partial_{d})=\sum_{j=1}^{d}\Phi_{j}(x_{1},\ldots,x_{d},y_{1},\ldots,y_{d})\partial_{j}$

where each $\Phi_{j}$ is a power series in the ring ${\mathcal{O}}\langle
x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}\rangle$ of strictly convergent power
series in $x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}$. Consider
$L_{H}^{\vee}:={\rm Hom}_{{\mathcal{O}}_{F}}(L_{H},{\mathcal{O}}_{F})$ and let
$(\partial_{j}^{\vee})_{j}$ be the dual basis to $(\partial_{j})_{j}$, i.e.,
$\partial_{j}^{\vee}(\partial_{i})=\delta_{i,j}$. Then the $p$-adic completion
${\rm Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})^{\wedge}$ of the symmetric
algebra ${\rm
Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})\cong{\mathcal{O}}_{F}[\partial_{1}^{\vee},\ldots,\partial_{d}^{\vee}]$
is ismormorphic to
${\mathcal{O}}_{F}\langle\partial_{1}^{\vee},\ldots,\partial_{d}^{\vee}\rangle$.
Via those isomorphisms we define a comultiplication

(3.1) ${\rm Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})^{\wedge}\longrightarrow{\rm
Sym}_{{\mathcal{O}}_{F}}\Big{(}(L_{H}\oplus L_{H})^{\vee}\Big{)}^{\wedge}$

by sending $\partial_{j}^{\vee}$ to
$\Phi_{j}(\partial^{\vee}_{1},\ldots,\partial^{\vee}_{d},\partial^{\vee}_{1},\ldots,\partial^{\vee}_{d})$.
This gives the affinoid space

${\mathbb{H}}={\rm Sp}\left({\rm
Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})^{\wedge}\otimes_{{\mathcal{O}}_{F}}F\right)$

the structure of a rigid analytic group, which is independent of the choice of
the basis $(\partial_{j})_{j}$ made above. More generally, if ${\varpi}$ is a
uniformizer of $F$, then we denote by ${\mathbb{H}}_{n}\subset{\mathbb{H}}$
the rigid analytic affinoid subgroup that one obtains by replacing $L_{H}$ by
${\varpi}^{n}L_{H}$. If $\exp_{H}:L_{H}\rightarrow H$ is the exponential map
as in 3.2, then the group of $F$-valued points of ${\mathbb{H}}_{n}$ is
$\exp_{H}({\varpi}^{n}L_{H})$.

It follows from the discussion in [5, p. 100] that for any $n\geq 0$ the rigid
analytic group ${\mathbb{H}}_{n}$ is a good analytic subgroup of $H$, using
the terminology introduced by Emerton in [5]. Thus the subsequent
constructions of distribution algebras, recalled in the next section, apply in
our setting.

######  3.7.

The wide open groups ${\mathbb{H}}_{n}^{\circ}$. Because the comultiplication
map 3.1 is defined integrally, it equips the formal scheme
${\mathfrak{H}}={\rm Spf}\Big{(}{\rm
Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})^{\wedge}\Big{)}$ with the structure of
a formal group scheme over ${\rm Spf}({\mathcal{O}}_{F})$. Let
${\mathfrak{H}}^{\circ}$ be the completion of ${\mathfrak{H}}$ along its unit
section $e_{\mathfrak{H}}:{\rm
Spf}({\mathcal{O}}_{F})\rightarrow{\mathfrak{H}}$. This is a affine formal
group scheme over ${\rm Spf}({\mathcal{O}}_{F})$ whose corresponding
${\mathcal{O}}_{F}$-algebra ${\mathcal{O}}({\mathfrak{H}}^{\circ})$ is the
completion of ${\rm Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})$ with respect to
the augmentation ideal $I:=\ker\Big{(}{\rm
Sym}_{{\mathcal{O}}_{F}}(L_{H}^{\vee})\xrightarrow{e_{\mathfrak{H}}^{*}}{\mathcal{O}}_{F}\Big{)}$.
Using the basis $(\partial_{j}^{\vee})_{j}$ introduced above, one has an
isomorphism
${\mathcal{O}}({\mathfrak{H}}^{\circ})\cong{\mathcal{O}}_{F}[[\partial_{1}^{\vee},\ldots,\partial_{d}^{\vee}]]$.
We then let ${\mathbb{H}}^{\circ}$ be the rigid analytic group associated to
the group scheme ${\mathfrak{H}}^{\circ}$ (by Berthelot’s construction [2,
sec. 7]). In the same way we construct the rigid analytic group
${\mathbb{H}}_{n}^{\circ}$ starting from the ${\mathbb{Z}_{p}}$-Lie algebra
${\varpi}^{n}L_{H}$, which we call wide open. One has
${\mathbb{H}}_{n+1}\subset{\mathbb{H}}_{n}^{\circ}\subset{\mathbb{H}}_{n}$ and
${\mathbb{H}}_{n}^{\circ}(F)={\mathbb{H}}_{n+1}(F)$.

##  4\. Distribution Algebras (summary of results)

We recall that $E/F$ denotes a finite field extension. Given a compact,
locally $F$-analytic group $G$ we let $C^{\rm la}(G,E)$ be the locally convex
$E$-vector space of locally $F$-analytic functions $G\rightarrow E$. The space
$C^{\rm la}(G,E)$ is of compact type and is a locally $F$-analytic
representation of $G$ via the right regular action of $G$, i.e.,
$g.f(x)=f(x.g^{-1})$ (cf. [9, sec. 2] for details of the construction and the
topology on the space). The strong dual $D^{\rm la}(G,E):=(C^{\rm
la}(G,E)^{\prime}_{b})$ of $C^{\rm la}(G,E)$ is called the locally
$F$-analytic distribution algebra of $G$ (with coefficients in $E$). $D^{\rm
la}(G,E)$ is an $E$-Fréchet space and it has an $E$-algebra structure with
convolution product as multiplication. For future use we note that for each
$g\in G$ we have the delta distribution $\delta_{g}\in D^{\rm la}(G,E)$
defined by $\delta_{g}(f)=f(g)$.

From now on we fix an $F$-uniform pro-$p$ group. Given a wide open rigid
analytic group ${\mathbb{H}}_{n}^{\circ}$ as above, we let ${\mathcal{D}}^{\rm
an}({\mathbb{H}}_{n}^{\circ},E)={\rm Hom}^{\rm
cont}_{F}({\mathcal{O}}({\mathbb{H}}_{n}^{\circ}),E)^{\prime}_{b}$ be the
analytic distribution algebra of Emerton, cf. [5, 2.2.2]. Moreover, we
consider the $E$-algebra

$D({\mathbb{H}}^{\circ}_{n},H):=\big{(}C^{{\rm
la}}(H,E)_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}\big{)}^{\prime}_{b}$

which is the strong dual of the ${\mathbb{H}}_{n}^{\circ}$-rigid analytic
vectors of $C^{{\rm la}}(H,E)$. It is shown in [5, proof of 5.3.1] that

$D({\mathbb{H}}^{\circ}_{n},H)=\oplus_{h\in
H/H_{n+1}}\;\delta_{h}\cdot{\mathcal{D}}^{\rm
an}({\mathbb{H}}_{n}^{\circ},E)\,,$

where $H_{n+1}={\mathbb{H}}_{n+1}(F)={\mathbb{H}}_{n}^{\circ}(F)$. As in 3.5,
we choose an ${\mathcal{O}}_{F}$-basis $\partial_{1},\ldots,\partial_{d}$ of
the Lie algebra $L_{H}$ corresponding to $H$. As in [5, p. 102] we have the
following ${\mathcal{O}}_{E}$-subalgebra of the universal enveloping algebra
$U(L_{H})$

$A^{(m)}:=\Big{\\{}\sum_{I}b_{I}\frac{q_{i_{1}}^{(m)}!\ldots
q_{i_{d}}^{(m)}!}{i_{1}!\ldots
i_{d}!}\partial_{1}^{i_{1}}\ldots\partial_{d}^{i_{d}}\,\Big{|}\;b_{I}\in{\mathcal{O}}_{E},\\\
b_{I}=0\;\text{for\; almost\; all}\;I\Big{\\}}$

where $q^{(m)}_{i}=\left\lfloor\frac{i}{p^{m}}\right\rfloor$. Let
$\widehat{A^{(m)}}$ be the $p$-adic completion of $A^{(m)}$. We recall the
definitions of [5, p. 97, 107]

${\mathcal{D}}^{\rm
an}({\mathbb{H}}_{n}^{\circ},E)^{(m)}:=E\otimes_{{\mathcal{O}}_{E}}\widehat{A^{(m)}}$

and

$\begin{array}[]{rl}D({\mathbb{H}}^{\circ}_{n},H)^{(m)}&:=E[H]\otimes_{E[H_{n+1}]}D^{an}({\mathbb{H}}_{n}^{\circ},E)^{(m)}\\\
&\\\ &=\oplus_{h\in H/H_{n+1}}\;\delta_{h}\cdot{\mathcal{D}}^{\rm
an}({\mathbb{H}}_{n}^{\circ},E)^{(m)}\end{array}$

It is shown in the proof of [5, 5.3.19] that for every $n\geq 0$ there exists
a $m_{n}\in{\mathbb{N}}$ such that the natural map
$D({\mathbb{H}}_{n+1}^{\circ},H)\rightarrow D({\mathbb{H}}_{n}^{\circ},H)$
factors as

(4.1) $D({\mathbb{H}}_{n+1}^{\circ},H)\longrightarrow
D({\mathbb{H}}_{n}^{\circ},H)^{(m_{n})}\longrightarrow D^{\rm
an}({\mathbb{H}}_{n}^{\circ},H)\;.$

Moreover we have the following important results.

######  Theorem 4.2.

[5, 5.3.11, 5.2.6, and remark before 5.2.6] (i)
$D({\mathbb{H}}_{n}^{\circ},H)^{(m)}$ is a Noetherian Banach algebra for each
$m,n\in{\mathbb{N}}$.

(ii) There exists a natural isomorphism of topological $E$-algebras

$\varinjlim_{m}D({\mathbb{H}}_{n}^{\circ},H)^{(m)}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}D({\mathbb{H}}_{n}^{\circ},H)\,.$

This isomorphism makes $D({\mathbb{H}}_{n}^{\circ},H)$ an $E$-algebra of
compact type. ∎

######  Theorem 4.3.

(i) For an $F$-uniform pro-$p$ group $H$ the distribution algebra $D^{\rm
la}(H,E)$ is a weak Fréchet-Stein algebra with a weak Fréchet-Stein structure

$D^{\rm
la}(H,E)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}D({\mathbb{H}}_{n}^{\circ},H)\,.$

(ii) The inclusions $D({\mathbb{H}}_{n}^{\circ},H)^{(m)}\hookrightarrow D^{\rm
an}({\mathbb{H}}_{n}^{\circ},H)$ are flat for all $n$ and $m$.

(iii) The inclusions in (ii) induce an isomorphism

$D^{\rm
la}(H,E)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}D({\mathbb{H}}_{n}^{\circ},H)^{(m_{n})}\,,$

which equip $D^{\rm la}(H,E)$ with a Fréchet-Stein structure.

###### Proof.

(i) [5, 5.3.1].

(ii) This is stated after equation (5.3.20) in the proof of [5, 5.3.19].

(iii) [5, 5.3.19]. ∎

As a consequence of the previous theorem we obtain the following flatness
results.

######  Theorem 4.4.

(i) For every $n\geq 0$ the homomorphism of $E$-algebras

${\rm pr}^{{D^{\rm la}(H,E)}_{\bullet}}_{n}:{D^{\rm la}(H,E)}\longrightarrow
D({\mathbb{H}}_{n}^{\circ},H)^{(m_{n})}\;,$

obtained from the canonical projection ${D^{\rm la}(H,E)}\rightarrow
D({\mathbb{H}}_{n+1}^{\circ},H)$, followed by the inclusion
$D({\mathbb{H}}_{n+1}^{\circ},H)\hookrightarrow
D({\mathbb{H}}_{n}^{\circ},H)^{(m_{n})}$ in 4.1, is flat.

(ii) The natural map

${D^{\rm la}(H,E)}\rightarrow D({\mathbb{H}}_{n}^{\circ},H)$

is flat for every $n\geq 1$.

###### Proof.

(i) Because $D^{\rm
la}(H,E)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}D({\mathbb{H}}_{n}^{\circ},H)^{(m_{n})}$
gives $D^{\rm la}(H,E)$ a Fréchet-Stein structure, cf. 4.3 (iii), the map
${\rm pr}^{{D^{\rm la}(H,E)}_{\bullet}}_{n}$ is flat by [10, Rem. 3.2].

(ii) This follows from part (i) and 4.3 (ii). ∎

##  5\. Application to rigid analytic vectors

######  5.1.

Admissible representations and rigid analytic vectors. Let $G$ be a locally
$F$-analytic group, and let $V$ be an admissible locally analytic
representation of $G$ on an $E$-vector space $V$, as introduced in [10]. This
means that the dual space $M:=V^{\prime}_{b}$, equipped with the strong
topology, is a coadmissible module over the locally analytic distribution
algebra ${D^{\rm la}(H,E)}$ for any compact open subgroup $H\subset G$. In the
following we will assume that $H$ is an $F$-uniform pro-$p$ group.666Any such
$G$ has a fundamental system of open neighborhoods of the identity element
which are $F$-uniform pro-$p$ groups. Whereas the definition of ”coadmissible”
in [10] is based on the key result that ${D^{\rm la}(H,E)}$ is a Fréchet-Stein
algebra, we are here rather interested in the weak Fréchet-Stein structure
given by $D^{\rm
la}(H,E)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\varprojlim_{n}D({\mathbb{H}}_{n}^{\circ},H)$,
cf. 4.2. In this case, a $\\{D({\mathbb{H}}_{n}^{\circ},H)\\}_{n}$-sequence
for $M$, cf. 2.5, is given by

$M_{n}=\left(V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}\right)^{\prime}_{b}$

as follows from [5, 6.1.19] and its proof. We note here, that even though
there is a natural continuous injection from $V_{{\mathbb{H}}_{n}^{\circ}-{\rm
an}}\rightarrow V$, the space $V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$ is
topologized as a closed subspace of $C^{\rm an}({\mathbb{H}},V)$, where
$C^{\rm an}({\mathbb{H}},V)$ is the space of $V$-valued $F$-rigid analytic
functions on ${\mathbb{H}}$, cf. [5, 3.3.1, 3.3.13]. Usually this intrinsic
topology on $V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$ does not conincide with
the subspace topology induced on it as a subspace of $V$.

Therefore, by 2.9, the canonical map
$D({\mathbb{H}}_{n}^{\circ},H){\widehat{\otimes}}_{D^{\rm
la}(H,E)}M\rightarrow M_{n}$ is an isomorphism of
$D({\mathbb{H}}_{n}^{\circ},H)$-modules. The next result shows that the same
is true for the ordinary tensor product instead of the completed tensor
product.

######  Theorem 5.2.

Let $M=V^{\prime}_{b}$ be a coadmissible $D^{\rm la}(H,E)$-module, and let
$M_{n}=\left(V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}\right)^{\prime}_{b}$ be as
above. Then the canonical map

$D({\mathbb{H}}_{n}^{\circ},H)\otimes_{D^{\rm la}(H,E)}M\longrightarrow
M_{n}\;,$

is an isomorphism of topological $D({\mathbb{H}}_{n}^{\circ},H)$ modules for
all $n\geq 1$.

###### Proof.

Set $A={D^{\rm la}(H,E)}$, $A_{n}=D({\mathbb{H}}_{n}^{\circ},H)$, and
$B_{n}=D({\mathbb{H}}_{n}^{\circ},H)^{(m_{n})}$, with $m_{n}$ as in 4.1. Then,
because of 4.2 and 4.3 we can apply 2.11, and deduce the assertion above. ∎

######  Theorem 5.3.

Let $H$ be an $F$-uniform pro-$p$ group, and let ${\mathbb{H}}_{n}^{\circ}$ be
defined as in 3.7. The functor from the category of locally analytic
admissible representations of $H$ over $E$ to the category of $E$-vector
spaces given by

$V\rightsquigarrow V_{{\mathbb{H}}_{n}^{\circ}-{\rm an}}$

is exact for every $n\geq 1$.

###### Proof.

Denote by ${\mathcal{A}}_{H}$ (${\mathcal{C}}_{D^{\rm la}(H,E)}$, ${\rm
Mod}^{\rm f.g.}_{D({\mathbb{H}}_{n}^{\circ},H)}$, ${\rm Mod}_{E}$, resp.) the
category of admissible locally analytic $H$-representations on $E$-vector
spaces (coadmssible $D^{\rm la}(H,E)$-modules, finitely generated
$D({\mathbb{H}}_{n}^{\circ},H)$-modules, $E$-vector spaces, resp.). The
functor in question is a composition of three functors

(5.1) ${\mathcal{A}}_{H}\;\;\xrightarrow{V\rightsquigarrow
V^{\prime}_{b}}\;\;{\mathcal{C}}_{D^{\rm
la}(H,E)}\;\;\xrightarrow{M\rightsquigarrow M_{n}}\;\;{\rm Mod}^{\rm
f.g.}_{D({\mathbb{H}}_{n}^{\circ},H)}\;\;\xrightarrow{X\rightsquigarrow
X^{\prime}}\;\;{\rm Mod}_{E}\;,$

where $M_{n}=D({\mathbb{H}}_{n}^{\circ},H)\otimes_{D^{\rm la}(H,E)}M$. Objects
in the category ${\rm Mod}^{\rm f.g.}_{D({\mathbb{H}}_{n}^{\circ},H)}$ carry a
canonical $D({\mathbb{H}}_{n}^{\circ},H)$-module topology with respect to
which they are $E$-vector spaces of compact type, by 4.2 (ii) and [6, 5.1.1].

By [10, 6.3] the first functor is an anti-equivalence of categories. A quasi-
inverse functor is given by $M\rightsquigarrow M^{\prime}_{b}$. We show that
this functor is exact. Let

$0\longrightarrow
M_{1}\xrightarrow{\iota}M_{2}\xrightarrow{\pi}M_{3}\longrightarrow 0$

be an exact sequence of coadmissible modules. Consider the sequence of dual
spaces

(5.2) $0\longrightarrow
M_{3}^{\prime}\xrightarrow{\pi^{\prime}}M_{2}^{\prime}\xrightarrow{\iota^{\prime}}M_{1}^{\prime}\longrightarrow
0\;.$

$\pi^{\prime}$ is clearly injective, and $\iota^{\prime}\circ\pi^{\prime}=0$.
If $\iota^{\prime}(\lambda)=0$, then $\lambda$ descends to a continuous linear
form $\overline{\lambda}$ on $M_{2}/M_{1}$, equipped with the quotient
topology. By the open mapping theorem, the continuous bijection
$M_{2}/M_{1}\rightarrow M_{3}$ is a homeomorphism, and $\overline{\lambda}$ is
a continuous linear form $\nu$ on $M_{3}$, and $\pi^{\prime}(\nu)=\lambda$.
Since any morphsim in ${\mathcal{C}}_{D^{\rm la}(H,E)}$ is strict (proof of
[10, 3.6]), so is $M_{1}\xrightarrow{\iota}M_{2}$, and any continuous linear
form on $M_{1}$ extends therefore to a continuous linear form on $M_{2}$ (cf.
[7, 9.4]). This shows that 5.2 is exact.

Since $M\rightsquigarrow M^{\prime}_{b}$ is an anti-equivalence of categories,
any continuous $H$-morphism between admissible locally analytic
representations is strict, and the maps in 5.2 are thus strict too, i.e., 5.2
is an exact sequence of topological vector spaces.

The functor in the middle of 5.1 is exact by 4.4 (ii). Now consider an exact
sequence

$0\longrightarrow
M_{1}\xrightarrow{\iota}M_{2}\xrightarrow{\pi}M_{3}\longrightarrow 0$

of $D({\mathbb{H}}_{n}^{\circ},H)$-modules. By 2.10, the maps in this exact
sequence are strict with closed image. It then follows from [9, 1.2] that the
sequence of dual spaces

$0\longrightarrow
M_{3}^{\prime}\xrightarrow{\pi^{\prime}}M_{2}^{\prime}\xrightarrow{\iota^{\prime}}M_{1}^{\prime}\longrightarrow
0\;.$

is exact too. ∎

##  6\. Example: a weak Fréchet-Stein structure on $D^{\rm
la}({\mathbb{Z}_{p}},E)$

Let us consider the locally analytic functions $C^{\rm
la}({\mathbb{Z}_{p}},E)$. This is a locally convex $E$-vector space with a
compact-type topology and is naturally a ${{\mathbb{Q}}_{p}}$-analytic
representation over an $E$-vector space of ${\mathbb{Z}_{p}}$. The dual
$D^{\rm la}({\mathbb{Z}_{p}},E)$ naturally inherits the structure of a Fréchet
space. Let $X$ be the rigid analytic open unit disc centered at $0$ over $E$,
then we can show for any $z\in X({\mathbb{C}}_{p})$ and $a\in{\mathbb{Z}_{p}}$
the function $\kappa_{z}(a):=(1+z)^{a}$ defines a locally
${\mathbb{Q}}_{p}$-analytic character of ${\mathbb{Z}_{p}}$. Given $\lambda\in
D^{\rm la}({\mathbb{Z}_{p}},E)$ we obtain a function
$F_{\lambda}(z):=\lambda(\kappa_{z})$ on $X({\mathbb{C}}_{p})$.
$F_{\lambda}(z)$ is a rigid analytic function on $X$ and can thus be expanded
as a power series $\sum_{n=0}^{\infty}a_{n}z^{n}$ where $a_{n}\in E$ and
$\lim_{n\rightarrow\infty}|a_{n}|r^{n}=0$ for all $r<1$. If we denote the
$E$-valued rigid analytic functions on $X$ by
${\mathcal{O}}(X):=\Big{\\{}\sum_{n=0}^{\infty}a_{n}T^{n}\,\Big{|}a_{n}\in
E\;\text{s.t}\;\lim_{n\rightarrow\infty}a_{n}r^{n}\rightarrow
0\;\forall\;r<1\Big{\\}}$ then it is known [1, Thm. 1.3, Section. 2.3.4] that

$\displaystyle D^{\rm la}({\mathbb{Z}_{p}},E)$
$\displaystyle\rightarrow{\mathcal{O}}(X)$ $\displaystyle\lambda$
$\displaystyle\rightarrow F_{\lambda}$

is an isomorphism.

Let us consider, $C^{\rm
la}({\mathbb{Z}_{p}})_{p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}-{\rm an}}$, this
is the space of locally analytic functions such that around any point on
${\mathbb{Z}_{p}}$ the function is rigid analytic on an open unit disk of
radius $|p|^{n}$, i.e.,

$C^{\rm la}({\mathbb{Z}_{p}},E)_{p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}-{\rm
an}}=\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})$

where
${\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}):=\Big{\\{}\sum_{n}c_{n}(T-a)^{n}\,\Big{|}\;c_{n}\in
E,\lim_{n\rightarrow\infty}c_{n}r^{n}=0\;\forall\;r<|p|^{n}\Big{\\}}$. By
compactness of ${\mathbb{Z}_{p}}$ we see that

$C^{\rm la}({\mathbb{Z}_{p}},E)=\varinjlim_{n\geq
0}\Big{(}\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\Big{)}\;,$

and dually (cf. [7, 16.10 (iii)])

$D^{\rm la}({\mathbb{Z}_{p}},E)=\varprojlim_{n\geq
0}\Big{(}\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\Big{)}^{\prime}_{b}\;.$

Our main objective is to analyze,
$\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}\Big{(}a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}\Big{)}$.

######  Lemma 6.1.

Let $b\in{\mathbb{Z}}$ be a representative of an element of
${\mathbb{Z}}/p^{n+1}{\mathbb{Z}}$. Let $a=b+p^{n+1}c$ with $|c|=1$ and
$\frac{b-i}{p^{n+1}}\neq-c\,(\mbox{\,\rm mod\;}p)\;,0\leq i<p^{n+1}$, then

$\max_{b\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}\Big{|}{a\choose
k}\Big{|}=\Big{|}\lfloor\frac{k}{p^{n+1}}\rfloor!\Big{|}^{-1}\;.$

###### Proof.

We have,

(6.1) $|a-i|=\begin{cases}|b-i|\hskip 50.0pti\neq b\,(\mbox{\,\rm
mod\;}p^{n+1}{\mathbb{Z}_{p}})\\\ |p^{n+1}|\hskip 50.0pti=b\,(\mbox{\,\rm
mod\;}p^{n+1}{\mathbb{Z}_{p}})\end{cases}$

Let $\alpha_{b,j}:=\\#\\{0\leq i<k\,|\;|b-i|\leq|p|^{j}\\}\,,$ note with our
choices of $a$ and $b$,

(6.2) $\alpha_{b,n+1}=\\{0\leq i<k\,|\;|b-i|=|p|^{n+1}\;.\\}$

Using (6.1) and (6.2) we get that for a fixed $b$,

$\displaystyle|a(a-1)\ldots(a-k+1)|=\prod_{\begin{subarray}{c}i=b(\,(\mbox{\,\rm
mod\;}p^{n+1})\\\ 0\leq
i<k\end{subarray}}|p|^{n+1}\prod_{\begin{subarray}{c}i\neq b(\,(\mbox{\,\rm
mod\;}p^{n+1})\\\ 0\leq i<k\end{subarray}}|a-i|$
$\displaystyle=|p|^{(n+1)\alpha_{b,n+1}}|p|^{n(\alpha_{b,n}-\alpha_{b,n+1})}|p|^{(n-1)(\alpha_{b,n-1}-\alpha_{b,n})}\ldots|p|^{(\alpha_{b,1}-\alpha_{b,2})}$
$\displaystyle=|p|^{\sum_{j=0}^{n+1}\alpha_{b,j}}$

We note that $\alpha_{b,j}=\lfloor\frac{k}{p^{j}}\rfloor$ or
$\lfloor\frac{k}{p^{j}}\rfloor+1$ and if we write $k=m.p^{n+1}+b_{0}$, then
$\alpha_{b_{0},j}=\lfloor\frac{k}{p^{j}}\rfloor\;\forall\;1\leq j\leq n+1$.

So we obtain,

$\max_{b\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}|a(a-1)\ldots(a-k+1)|=|p|^{\sum_{j=1}^{n+1}\lfloor\frac{k}{p^{j}}\rfloor}=\Big{|}\frac{k!}{\lfloor\frac{k}{p^{n+1}}\rfloor!}\Big{|}$

Or equivalently,

$\max_{b\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}\Big{|}{a\choose
k}\Big{|}=\Big{|}\Big{\lfloor}\frac{k}{p^{n+1}}\Big{\rfloor}!\Big{|}^{-1}$

∎

######  Theorem 6.4.

Let us define $r_{n}:=|p|^{\frac{1}{(p-1)p^{n}}}$ and

${\mathcal{F}}_{n}:=\Big{\\{}\sum_{k=0}^{\infty}c_{k}{x\choose
k}\;\Big{|}\lim_{k\rightarrow\infty}c_{k}r_{n}^{k}\epsilon^{k}\rightarrow
0\;\mbox{for all}\,\epsilon<1\Big{\\}}$

Given
$f\in\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\,,$
the map $ev_{f}:{\mathbb{Z}_{p}}\rightarrow E$ defined by restricting $f$ to
${\mathbb{Z}_{p}}$ is locally analytic and its Mahler series lies in
${\mathcal{F}}_{n}$. Conversely, every element of ${\mathcal{F}}_{n}$ is
obtained as the Mahler series of some
$g\in\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})$.
The resulting map
$\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{Z}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})\rightarrow{\mathcal{F}}_{n}$
is a topological isomorphism.

###### Proof.

The crux of the proof lies in estimating $p$-adic norms of ${x\choose k}$ for
$x\in a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}$. For such an $x$ we write
$x=a+p^{n}y$, maximum-modulus principle allowing us to choose $|p|<|y|<1$.

If we choose representatives of ${\mathbb{Z}_{p}}/p^{n+1}{\mathbb{Z}_{p}}$ in
${\mathbb{Z}_{p}}$ as $a=b+p^{n+1}c$ with $|c|=1$ then we have,

(6.3) $|x-i|=\begin{cases}|a-i|\hskip 50.0pti\notin
a+p^{n+1}{{\mathcal{O}}_{{\mathbb{C}}_{p}}}\\\ |p^{n}y|\hskip 54.0pti\in
a+p^{n+1}{\mathcal{O}}_{{\mathbb{C}}_{p}}p\end{cases}$

Using (6.3) we get,

$\displaystyle|x(x-1)\ldots(x-k+1)|=|(a+p^{n}y)(a-1+p^{n}y)\ldots(a-k+1+p^{n}y)|$
$\displaystyle=\prod_{\begin{subarray}{c}a-i\in
p^{n+1}{\mathcal{O}}_{{\mathbb{C}}_{p}}\\\ 0\leq
i<k\end{subarray}}|p^{n}y|\prod_{\begin{subarray}{c}a-i\notin
p^{n+1}{\mathcal{O}}_{{\mathbb{C}}_{p}}\\\ 0\leq
i<k\end{subarray}}|a-i|=\prod_{0\leq
i<k}|a-i|\prod_{\begin{subarray}{c}\nu_{p}(a-i)=n+1\\\ 0\leq
i<k\end{subarray}}|\frac{y}{p}|$ $\displaystyle=\Big{|}{a\choose
k}k!\Big{|}\prod_{\begin{subarray}{c}\nu_{p}(a-i)=n+1\\\ 0\leq
i<k\end{subarray}}|\frac{y}{p}|$

By Thm.6.1,

$\max\Big{\\{}\Big{|}{a\choose k}\Big{|}_{{\mathbb{B}}_{\leq
p^{n+1}}(b)}\,\Big{|}\;b\in{\mathbb{Z}_{p}}/p^{n+1}{\mathbb{Z}_{p}}\Big{\\}}=\Big{|}\lfloor\frac{k}{p^{n+1}}\rfloor!\Big{|}^{-1}$

where ${\mathbb{B}}_{\leq p^{n+1}}(b)$ is the closed disk of radius $p^{n+1}$
around $b$. And
$\\#\\{\nu_{p}(a-i)=n+1\\}\leq\lceil\frac{k-b}{p^{n+1}}-1\rceil$ with the
bound being sharp. Combining all of these together we get for $x\in
a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}$

$\displaystyle\Big{|}{x\choose k}\Big{|}$
$\displaystyle=|\lfloor\frac{k}{p^{n+1}}\rfloor!|^{-1}|\frac{y}{p}|^{\lceil\frac{k-b}{p^{n+1}}-1\rceil}\text{\;with\,
equality\, for\, certain $a$ and $b$},$
$\displaystyle\leq|p|^{-\frac{k}{(p-1)p^{n+1}}-\frac{k}{p^{n+1}}}|y|^{\frac{k}{p^{n+1}}}=r_{n}^{-k}(|y|^{\frac{1}{p^{n+1}}})^{k}\text{\;(asymptotically)}$
$\displaystyle\leq r_{n}^{-k}\epsilon^{k}\;\;\forall\;\epsilon<1.$

The second line follows from the facts $|n!|\sim|p|^{\frac{n}{p-1}}$ and
$\lceil\frac{k-b}{p^{n+1}}-1\rceil\sim\frac{k}{p^{n+1}}$ asymptotically.

Let
$f\in\bigoplus_{a\in{\mathbb{Z}_{p}}/p^{n+1}{\mathbb{Z}_{p}}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})$
(considered as an element of $C^{\rm la}({\mathbb{Z}_{p}},E)$) be written as
$f=\sum_{k}c_{k}{x\choose k}$ (Mahler series expansion of $f$). Convergence of
this expression for $x\in a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}$ gives
$\lim_{k\rightarrow\infty}c_{k}r_{n}^{-k}\epsilon^{k}\rightarrow 0$ for every
$\epsilon<1$.

That there are no $0<r<r_{n}$ such that
$\lim_{k\rightarrow\infty}c_{k}r^{k}\epsilon^{k}\rightarrow 0$ can be deduced
from the logarithmic (in $k$) growth of
$\frac{|\lfloor\frac{k}{p^{n+1}}\rfloor!|^{-1}|\frac{y}{p}|^{\lceil\frac{k-b}{p^{n+1}}-1\rceil}}{|p|^{-\frac{k}{(p-1)p^{n+1}}-\frac{k}{p^{n+1}}}|y|^{\frac{k}{p^{n+1}}}}$
∎

Now we prove the following theorem

######  Theorem 6.6.

Let a space of sequences $(c_{k})_{k},c_{k}\in E$ be defined as

${\mathcal{F}}_{n}:=\Big{\\{}\sum_{k=0}^{\infty}c_{k}{x\choose
k}\,\Big{|}\lim_{k\rightarrow\infty}c_{k}r_{n}^{-k}\epsilon^{k}\rightarrow
0\,\mbox{for all}\,\epsilon<1\Big{\\}}\,.$

Let us define for a fixed $\epsilon<1$ the normed space

${\mathcal{F}}_{n,\epsilon}:=\Big{\\{}\sum_{k=0}^{\infty}c_{k}{x\choose
k}\;\mbox{where\;}\,\Big{|}\sum_{k}c_{k}{x\choose
k}\Big{|}_{\epsilon}:=\sup\\{|c_{k}|r_{n}^{-k}\epsilon^{k}\\}_{k}\Big{\\}}\,.$

We have
${\mathcal{F}}_{n}=\varprojlim_{\epsilon<1}{\mathcal{F}}_{n,\epsilon}$. Then
the continuous dual of this space is the space of sequence $(d_{k})_{k}$,
formally written as $\sum d_{k}T^{k}$ can be explicitly described as

${\mathcal{G}}_{n}:=\Big{\\{}\sum_{k=0}^{\infty}d_{k}T^{k}\,\Big{|}\;\exists\;R>r_{n}\;s.t.\lim_{k\rightarrow\infty}d_{k}R^{k}\rightarrow
0\Big{\\}}\,.$

###### Proof.

Let $\phi$ be a continuous linear functional from
${\mathcal{F}}_{n}\rightarrow E$. By continuity, there exists $\epsilon<1$ and
$C>0$ such that for each $f(x):=\sum_{k=0}^{\infty}c_{k}{x\choose k}$ we have
$|\phi(f(x))|\leq C|f(x)|_{\epsilon}$. Now if we write $\phi({x\choose
k})=d_{k}$, then we get

$\displaystyle|d_{k}|\leq Cr_{n}^{-k}\epsilon^{k}$ $\displaystyle\implies$
$\displaystyle|d_{k}|r_{n}^{k}(\frac{1}{\epsilon})^{k}\leq C$
$\displaystyle\implies$
$\displaystyle\lim_{k\rightarrow\infty}|d_{k}|R^{k}=0\;\forall\;R<\frac{r_{n}}{\epsilon}$

∎

As a consequence of the previous theorem we get a neat description of the dual
of
$\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{C}}_{p}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}})$

######  Corollary 6.7.

$(\bigoplus_{a\in{\mathbb{Z}}/p^{n+1}{\mathbb{C}}_{p}}{\mathcal{O}}(a+p^{n}{\mathfrak{m}}_{{\mathbb{C}}_{p}}))^{\prime}_{b}\cong{\mathcal{O}}_{r_{n}}(X)^{\dagger}$,
where ${\mathcal{O}}_{r_{n}}(X)^{\dagger}$ is the space of overconvergent
functions on a closed disk of radius $r_{n}$

###### Proof.

On the level of sets this immediately follows from the above theorem once we
explicitly recall the description of

${\mathcal{O}}_{r_{n}}(X)^{\dagger}:=\Big{\\{}\sum_{n=0}^{\infty}a_{n}T^{n}\;\Big{|}\;\lim_{n\rightarrow\infty}a_{n}R^{n}\rightarrow
0\mbox{ for some }R>r_{n}\Big{\\}}\;.$

In fact it can be shown that the isomorphism is indeed an isomorphism of
topological vector spaces. ∎

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