We say that an $E$-representation $V$ of ${\mathcal{G}}_{K}$ is split
trianguline if ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$ is a successive
extension of $(\varphi,\Gamma_{K})$-modules of rank $1$ over $E\otimes_{{{\bf
Q}_{p}}}{\bf B}_{{\mathrm{rig}},K}^{\dagger}$.

We say that an $L$-representation $V$ of ${\mathcal{G}}_{K}$ is trianguline if
there exists a finite extension $E$ of $L$ such that the $E$-representation
$E\otimes_{L}V$ is split trianguline.

We say that an $E$-representation $V$ of ${\mathcal{G}}_{K}$ is potentially
split trianguline (resp. potentially trianguline) if there exists a finite
extension $K^{\prime}$ of $K$ such that $V_{|{\mathcal{G}}_{K^{\prime}}}$ is
split trianguline (resp. trianguline).

######  .

Definition 7.1 can be equivalently stated in terms of $B$-pairs: an
$E$-representation $V$ of ${\mathcal{G}}_{K}$ is split trianguline if the
attached ${{\bf B}_{|K}^{\otimes E}}$-pair is a successive extension of rank
$1$ ${{\bf B}_{|K}^{\otimes E}}$-pairs.

######  .

Let $V$ be a $K$-analytic representation. Then the following are equivalent:

  1. 1.

$V$ is split trianguline.

  2. 2.

The Lubin-Tate $(\varphi_{q},\Gamma_{K})$-module ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$ is a successive extension of $F$-analytic
Lubin-Tate $(\varphi_{q},\Gamma_{K})$-modules of rank $1$.

###### Proof.

See [Poy20, Thm. 4.11]. ∎

For a ${\bf B}_{e,E}$-representation, we say that it is split triangulable if
it is a successive extension of rank $1$ ${\bf B}_{e,E}$-representations.

######  .

An $E$-representation $V$ of ${\mathcal{G}}_{K}$ is split trianguline if and
only if the corresponding ${\bf B}_{e,E}$-representation is split triangulable
as a ${\bf B}_{e,E}$-representation of ${\mathcal{G}}_{K}$.

###### Proof.

See [BDM19, Coro. 3.2]. ∎

######  .

The categories of split trianguline representations and of trianguline
representations are stable by subobjects, quotients, direct sums and tensor
products.

###### Proof.

The fact that it is stable by quotients and subobjects follows from [BDM19,
Prop. 3.3]. For direct sums and tensors products it is a straightforward
consequence of definition 7.1. ∎

Let $D$ be a $(\varphi,\Gamma_{K})$-module of rank $d$ over $E\otimes_{{{\bf
Q}_{p}}}{\bf B}_{{\mathrm{rig}},K}^{\dagger}$ and equipped with a strictly
increasing filtration $({\mathrm{Fil}}_{i}(D))_{i=0\dots d}$ :

${\mathrm{Fil}}_{0}(D):=\\{0\\}\subsetneq{\mathrm{Fil}}_{1}(D)\subsetneq\cdots\subsetneq{\mathrm{Fil}}_{i}(D)\subsetneq\cdots\subsetneq{\mathrm{Fil}}_{d-1}(D)\subsetneq{\mathrm{Fil}}_{d}(D):=D,$

of $(\varphi,\Gamma_{K})$-submodules which are direct summand as
$E\otimes_{{{\bf Q}_{p}}}{\bf B}_{{\mathrm{rig}},K}^{\dagger}$-modules. We
call such a $D$ a triangular $(\varphi,\Gamma_{K})$-module over
$E\otimes_{{{\bf Q}_{p}}}{\bf B}_{{\mathrm{rig}},K}^{\dagger}$, and the
filtration $\mathcal{T}:=({\mathrm{Fil}}_{i}(D))$ a triangulation of $D$ over
$E\otimes_{{{\bf Q}_{p}}}{\bf B}_{{\mathrm{rig}},K}^{\dagger}$.

Let $D$ be a triangular $(\varphi,\Gamma_{K})$-module. By proposition 3.1 of
[Col08b], each

$\mathrm{gr}_{i}(D):={\mathrm{Fil}}_{i}(D)/{\mathrm{Fil}}_{i-1}(D),\,\,\,1\leq
i\leq d,$

is isomorphic to the $(\varphi,\Gamma_{K})$-module on $E\otimes_{{{\bf
Q}_{p}}}{\bf B}_{{\mathrm{rig}},K}^{\dagger}$ attached to a character
$\delta_{i}$ for some unique $\delta_{i}:K^{\times}{\rightarrow}E^{\times}$.
Following [BC09a, 2.3.2], we define the parameter of the triangulation to be
the continuous homomorphism

$\delta:=(\delta_{i})_{i=1,\cdots,d}:K^{\times}{\rightarrow}(E^{\times})^{d}.$

When $K={{\bf Q}_{p}}$, the parameter of a triangular
$(\varphi,\Gamma_{K})$-module refines the data of its Sen polynomial:

######  .

Let $D$ be a triangular $(\varphi,\Gamma)$-module over $E\otimes_{{{\bf
Q}_{p}}}{\bf B}_{{\mathrm{rig}},{{\bf Q}_{p}}}^{\dagger}$ and $\delta$ the
parameter of a triangulation of $D$. Then the Sen polynomial of $D$ is

$\prod_{i=1}^{d}(T-w(\delta_{i})).$

###### Proof.

See [BC09a, Prop. 2.3.3]. ∎

We now recall the notion of refinements for crystalline trianguline
representations of ${\mathcal{G}}_{{{\bf Q}_{p}}}$ as in [BC09a, §2.4]. Let
$V$ be finite, $d$-dimensional, continuous, $E$-representation of
${\mathcal{G}}_{{{\bf Q}_{p}}}$. We will assume that $V$ is crystalline and
that the crystalline Frobenius $\varphi$ acting on ${\bf
D}_{{\mathrm{crys}}}(V)$ has all its eigenvalues in $E^{\times}$.

By a refinement of $V$, using the definition of [Maz00, §3], we mean the data
of a full $\varphi$-stable $E$-filtration
${\mathcal{F}}=({\mathcal{F}}_{i})_{i=0,\dots,d}$ of ${\bf
D}_{{\mathrm{crys}}}(V)$:

${\mathcal{F}}_{0}=0\subsetneq{\mathcal{F}}_{1}\subsetneq\cdots\subsetneq{\mathcal{F}}_{d}={\bf
D}_{{\mathrm{crys}}}(V).$

As in [BC09a, 2.4.1], we remark that any refinement ${\mathcal{F}}$ determines
two orderings:

  1. 1.

It determines an ordering $(\varphi_{1},\cdots,\varphi_{d})$ of the
eigenvalues of $\varphi$, defined by the formula

$\det(T-\varphi_{|{\mathcal{F}}_{i}})=\prod_{j=1}^{i}(T-\varphi_{j}).$

If all these eigenvalues are distinct then such an ordering conversely
determines ${\mathcal{F}}$.

  2. 2.

It determines also an ordering $(s_{1},\cdots,s_{d})$ on the set of Hodge-Tate
weights of $V$, defined by the property that the jumps of the weight
filtration of ${\bf D}_{{\mathrm{crys}}}(V)$ induced on ${\mathcal{F}}_{i}$
are $(s_{1},\cdots,s_{i})$

The theory of refinements has a simple interpretation in terms of
$(\varphi,\Gamma)$-modules: let $D$ be a crystalline $(\varphi,\Gamma)$-module
as above and let ${\mathcal{F}}$ be a refinement of $D$. We can construct from
${\mathcal{F}}$ a filtration $({\mathrm{Fil}}_{i}(D))_{i=0,\cdots,d}$ of $D$
by setting

${\mathrm{Fil}}_{i}(D):=(E\otimes_{{{\bf Q}_{p}}}{\bf B}_{{\mathrm{rig}},{{\bf
Q}_{p}}}^{\dagger}[1/t]{\mathcal{F}}_{i})\cap D,$

which is a finite type saturated $E\otimes_{{{\bf Q}_{p}}}{\bf
B}_{{\mathrm{rig}},{{{\bf Q}_{p}}}}^{\dagger}$-submodule of $D$.

######  .

The map defined above $({\mathcal{F}}_{i})\mapsto({\mathrm{Fil}}_{i}(D))$
induces a bijection between the set of refinements of $D$ and the set of
triangulations of $D$, whose inverse is
${\mathcal{F}}_{i}:={\mathrm{Fil}}_{i}(D)[1/t]^{\Gamma}$. In the bijection
above, for $i=1,\dots,d$, the graded piece
${\mathrm{Fil}}_{i}(D)/{\mathrm{Fil}}_{i-1}(D)$ is isomorphic to the
$(\varphi,\Gamma)$-module on $E\otimes_{{{\bf Q}_{p}}}{\bf
B}_{{\mathrm{rig}},{{{\bf Q}_{p}}}}^{\dagger}$ attached to $\delta_{i}$ where
$\delta_{i}(p)=\varphi_{i}p^{-s_{i}}$ and
${\delta_{i}}_{|\Gamma}=\chi^{-s_{i}}$, where the $\varphi_{i}$ and $s_{i}$
are defined by items $1$ and $2$ above.

###### Proof.

See [BC09a, Prop. 2.4.1]. ∎

######  .

In particular, Proposition 7.1 shows that crystalline representations are
trianguline, and that the set of their triangulations is in natural bijection
with the set of their refinements.

We now finish this section with a result regarding trianguline representations
that we were not able to find in the litterature but will prove useful in the
next sections.

######  .

Let $V$ be an $L$-representation of ${\mathcal{G}}_{K}$. Then $V$ is
trianguline if and only if the underlying ${{\bf Q}_{p}}$-representation of
$V$ is trianguline.

###### Proof.

Let $V$ be an $L$-representation of ${\mathcal{G}}_{K}$ and let $E$ be a
finite extension of $L$, containing all the images of the embeddings
$\tau:L\to\overline{K}$ and such that $E\otimes_{L}V$ is split trianguline.
Then $E\otimes_{{{\bf Q}_{p}}}V=(E\otimes_{{{\bf
Q}_{p}}}L)\otimes_{L}V=\oplus_{\tau\in\Sigma}(E\otimes_{L}V)_{\tau}$ where
$\Sigma={\mathrm{Emb}}(L,\overline{K})$.

In particular, $E\otimes_{L}V$ is a subrepresentation of $E\otimes_{{{\bf
Q}_{p}}}V$ and this concludes the first half of the proof by proposition 7.1.
For the other direction, let $W=W_{e}(E\otimes_{L}V)$ the corresponding ${\bf
B}_{e,E}$-representation and let $W_{0}=0\subset W_{1}\subset\ldots W_{d}=W$ a
triangulation of $W$. For $\tau\in\Sigma$, let ${\bf
B}_{e,E,\tau}=E\otimes_{L,\tau}{\bf B}_{e,E}$. For $\tau\in\Sigma$ and $1\leq
i\leq d$, let $W_{i,\tau}={\bf B}_{e,E,\tau}\otimes_{{\bf B}_{e,E}}W_{i}$. By
construction

$0\subset W_{1,\tau}\subset\ldots\subset W_{d,\tau}$

is a triangulation of $W((E\otimes_{L}V)_{\tau})$ and thus $E\otimes_{{{\bf
Q}_{p}}}V$ is trianguline. ∎

###  7.2 Discussion on a ring of periods for trianguline representations

By proposition 7.1, we know that the category of (split) trianguline
representations of ${\mathcal{G}}_{K}$ is tannakian. Therefore, it appears
reasonable to look for a ring $B$ such that trianguline representations are
exactly the representations which are $B$-admissible in the sense of Fontaine.

Recall that the notion of admissibility in the sense of Fontaine is defined
for what he called regular rings and is as follows (we only recall the
definitions of [Fon94b] in the particular case of ${{\bf
Q}_{p}}$-representations because that’s all we need here).

Let $B$ be a topological ${{\bf Q}_{p}}$-algebra endowed with an action of a
group $G$. For any ${{\bf Q}_{p}}$-representation of $G$, we let
$D_{B}(V):=(B\otimes_{{{\bf Q}_{p}}}V)^{G}$. We let $\alpha_{B}(V)$ denote the
$B$-linear map $B\otimes_{B^{G}}D_{B}(V)\longrightarrow B\otimes_{{{\bf
Q}_{p}}}V$ deduced from the inclusion $D_{B}(V)\subset B\otimes_{{{\bf
Q}_{p}}}V$ by extending the scalars to $B$. The ring $B$ is said to be
$G$-regular if the following hold:

  1. 1.

$B$ is reduced;

  2. 2.

for any $p$-adic representation $V$ of $G$, the map $\alpha_{V}$ is injective;

  3. 3.

any element $b$ of $B$ which is nonzero and is such that the ${{\bf
Q}_{p}}$-line generated by $B$ is $G$-stable is invertible.

The last condition implies in particular that $B^{G}$ is a field. If $B$ is
$G$-regular, a representation $V$ of $G$ is said to be $B$-admissible if
$\alpha_{B}(V)$ is an isomorphism, which is equivalent as saying that
$\dim_{B^{G}}D_{B}(V)=\dim_{{{\bf Q}_{p}}}V$.

Unfortunately, it seems to us that in the case we consider, the last condition
is too strong and thus we extend the notion of $G$-regularity as follows: we
say that $B$ is $G$-regular if the following conditions are met:

  1. 1.

$B$ is reduced;

  2. 2.

for any $p$-adic representation $V$ of $G$, $D_{B}(V)$ is a free
$B^{G}$-module;

  3. 3.

the map $\alpha_{V}$ is injective.

It is clear that $G$-regular rings in the sense of Fontaine are $G$-regular
for us, but that the converse does not hold.

In the rest of the paper, $G$-regularity and admissibility are to be
understood in our sense.

We now explain exactly what we mean by a ring of trianguline periods.

######  .

A ${\mathcal{G}}_{K}$-regular ring $B$ is said to be a trianguline periods
ring for ${\mathcal{G}}_{K}$ if trianguline representations of
${\mathcal{G}}_{K}$ are $B$-admissible, and if $B$-admissible representations
of ${\mathcal{G}}_{K}$ are trianguline.

######  .

Let $B$ be a ${\mathcal{G}}_{K}$-regular ring and let $V$ be an
$L$-representation of ${\mathcal{G}}_{K}$. Then $V$ is $B$-admissible if and
only if there exists a finite extension $E$ of $L$ such that $V\otimes_{L}E$
is $B$-admissible.

###### Proof.

It’s clear that if $V$ is $B$-admissible, then there exists a finite extension
$E$ of $L$ such that $V\otimes_{L}E$ is $B$-admissible. To show the reverse,
first note that the admissibility of an $E$ representation $V$ does not depend
on wether one considers it as a ${{\bf Q}_{p}}$-representation or as an
$E$-representation (the $B^{{\mathcal{G}}_{K}}$-module
$D_{B}(V)=(V\otimes_{{{\bf Q}_{p}}}B)^{{\mathcal{G}}_{K}}$ is always the
same). Now because the category of $B$-admissible representations is clearly
stable by subobjects, it suffices to prove the proposition to note that $V$ is
a sub-${{\bf Q}_{p}}$-representation of $V\otimes_{L}E$. ∎

Unlike in the crystalline or semistable case, if such a ring exists, it has to
depend on $K$:

######  .

There is no ring $B$ satisfying the properties above such that, for any finite
extension $K$ of ${{\bf Q}_{p}}$, $B$ is a trianguline periods ring for
${\mathcal{G}}_{K}$.

Proposition 7.2 is a consequence of the following result:

######  .

Let $L/K$ be any finite extension. Then there exists a representation $V$ of
${\mathcal{G}}_{K}$ such that $V$ is trianguline as a representation of
${\mathcal{G}}_{L}$ but is not trianguline as a representation of
${\mathcal{G}}_{K}$.

###### Proof.

Let $\eta:{\mathcal{G}}_{L}\to L^{\times}$ be a character such that there
exists $\tau_{1}\neq\tau_{2}\in{\mathrm{Emb}}(L,{\overline{{\bf Q}}_{p}})$
with $(\tau_{1})_{|K}=(\tau_{2})_{|K}$, and such that $\eta$ is $\tau_{1}$-de
Rham but not $\tau_{2}$-de Rham in the sense of [Din14]. Our claim is that
such a character can’t possibly extend to ${\mathcal{G}}_{K}$ and neither can
any of its conjugate, i.e. there is no character $\rho:{\mathcal{G}}_{K}\to
L^{\times}$ such that $\rho_{|{\mathcal{G}}_{L}}=\sigma(\eta)$ for some
$\sigma\in{\mathrm{Emb}}(L,{\overline{{\bf Q}}_{p}})$. Indeed, if such a
$\rho$ existed, then the dimension of ${\bf
D}_{{\mathrm{dR}},\sigma}(\sigma(\eta))$ would only depend of the dimension of
${\bf D}_{{\mathrm{dR}},\sigma_{|K}}(\rho)$, which is not the case because of
the assumption on $\tau_{1}$ and $\tau_{2}$.

We now let $V=\mathrm{ind}_{{\mathcal{G}}_{L}}^{{\mathcal{G}}_{K}}\eta$. This
is a $p$-adic representation of ${\mathcal{G}}_{K}$, whose restriction to
${\mathcal{G}}_{L}$ is the sum of the conjugates of $\eta$, so that it clearly
is trianguline as a representation of ${\mathcal{G}}_{L}$. Let us assume that
it also is trianguline as a representation of ${\mathcal{G}}_{K}$. Let $W$ be
the ${\bf B}_{|K}^{\otimes L}$-pair attached to $V$. As a ${\bf
B}_{|L}^{\otimes L}$-pair, we can write

$W=\oplus_{\sigma}W(\sigma(\eta)).$

Since we assumed that $W$ is trianguline as a ${\bf B}_{|K}^{\otimes L}$-pair,
there exists $W_{1}\subset W$ a direct summand of rank $1$. For
$\tau\in{\mathrm{Emb}}(L,{\overline{{\bf Q}}_{p}})$, we have the following
exact sequence

$0{\rightarrow}\oplus_{\tau\neq\sigma}W(\sigma(\eta)){\rightarrow}W{\rightarrow}W(\tau(\eta)){\rightarrow}0$

so that, since $W_{1}$ is a direct summand of rank $1$ of $W$ and by
proposition 2.4 of [BDM19], we either have $W_{1}=W(\tau(\eta))$ or
$W_{1}\subset\oplus_{\tau\neq\sigma}W(\sigma(\eta))$. By induction, $W_{1}$
has to be equal to one of the $W(\tau(\eta))$. Therefore, one of the
conjugates of $\eta$ has to extend to ${\mathcal{G}}_{K}$, which we have
proven is not possible. ∎

We can now give a proof of proposition 7.2:

###### Proof.

By the results of §5, the periods of any representation live in
${\widetilde{\bf{B}}}^{I}{\\{\\{u\\}\\}}$, for $I$ any compact subinterval of
$[r_{0};+\infty[$, and in particular so do the periods of any trianguline
representation $V$ of ${\mathcal{G}}_{K}$, so that we can assume that
$B\subset{\widetilde{\bf{B}}}^{I}{\\{\\{u\\}\\}}$. Since every unramified
representation of ${\mathcal{G}}_{K}$ is trianguline, we can assume that
$B\supset\widehat{{{\bf Q}_{p}}^{nr}}$. Moreover, if $L/K/{{\bf Q}_{p}}$ are
unramified then it is easy to see that
$({\widetilde{\bf{B}}}^{I}{\\{\\{u\\}\\}})^{{\mathcal{G}}_{L}}=L\otimes_{K}({\widetilde{\bf{B}}}^{I}{\\{\\{u\\}\\}})^{{\mathcal{G}}_{K}}$,
and thus we can assume that
$B^{{\mathcal{G}}_{L}}=L\otimes_{K}B^{{\mathcal{G}}_{K}}$.

Assume that $B$ is a ring satisfying the properties, and such that for any
finite extension $K/{{\bf Q}_{p}}$, $B$ is a trianguline periods ring for
${\mathcal{G}}_{K}$. Let $K$ be a finite unramified extension of ${{\bf
Q}_{p}}$, let $L$ be a finite unramified extension of $K$. Let $V$ be a
$p$-adic representation of ${\mathcal{G}}_{K}$ which becomes trianguline over
${\mathcal{G}}_{L}$, which exists by the previous proposition. Let
$D_{L}=(B\otimes V)^{{\mathcal{G}}_{L}}$. It is a
$B^{{\mathcal{G}}_{L}}$-module, endowed with a semilinear action of
${\mathrm{Gal}}(L/K)$. Let $D=D_{L}^{{\mathrm{Gal}}(L/K)}$. By Speiser’s
lemma, $D_{L}\simeq L\otimes_{K}D$ and thus
$D_{L}=B^{{\mathcal{G}}_{L}}\otimes_{B^{{\mathcal{G}}_{K}}}D$. Thus, $V$ is
$B$-admissible as a representation of ${\mathcal{G}}_{K}$. ∎

###  7.3 The ring ${{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u\\}\\}}$

We now study the ring
${{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u\\}\\}}$ as a potential ring
of trianguline periods for ${\mathcal{G}}_{{{\bf Q}_{p}}}$. In a previous
version of this paper, we claimed that trianguline representations were
admissible for this ring but there was a gap in the proof and the claim is
actually not true. We do expect though that if such a ring exists then it has
to be some intermediate ring $B$ between
${{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u\\}\\}}$ and the rings
${\widetilde{\bf{B}}}^{I}{\\{\\{u\\}\\}}$, but it is not clear at all “how
much periods we have to add” to
${{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u\\}\\}}$. More generally, we
extend these constructions to the $F$-analytic Lubin-Tate case.

If $T$ is a variable and $L$ is a finite extension of ${{\bf Q}_{p}}$, we let
$L\langle\langle T\rangle\rangle$ denote the set of power series in $T$ with
coefficients in $L$ and with infinite radius of convergence.

######  .

For any $r\geq 0$, we have
$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger,r}{\\{\\{u_{K}\\}\\}}_{n})^{{\mathcal{G}}_{K_{n}}}=K{\langle\langle
x_{K}\rangle\rangle}$.

###### Proof.

Let
$z\in({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger,r}{\\{\\{u_{K}\\}\\}}_{n})^{{\mathcal{G}}_{K_{n}}}$.
We can write $z=\sum_{k\geq 0}a_{k}u_{K}^{k}$ with the $a_{k}\in
K\otimes_{K_{0}}{{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger,r}$, and
actually in
$K\otimes_{K_{0}}{\widetilde{\bf{B}}}_{{\mathrm{rig}},K}^{\dagger,r}$ since
$u_{K}$ is left invariant by $H_{K}$. The fact that $z$ is left invariant by
the action of ${\mathcal{G}}_{K_{n}}$ means that, for any
$g\in{\mathcal{G}}_{K_{n}}$, we have

$g(a_{i})=\sum_{j\geq 0}a_{i+j}\binom{i+j}{i}(-c(g))^{j},$

where $c(g)=\log\chi_{\pi}(g)$. Therefore, for any compact $I=[r;s]$, the
$a_{i}$ are $\Gamma_{n}$-$K$-anaytic vectors of ${\widetilde{\bf{B}}}^{I}$. By
item 1. of [Ber16b, Thm. 4.4], we have that the images of the $a_{i}$ in
${\widetilde{\bf{B}}}^{I}$ all belong to ${\bf B}_{K}^{I}$ as long as $s$ is
such that $r_{n}\leq s$. Taking the inverse limit, this implies that the
$a_{i}$ all belong to ${\bf B}_{{\mathrm{rig}},K}^{\dagger,r}$.

Since $\varphi$ commutes with the Galois action, the reasoning above also
applies to $\varphi_{q}^{-1}(z)$. Therefore, for all $k,i\geq 0$,
$\varphi_{q}^{-k}(a_{i})\in{\bf B}_{{\mathrm{rig}},K}^{\dagger}$. This implies
that the $a_{i}$ all belong to the ring $K\langle\langle
t_{\pi}\rangle\rangle$. Indeed, in the cyclotomic case this is [Col14, Prop.
3.9], and for the general case this follows from the same arguments, using the
dictionary developped by Colmez in the Lubin-Tate case in [Col16, §2].

Note that the ring $K\langle\langle t_{\pi}\rangle\rangle$ corresponds to the
subset of $K_{n}[\\![t_{\pi}]\\!]$ consisting of elements $y=\sum_{k\geq
0}y_{k}t_{\pi}^{k}$ with $y_{k}\in K$ for which the sum converges in
${{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}$. Therefore, by proposition
4.2, we can write $z$ as $z=\sum_{k\geq 0}b_{k}x_{K}^{k}$ and the sum
converges in
${{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}$, so
that the series $\sum_{k\geq 0}b_{k}x_{K}^{k}\in K{\langle\langle
x_{K}\rangle\rangle}$. To conclude the proof, it suffices to see that the
elements of $K{\langle\langle x_{K}\rangle\rangle}$ left invariants by
${\mathcal{G}}_{K_{n}}$ belong to $K{\langle\langle x_{K}\rangle\rangle}$, and
that any element of $K{\langle\langle x_{K}\rangle\rangle}$ belong to
$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger,r}{\\{\\{u_{K}\\}\\}}_{n})^{{\mathcal{G}}_{K_{n}}}$,
which is straightforward. ∎

In order to simplify the notations, we let ${\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}:={{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u_{K}\\}\\}}_{n}$
and ${\bf B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}:=\bigcup_{n\geq 1}{\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}$.

######  .

The rings
$(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$
are principal.

###### Proof.

We first prove that
$(({\widetilde{\bf{B}}}_{K}^{I})^{K-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}=\left\\{\sum_{k=0}^{\infty}\frac{(-1)^{k}\nabla^{k}(z)}{k!}u_{K}^{k},z\in\varphi_{q}^{-n}({\bf
B}_{K}^{q^{n}I})\right\\}$.

Let
$y\in(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$.
Recall [Ber02, §2.2] that there is a ${\mathcal{G}}_{K_{k}}$-equivariant
injective map
$\iota_{k}:{\widetilde{\bf{B}}}_{K}^{I}{\rightarrow}{\bf{B}_{\mathrm{dR}}^{+}}$
which induces an injective map
$\iota_{k}:(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n}{\rightarrow}({\bf{B}_{\mathrm{dR}}^{+}})^{H_{K}}{\\{\\{u_{K}\\}\\}}_{n}$
which is $\Gamma_{m}$-equivariant for $m=\max(k,n)$.

The image of $y$ in
$({\bf{B}_{\mathrm{dR}}^{+}})^{H_{K}}{\\{\\{u_{K}\\}\\}}_{n}$ is therefore
pro-analytic and killed by $\nabla=\frac{\log g}{\log(\chi_{\pi}(g))}$ for $g$
close enough to $1$, so that it is invariant by $\Gamma_{N}$ for $N\gg 0$. By
proposition 4.2, this implies that the image of $y$ in
${\bf{B}_{\mathrm{dR}}^{+}}{\\{\\{u_{K}\\}\\}}_{n}$ belongs to
$K_{N}[\\![x_{K}]\\!]$, and we can even say that it belongs to
$K_{m}[\\![x_{K}]\\!]$ since $\iota_{k}$ is $\Gamma_{m}$-equivariant. Let us
write $\iota_{k}(y)=\sum_{k=0}^{\infty}b_{k}x_{K}^{k}$ with $b_{k}\in K_{k}$.
We also know that
$\iota_{k}(y)\in{\bf{B}_{\mathrm{dR}}^{+}}{\\{\\{u_{K}\\}\\}}_{n}$ can be
written as $\iota_{k}(y)=\sum_{k\geq 0}a_{k}u_{K}^{k}$ with the
$a_{k}\in{\bf{B}_{\mathrm{dR}}^{+}}$. The equality
$\sum_{k=0}^{\infty}b_{k}x_{K}^{k}=\sum_{k=0}^{\infty}a_{k}u_{K}^{k}$ tells us
that $a_{k}=\frac{(-1)^{k}}{k!}\sum_{j=0}^{\infty}t_{\pi}^{j}b_{j}j^{k}$.

Let $\alpha:=\sum_{j=0}^{\infty}t_{\pi}^{j}b_{j}$. Since $\nabla$ acts on
$K_{k}[\\![x_{K}]\\!]$ by $t_{\pi}\frac{d}{dx_{K}}$, we can write
$a_{\ell}=\frac{(-1)^{\ell}}{\ell!}\nabla^{\ell}(\alpha)$. Let us write
$y=\sum_{\ell=0}^{\infty}\beta_{\ell}u_{K}^{\ell}$ in
$(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n}$.
By definition of $\iota_{k}$, we get that $\beta_{0}$ is such that
$\iota_{k}(\beta_{0})=\alpha$, and since it is $\Gamma_{k}$-equivariant, we
obtain that $\beta_{j}=\frac{(-1)^{j}}{j!}\nabla^{j}(\beta_{0})$ in
$(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}})$. But by definition of
$(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n}$,
this implies that the series $\sum_{j\geq
0}\frac{(-1)^{j}}{j!}\pi^{nj}\nabla^{j}(\beta_{0})$ converges in
$({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}$, so that $\beta_{0}$ is a
$\Gamma_{n}$-analytic vector. By [Ber16b, Thm. 4.4], this implies that
$\beta_{0}\in\varphi_{q}^{-n}({\bf B}_{K}^{q^{n}I})$.

Reciprocally, if $z$ is in $\varphi_{q}^{-n}{\bf B}_{K}^{q^{n}I}$, then it is
$\Gamma_{n}$-F-analytic by [Ber16b, Prop. 4.1], so that the series
$\sum_{j\geq 0}\frac{(-1)^{j}}{j!}\pi^{nj}\nabla^{j}(z)$ converges in
${\widetilde{\bf{B}}}_{K}^{I}$ and thus $y:=\sum_{j\geq
0}\frac{(-1)^{j}}{j!}\nabla^{j}(z)u_{K}^{j}$ belongs to
$(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n})$,
and a simple computation shows that $\nabla(y)=0$.

This implies that the map $f(u_{K})\mapsto f(0)$, where
$f(u_{K})=\sum_{k=0}^{\infty}a_{k}u_{K}\in(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$,
induces an isomorphism of $K$-algebras between
$(({\widetilde{\bf{B}}}_{K}^{I})^{F-{\mathrm{la}}}){\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$
and $\varphi_{q}^{-n}({\bf B}_{K}^{q^{n}I})$, and thus concludes the proof. ∎

######  .

The natural map $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}{\rightarrow}{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ is faithfully flat.

###### Proof.

This map is clearly injective, and since ${\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ is an integral domain, ${\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ is torsion free over $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$ which is a PID by corollary
7.3. Therefore, the map $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}{\rightarrow}{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ is flat. To show that it is faithfully
flat, it therefore suffices to prove that for any maximal ideal $\mathfrak{m}$
of $({\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$, $\mathfrak{m}{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}\neq{\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$.
But if $\mathfrak{m}$ is a maximal ideal of $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$, then by the proof of
proposition 7.3 it is generated by an element $f$ of $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}\simeq\varphi_{q}^{-n}({\bf
B}_{K}^{q^{n}I})$ which corresponds through this isomorphism to an element of
$\varphi_{q}^{-n}({\bf B}_{K}^{q^{n}I})$ which vanishes on some point $z$ of
the corresponding annulus. But then every element of $\mathfrak{m}{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ will vanish on $(z,0)$ after identifying
${\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ with the set of analytic functions on
the product of the annulus corresponding to $I$ with the disk
$D(0,|\pi|^{n})$, so that $\mathfrak{m}{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}\neq{\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$.
∎

######  .

Let $V$ be a $K$-analytic $E$-representation of $K$. Then

$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}})^{\Gamma_{K_{n}}}$

and

$\bigcup_{n\geq
1}({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}})^{\nabla=0}.$

###### Proof.

We have

$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}=({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V))^{{\mathcal{G}}_{K_{n}}}$

since ${\widetilde{\bf{B}}}_{{\mathrm{rig}}}^{\dagger}$ contains the periods
of $F$-analytic representations used to define the
$(\varphi_{q},\Gamma_{K})$-modules attached to such representations. Taking
the invariants under $H_{K}$, we get that

$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}=({\widetilde{\bf{B}}}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V))^{\Gamma_{K_{n}}}.$

Since the elements of
$({\widetilde{\bf{B}}}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V))^{\Gamma_{K_{n}}}$ are fixed by
$\Gamma_{K_{n}}$ they are in particular pro-analytic vectors of $\Gamma_{K}$.
By proposition 2 and by the fact that ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$
is composed of pro-analytic vectors of $\Gamma_{K}$ (this follows from example
from [Ber16b, Thm. 10.4]), we get using [Ber16b, Thm. 4.4] that

$({\widetilde{\bf{B}}}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V))^{\Gamma_{K_{n}}}=\left(\bigcup_{m\geq
1}\varphi_{q}^{-m}({\bf
B}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n})\otimes{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)\right)^{\Gamma_{K_{n}}}.$

Since the action of $\varphi_{q}$ and $\Gamma_{K_{n}}$ commute and since
$\varphi_{q}^{*}{\bf D}_{{\mathrm{rig}}}^{\dagger}(V)\simeq{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$, we obtain that

$({\widetilde{\bf{B}}}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V))^{\Gamma_{K_{n}}}=\bigcup_{m\geq
0}(\varphi_{q}^{-m}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n})^{\Gamma_{K_{n}}}.$

Using once again that $\varphi_{q}^{*}{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)\simeq{\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$,
we get that

$\bigcup_{m\geq 0}(\varphi_{q}^{-m}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n})^{\Gamma_{K_{n}}}=\bigcup_{m\geq
0}(\varphi_{q}^{-m}({\bf
B}_{{\mathrm{rig}},K}^{\dagger})^{\Gamma_{K_{n}}}\otimes_{K{\langle\langle
x_{K}\rangle\rangle}}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n})^{\Gamma_{K_{n}}}$

and this is equal to

$K{\langle\langle x_{K}\rangle\rangle}\otimes_{K{\langle\langle
x_{K}\rangle\rangle}}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n})^{\Gamma_{K_{n}}}=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n})^{\Gamma_{K_{n}}}.$

For the second point of the proposition, it suffices to see that an element
$z$ in a pro-$K$-analytic representation of $\Gamma_{K}$ is killed by $\nabla$
if and only if there exists $m\gg 0$ such that $z$ is invariant under
$\Gamma_{m}$. ∎

In particular, the module $\bigcup_{n\geq
1}({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}$
can be thought of as the module of solutions of the differential operator
$\nabla$ over a ring of power series in two variables, defined over a
polyannulus (here a product between a disk and an annulus). In the cyclotomic
case for example, the operator on the ring of power series in two variables
$X$ and $Y$ is given by $(1+X)\cdot\log(1+X)\cdot\frac{d}{dX}+\frac{d}{dY}$.

If $V$ is an $F$-analytic $E$-representation of ${\mathcal{G}}_{K}$, if $r\geq
0$ is such that ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$ and all its structures
are defined over ${\bf B}_{{\mathrm{rig}},K}^{\dagger,r}$ and if $I$ is a
compact subinterval of $[r,+\infty[$, we let $\mathcal{D}_{n}^{I}(V):=({\bf
B}_{K}^{I}\otimes_{{\bf B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$.

######  .

Let $V$ be an $F$-analytic $E$-representation of ${\mathcal{G}}_{K}$, let
$r\geq 0$ be such that ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$ and all its
structures are defined over ${\bf B}_{{\mathrm{rig}},K}^{\dagger,r}$ and let
$I$ be a compact subinterval of $[r,+\infty[$. Then $\mathcal{D}_{n}^{I}(V)$
is a free $({\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$-module of
finite type.

###### Proof.

Let $\mathcal{M}_{n}^{I}\subset{\bf B}_{K}^{I}\otimes_{{\bf
B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$ denote the image
of the map

$\alpha:{\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}^{\nabla=0}}\mathcal{D}_{n}^{I}(V){\rightarrow}{\bf
B}_{K}^{I}\otimes_{{\bf B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$

deduced from the inclusion $\mathcal{D}_{n}^{I}(V)\subset{\bf
B}_{K}^{I}\otimes_{{\bf B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$. Since ${\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}$ is noetherian, and since ${\bf
B}_{K}^{I}\otimes_{{\bf B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$ is free of rank
$\dim_{E}(V)$, we know that $\mathcal{M}_{n}^{I}$ is finitely generated.

In order to prove that $\alpha$ is injective, note that it suffices to prove
that

$\alpha^{\prime}:{\mathrm{Frac}}({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})\otimes_{{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}^{\nabla=0}}\mathcal{D}_{n}^{I}(V){\rightarrow}{\mathrm{Frac}}({\bf
B}_{K}^{I})\otimes_{{\bf B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$

deduced from the inclusion $\mathcal{D}_{n}^{I}(V)\subset{\mathrm{Frac}}({\bf
B}_{K}^{I})\otimes_{{\bf B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$ is injective,
because the natural map

${\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}^{\nabla=0}}\mathcal{D}_{n}^{I}(V){\rightarrow}{\mathrm{Frac}}({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})\otimes_{{\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}^{\nabla=0}}\mathcal{D}_{n}^{I}(V)$

is injective since $\mathcal{D}_{n}^{I}(V)$ is flat over $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$ (it is torsion free over $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$ which is a PID by corollary
7.3).

In order to prove that $\alpha^{\prime}$ is injective, we are reduced to check
that, given a family of elements $(u_{1},\cdots,u_{d})$ linearly independent
over $({\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$, they are linearly
independent over ${\mathrm{Frac}}({\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})$. We
prove the result by induction on $d$. The result is trivial when $d=1$ because
$({\mathrm{Frac}}({\bf B}_{K}^{I})\otimes_{{\bf
B}_{{\mathrm{rig}}}^{\dagger,r}}{\bf
D}_{{\mathrm{rig}}}^{\dagger,r}(V)){\\{\\{u_{K}\\}\\}}_{n}$ is free over
${\mathrm{Frac}}({\bf B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})$. Assume now that
$d\geq 2$ is such that the result holds for $d-1$. Let
$\lambda_{1},\cdots,\lambda_{d}$ in ${\mathrm{Frac}}({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})$ be such that
$\sum_{i=1}^{d}\lambda_{i}u_{i}=0$. By induction, we may assume that
$\lambda_{1}\neq 0$. Dividing by $\lambda_{1}$, we obtain that
$u_{1}+\sum_{i=2}^{d}\lambda_{i}^{\prime}u_{i}=0$ where
$\lambda_{i}^{\prime}=\frac{\lambda_{i}}{\lambda_{1}}$. Applying $\nabla$, we
get that $\sum_{i=2}^{d}\nabla(\lambda_{i}^{\prime})u_{i}=0$ so that the
$\lambda_{i}^{\prime}$ are all zero by induction.

This now implies that $\alpha$ is injective, so that ${\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}}\mathcal{D}_{n}^{I}(V)$ is
finitely generated. By lemma 7.3 and descent for faitfully flat modules, we
deduce that $\mathcal{D}_{n}^{I}(V)$ is finitely generated, and thus free
since it is torsion free over $({\bf
B}_{K}^{I}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$ which is a PID. ∎

######  .

If $V$ is a $K$-analytic $E$-representation of ${\mathcal{G}}_{K}$, then
$\mathcal{D}(V)=\bigcup_{n\geq
1}({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}$
is a free $E{\langle\langle x_{K}\rangle\rangle}$-module of finite type, and
the map

$\alpha_{V}:{\widetilde{\bf{B}}}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}\otimes_{E{\langle\langle
x_{K}\rangle\rangle}}\mathcal{D}(V){\rightarrow}{{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}\otimes_{E}V$

is injective.

###### Proof.

This follows from the fact that if we let $\mathcal{D}_{n}(V)=({\bf
D}_{{\mathrm{rig}},K}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n})^{\nabla=0}$ then
$\mathcal{D}_{n}(V)=\varprojlim\limits_{I}\mathcal{D}_{n}^{I}(V)$. The
injectivity of $\alpha_{V}$ follows from the same arguments as in the proof of
proposition 7.3 above. ∎

###  7.4 Frobenius regularity

We now explain how to use the fact that our rings are embedded with a
Frobenius in order to show some regularity property. Namely, we will show that
$K$-analytic representation which are
${\widetilde{\bf{B}}}_{{\mathrm{rig}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}$-admissible
are actually ${\bf B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}$-admissible. This
section is in the same spirit as [Ber02, §3.1 and §3.2].

######  .

Let $h$ be a positive integer. Then

$\bigcap_{s=0}^{+\infty}\pi^{-hs}{\widetilde{\bf{A}}}^{\dagger,q^{-s}r}={\widetilde{\bf{A}}^{+}}\quad\textrm{and}\quad\bigcap_{s=0}^{+\infty}\pi^{-hs}{\widetilde{\bf{A}}}^{\dagger,q^{-s}r}_{{\mathrm{rig}}}\subset{{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}.$

###### Proof.

This is [Ber02, Lemm. 3.1] when $K={{\bf Q}_{p}}$. The generalization when $K$
is a finite extension of ${{\bf Q}_{p}}$ is straightforward. ∎

######  .

Let $h$ be a positive integer and let $n$ be an integer $\geq 1$. Then

$\bigcap_{s=0}^{+\infty}\pi^{-hs}{\widetilde{\bf{A}}}^{\dagger,q^{-s}r}{\\{\\{u_{K}\\}\\}}_{n}={\widetilde{\bf{A}}^{+}}{\\{\\{u_{K}\\}\\}}_{n}\quad\textrm{and}\quad\bigcap_{s=0}^{+\infty}\pi^{-hs}{\widetilde{\bf{A}}}^{\dagger,q^{-s}r}_{{\mathrm{rig}}}{\\{\\{u_{K}\\}\\}}_{n}\subset{\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}.$

###### Proof.

It just follows from the definitions of 3 for ${{\bf Z}_{p}}$-algebras, and
from lemma 7.4. ∎

######  .

Let $r,v$ be two positive integers, and let $A\in{\mathrm{M}}_{v\times
r}({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n})$.
Assume that there exists $P\in{\mathrm{GL}}_{v}(K{\langle\langle
x_{K}\rangle\rangle})$ such that $A=P\varphi_{q}^{-1}A$. Then
$A\in{\mathrm{M}}_{v\times r}({\bf B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}})$.

###### Proof.

Write $A$ as $(a_{ij})$ and $P$ as $(p_{ij})$. The assumption on the relation
between $P$ and $A$ can be translated as:

$p_{i1}\varphi^{-1}(a_{1j})+\cdots+p_{iv}\varphi^{-1}(a_{vj})=a_{ij}\quad\forall
i\leq v,j\leq r.$

Let us first show that, if $f\in K{\langle\langle x_{K}\rangle\rangle}$ and if
$r_{0}\geq 0$, then there exists $h\geq 0$ such that $\pi^{h}\cdot
f\in{\widetilde{\bf{A}}}_{{\mathrm{rig}}}^{\dagger,r}{\\{\\{u_{K}\\}\\}}_{n}$
for all $0\leq r\leq r_{0}$. Indeed, write $f$ as $f=\sum_{k\geq
0}a_{k}x_{K}^{k}$. This can be also written as $f=\sum_{k\geq
0}b_{k}u_{K}^{k}$, with $b_{k}=\frac{(-1)^{k}}{k!}\sum_{j\geq
0}j^{k}a_{j}t_{\pi}^{j}$. We know that the $a_{j}$ tend to $0$ exponentially
and we also know that $-\infty<V([r_{0};r_{0}],t)\leq V([r;r],t_{\pi})$ for
$0\leq r\leq r_{0}$, hence the result follows.

Now let $c,r_{0}$ be such that all the $a_{ij}$ belong to
$\pi^{-c}{\widetilde{\bf{A}}}_{{\mathrm{rig}}}^{\dagger,r_{0}}{\\{\\{u_{K}\\}\\}}_{n}$
and let $h$ be as above, so that the $\pi^{h}p_{ij}$ belong to
${\widetilde{\bf{A}}}_{{\mathrm{rig}}}^{\dagger,r}{\\{\\{u_{K}\\}\\}}_{n}$ for
all $r\leq r_{0}$. Since
$\varphi_{q}^{-1}(a_{ij})\in\pi^{-c}{\widetilde{\bf{A}}}_{{\mathrm{rig}}}^{\dagger,r_{0}/q}$,
we get that the $a_{ij}$ belong to
$\pi^{-h-c}{\widetilde{\bf{A}}}_{{\mathrm{rig}}}^{\dagger,r_{0}/q}{\\{\\{u_{K}\\}\\}}_{n}$.
By iterating, we see that the $a_{ij}$ actually belong to
$\cap_{s=0}^{+\infty}\pi^{-hs-c}{\widetilde{\bf{A}}}_{{\mathrm{rig}}}^{\dagger,r_{0}q^{-s}}{\\{\\{u_{K}\\}\\}}_{n}$
and thus, by lemma 7.4, this proves the proposition. ∎

######  .

Let $V$ be an $E$-representation of ${\mathcal{G}}_{K}$. Then the morphism

${\bf
D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)\rightarrow({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}$

induced by the inclusion
${{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u_{K}\\}\\}}_{n}\subset{{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}$,
is an isomorphism of $(\varphi_{q},\nabla_{u})$-modules on $E{\langle\langle
x_{K}\rangle\rangle}$.

###### Proof.

Let $(v_{1},\ldots,v_{r})$ and $(d_{1},\ldots,d_{v})$ be respectively an
$E$-basis of $V$ and an $E{\langle\langle x_{K}\rangle\rangle}$-basis of
$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}$.
There exists $A\in{\mathrm{M}}_{r\times v}$ such that $(d_{i})=A(v_{i})$. Let
$P\in{\mathrm{GL}}_{v}(E{\langle\langle x_{K}\rangle\rangle})$ be the matrix
of $\varphi_{q}$ in the basis $(d_{i})$. We then have $\varphi_{q}(A)=PA$ and
thus $A=\varphi_{q}^{-1}(P)\varphi_{q}^{-1}(A)$. By proposition 7.4, we have
$A\in M_{r\times v}({\bf B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}})$ and hence

$({{\widetilde{\bf{B}}}_{{\mathrm{rig}}}}^{\dagger}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}\subset({{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}{\\{\\{u_{K}\\}\\}}_{n}\otimes_{E}V)^{{\mathcal{G}}_{K_{n}}}={\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V).$

∎

######  .

The ring ${\bf B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}$ is
${\mathcal{G}}_{K}$-regular for $F$-analytic representations.

###### Proof.

This follows from the previous proposition and from corollary 7.3. ∎

###  7.5 $\varphi$-modules on $L{\langle\langle x_{K}\rangle\rangle}$

By proposition 7.3, to any $E$-representation $V$ of ${\mathcal{G}}_{K}$ we
can attach a module on $E{\langle\langle x_{K}\rangle\rangle}$, which is
endowed with a Frobenius $\varphi_{q}$ and an operator $\nabla_{u}$. Note that
$\varphi_{q}$ and $\nabla_{u}$ act on $E{\langle\langle x_{K}\rangle\rangle}$
by

$\varphi_{q}(x_{K})=\pi x_{K},\quad\nabla_{u}=x_{K}\frac{d}{dx_{K}}.$

As a matter of fact, $\varphi$-modules on $E{\langle\langle
x_{K}\rangle\rangle}$ were already studied in a different setting (yet
somewhat close) by Colmez in [Col14, 3.1] and the results proved by Colmez
show that $\varphi$-modules on $E{\langle\langle x_{K}\rangle\rangle}$ are not
as bad as one may think. Be careful that what we call $E{\langle\langle
x_{K}\rangle\rangle}$ corresponds in the notations of Colmez to
$E{\\{\\{x_{K}\\}\\}}$ (which to us means something different!). In this
section, we recall Colmez’s results on $\varphi$-modules on $E{\langle\langle
x_{K}\rangle\rangle}$.

######  .

A $(\varphi,\nabla_{u})$-module on $E{\langle\langle x_{K}\rangle\rangle}$ is
a finite free $E{\langle\langle x_{K}\rangle\rangle}$-module, endowed with
semilinear actions of $\varphi_{q}$ and $\nabla_{u}$ which commute one to
another and such that $\varphi_{q}$ is an isomorphism.

The ring $E{\langle\langle x_{K}\rangle\rangle}$ can be interpreted via
analytic functions, as it is the projective limit of the rings of analytic
functions on the disks $v_{p}(x)\geq-ne$ for $n\in{\bf N}$. Those rings are
principal Banach rings and therefore $E{\langle\langle x_{K}\rangle\rangle}$
is a Fréchet-Stein ring, which in particular implies that any closed submodule
of a free module of rank $d$ if free of rank $\leq d$ and that a submodule of
finite type of a free finite type module is closed and thus free. Moreover,
Newton polygons theory show that an element $f\in E{\langle\langle
x_{K}\rangle\rangle}$ does not vanish if and only if $f\in E^{\times}$, so
that $(E{\langle\langle x_{K}\rangle\rangle})^{\times}=E^{\times}$.

Note that we can actually define an $E$-linear action of ${\mathcal{G}}_{K}$
on $E{\langle\langle x_{K}\rangle\rangle}$ that factors through
$\Gamma_{K}={\mathrm{Gal}}(K_{\infty}/K)$, by putting $g(x)=\chi_{\pi}(g)\cdot
x_{K}$ but that this action is not induced by the one on ${\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}$. It is however somewhat induced by
the operator $\nabla_{u}$, as we now explain. Given a
$(\varphi_{q},\nabla_{u})$-module $D$ on $E{\langle\langle
x_{K}\rangle\rangle}$, there exists $n_{0}\geq 0$ such that for $n\geq n_{0}$,
the operator $T_{n}:=e^{\pi^{n}\nabla_{u}}$ converges on $E{\langle\langle
x_{K}\rangle\rangle}$ and therefore gives an action of $\Gamma_{n}$ by
$g(z)=e^{\log\chi_{\pi}(g)\nabla_{u}}$ which is $K$-analytic. We can thus
endow $D$ with an $E$-linear action of ${\mathcal{G}}_{K_{n}}$ that factors
through $\Gamma_{n}$. Conversely, let $D$ be a $\varphi_{q}$-module on
$E{\langle\langle x_{K}\rangle\rangle}$ endowed with a $K$-analytic action of
$\Gamma_{n}$ for some $n\geq 0$ that commutes with $\varphi$. Then the
operator $\frac{\log(g)}{\log\chi_{\pi}(g)}$ converges for $g$ close enough to
$1$ to a connexion over $\nabla_{u}:E{\langle\langle
x_{K}\rangle\rangle}{\rightarrow}E{\langle\langle x_{K}\rangle\rangle}$ which
gives it a structure of a $(\varphi_{q},\nabla_{u})$-module.

######  .

A finite type ideal of $E{\langle\langle x_{K}\rangle\rangle}$ which is stable
by either $\varphi_{q}$ or a finite index subgroup of $\Gamma_{K}$ is of the
form $(x_{K}^{k})$ with $k\in{\bf N}$.

###### Proof.

The proof in the cyclotomic case for $\varphi$ and $\Gamma$ is done in [Col14,
Lemm. 3.1] and the extension of the proof to $K$ and to finite index subgroups
of $\Gamma_{K}$ is straightforward. ∎

######  .

If $\alpha\in E$, then $\varphi_{q}-\alpha$ induces an isomorphism on
$E{\langle\langle x_{K}\rangle\rangle}$, unless $\alpha=\pi^{i}$ with
$i\in{\bf N}$, in which case the kernel of $\varphi_{q}-\alpha$ is
$Ex_{K}^{i}$ and $\varphi_{q}-\alpha$ induces an isomorphism on
$\\{\sum_{k\geq 0}a_{k}x_{K}^{k},a_{i}=0\\}$.

###### Proof.

See [Col14, Lemm. 3.2]. ∎

######  .

Let $M$ be a rank $d$ $\varphi_{q}$-module on $E{\langle\langle
x_{K}\rangle\rangle}$ and let $v\in M$ be such that there exists $\alpha\in
E^{\times}$ such that $\varphi_{q}(v)=\alpha v$. Then there exists $k\in{\bf
N}$ such that $t_{\pi}^{-k}v\in M$ and $M/E{\langle\langle
x_{K}\rangle\rangle}t_{\pi}^{-k}v$ is free of rank $d-1$ on $E{\langle\langle
x_{K}\rangle\rangle}$.

###### Proof.

See [Col14, Lemm. 3.4]. ∎

Let $M$ be a $\varphi_{q}$-module on $E{\langle\langle x_{K}\rangle\rangle}$
and let $\overline{M}=M/x_{K}M$. If $P\in E[X]$ is unitary of degree $d$ and
irreducible, then we let $M_{P}$ (resp. $\overline{M}_{P}$) denote the set of
elements $v\in M$ (resp. in $\overline{M}$) such that $P(\varphi_{q})^{n}\cdot
v=0$ for $n\gg 0$ and if $k\in{\bf N}$, we let $P[k]$ be the polynomial
$\pi^{kd}P(X/\pi^{k})$.

######  .

The natural map $M_{P}{\rightarrow}\overline{M}_{P}$ is surjective.

###### Proof.

See [Col14, Lemm. 3.5]. ∎

######  .

If $M$ is a $\varphi_{q}$-module of rank $d$ on $E{\langle\langle
x_{K}\rangle\rangle}$, there exists a basis $e_{1},\ldots,e_{d}$ of $M$ in
which the matrix of $\varphi$ is $A+N$, where $A\in{\mathrm{GL}}_{d}(L)$ is
semisimple and invertible, and $N$ is nilpotent and commutes with $A$.
Moreover, $N$ splits into $N=N_{0}+x_{K}N_{1}+\ldots$, where $N_{i}\in
M_{d}(L)$ sends the kernel $M_{P}$ of $P(A)$ into the one $M_{P[-i]}$ of
$P(\pi^{i}A)$ for all $P$ (and thus in particular the sum is finite).

###### Proof.

See [Col14, Thm. 3.6]. ∎

Given a $(\varphi_{q},\Gamma_{n})$-module $D$ on $E{\langle\langle
x_{K}\rangle\rangle}$, we say that an element $v$ of $D$ is proper for the
action of $\varphi_{q}$ and $\Gamma_{n}$ if there exists
$\delta\in\widehat{\mathfrak{T}}(E)$ such that $\varphi(v)=\delta(\pi)v$ and
$g(v)=\delta(\chi_{\pi}(g))v$ for all $g\in\Gamma_{n}$.

######  .

Given a $(\varphi_{q},\Gamma_{n})$-module $D$ of rank $1$ on $E{\langle\langle
x_{K}\rangle\rangle}$, $F$-analytic, with basis $e$, then there exists
$\delta\in\widehat{\mathfrak{J}}(E)$ such that $e$ is proper for $\delta$.

###### Proof.

This just follows from the fact that a rank $1$
$(\varphi_{q},\Gamma_{n})$-module on $E{\langle\langle x_{K}\rangle\rangle}$
has a unique basis $e$, up to multiplication by an element of
$(E{\langle\langle x_{K}\rangle\rangle})^{\times}=E^{\times}$. ∎

######  .

Let $M$ be a $(\varphi_{q},\nabla_{u})$-module of rank $d$ on
$E{\langle\langle x_{K}\rangle\rangle}$. Then, up to extending the scalars to
some finite extension $L$ of $E$, there exists a filtration

$M_{0}=0\subsetneq M_{1}\subsetneq\ldots\subsetneq M_{d}=M$

of $M$ by saturated sub-$(\varphi_{q},\nabla_{u})$-modules.

###### Proof.

By the results above, there exists $n\gg 0$ such that $M$ has a structure of a
$(\varphi_{q},\Gamma_{n})$-module inducing the
$(\varphi_{q},\nabla_{u})$-structure of $M$. We prove the result for
$(\varphi_{q},\Gamma_{n})$-modules by induction. If $d=1$ there is nothing to
prove. Assume now that $d\geq 2$ and that the result holds for $d-1$.

By theorem 7.5 and lemma 7.5, upto a finite extension $E^{\prime}$ of $E$,
there exists $e_{1}$ proper for the action of $\varphi$ and $\Gamma_{n}$ such
that $E{\langle\langle x_{K}\rangle\rangle}\cdot e_{1}$ is saturated in $M$.
By induction, $M/(E{\langle\langle x_{K}\rangle\rangle}\cdot e_{1})$ admits a
full $(\varphi_{q},\Gamma_{n})$-stable filtration
$({\mathcal{F}}_{i})_{i=1}^{d-1}$. We let $M_{i+1}$ be a lift of
${\mathcal{F}}_{i}$ containing $E{\langle\langle x_{K}\rangle\rangle}\cdot
e_{1}$ and we put $M_{1}=E{\langle\langle x_{K}\rangle\rangle}\cdot e_{1}$. We
then have that $(M_{i})_{i=1}^{d}$ is a full $(\varphi_{q},\Gamma_{n})$-stable
filtration of $M$. ∎

###  7.6 Refinements of $F$-analytic ${\bf
B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}$-admissible representations

We now define a notion of refinements for $F$-analytic trianguline
representations of ${\mathcal{G}}_{K}$. We let $V$ be a ${\bf
B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}$-admissible $E$-representation of
dimension $d$ of ${\mathcal{G}}_{K}$. By a refinement of $V$, we mean the data
of a full $\varphi_{q}$\- and $\nabla_{u}$-stable $E{\langle\langle
x_{K}\rangle\rangle}$-filtration
${\mathcal{F}}=({\mathcal{F}}_{i})_{i=0,\dots,d}$ of ${\bf
D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)$:

${\mathcal{F}}_{0}=0\subsetneq{\mathcal{F}}_{1}\subsetneq\cdots\subsetneq{\mathcal{F}}_{d}={\bf
D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V).$

Note that, as in the crystalline case studied in [BC09a], the theory of
refinements has a simple interpretation in terms of
$(\varphi_{q},\Gamma_{K})$-modules: let ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$ be the triangulable
$(\varphi_{q},\Gamma_{K})$-module over $E\otimes_{K}{\bf
B}_{{\mathrm{rig}},K}^{\dagger}$ attached to $V$ and let ${\mathcal{F}}$ be a
refinement of ${\bf D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)$. We can
construct from ${\mathcal{F}}$ a filtration
$({\mathrm{Fil}}_{i}(D))_{i=0,\cdots,d}$ of ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$ by setting

${\mathrm{Fil}}_{i}({\bf D}_{{\mathrm{rig}}}^{\dagger}(V)):=((E\otimes_{K}{\bf
B}_{{\mathrm{rig}},K}^{\dagger})\otimes_{E{\langle\langle
x_{K}\rangle\rangle}}{\mathcal{F}}_{i})\cap{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V),$

which is a finite type saturated $E\otimes_{K}{\bf
B}_{{\mathrm{rig}},K}^{\dagger}$-submodule of ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$.

######  .

The map defined above $({\mathcal{F}}_{i})\mapsto({\mathrm{Fil}}_{i}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)))$ induces a bijection between the set of
refinements of $V$ and the set of triangulations of ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$, whose inverse is
${\mathcal{F}}_{i}:=\bigcup_{n\geq 0}{\mathrm{Fil}}_{i}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u\\}\\}}_{n})^{\Gamma_{K_{n}}}$.

###### Proof.

This is exactly as in the crystalline case. ∎

######  .

Let $V$ be a ${\bf B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}$-admissible $p$-adic
representation $V$. Then there exists $n\geq 0$ such that
$V_{|{\mathcal{G}}_{K_{n}}}$ is trianguline and moreover, the triangulation of
${\bf D}_{{\mathrm{rig}}}^{\dagger}(V_{|{\mathcal{G}}_{K_{n}}})$ extends to
${\mathcal{G}}_{K}$.

###### Proof.

By proposition 7.5 and proposition 7.6, there exists a finite extension $E$ of
$K$ and $n\geq 0$ such that ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V\otimes_{K}E)$ is a triangulable
$(\varphi_{q},\Gamma_{n})$-module over $E\otimes_{K}{\bf
B}_{{\mathrm{rig}},K}^{\dagger}$.

Moreover, we see from lemma 7.5 that the characters appearing in the
triangulation extend to ${\mathcal{G}}_{K}$ and are $K$-analytic. ∎

######  .

Let $V$ be an $E$-representation of ${\mathcal{G}}_{K}$ such that the attached
$(\varphi_{q},\Gamma_{K})$-module ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$ is
triangulable, and let $\delta:(K^{\times})^{d}{\rightarrow}(E^{\times})^{d}$
be the parameter of a triangulation of ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$. Then in an adapted basis for the refinement
of ${\bf D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)$ corresponding to $\delta$
by proposition 7.6, the matrices of $\nabla_{u}$ and $\varphi_{q}$ are
respectively of the form:

$\begin{pmatrix}w(\delta_{1})&*&\cdots&*\\\ 0&w(\delta_{2})&\cdots&*\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&w(\delta_{d})\end{pmatrix}\quad\textrm{and
}\begin{pmatrix}\delta_{1}(\pi)&*&\cdots&*\\\ 0&\delta_{2}(\pi)&\cdots&*\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&\delta_{d}(\pi)\end{pmatrix}.$

###### Proof.

We prove it by induction on $d$. For $d=1$, by proposition 3.1 of [Col08b]
there exists a basis $e_{\delta}$ of ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$
in which $g(e_{\delta})=\delta(\chi_{\pi}(g))e_{\delta}$ and
$\varphi_{q}(e_{\delta})=\delta(\pi)e_{\delta}$, so that there exists $n\geq
0$ and $\alpha\in E^{\times}$ such that $e^{\alpha u_{K}}e_{\delta}\in{\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n}$ and is left invariant
by $\Gamma_{K_{n}}$. Therefore $e^{\alpha u_{K}}e_{\delta}$ is a basis of
${\bf D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ and this basis satisfies the
result of the lemma. To see that it is unique note that since
$(E{\langle\langle x_{K}\rangle\rangle})^{\times}=E^{\times}$, the matrices of
$\nabla_{u}$ and $\varphi_{q}$ in an other basis of ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ would be the same.

Assume now that $d\geq 2$ is such that the result holds for $d-1$ and let
$({\mathrm{Fil}}_{i}({\bf D}_{{\mathrm{rig}}}^{\dagger}(V)))_{i=0,\ldots,d}$
be the filtration of ${\bf D}_{{\mathrm{rig}}}^{\dagger}(V)$ corresponding to
the triangulation. Since our constructions are stable by saturated sub-
objects, we get by induction that in an adapted basis for the refinement of
${\bf D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}({\mathrm{Fil}}_{d-1}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)))$, the matrices of $\nabla_{u}$ and
$\varphi$ are respectively of the form:

$\begin{pmatrix}w(\delta_{1})&*&\cdots&*\\\ 0&w(\delta_{2})&\cdots&*\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&w(\delta_{d-1})\end{pmatrix}\quad\textrm{and
}\begin{pmatrix}\delta_{1}(\pi)&*&\cdots&*\\\ 0&\delta_{2}(\pi)&\cdots&*\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&\delta_{d-1}(\pi)\end{pmatrix}.$

Since our constructions are also stable by quotients by saturated sub-objects
and using the proof in the rank $1$ case, we know that the matrices of
$\nabla_{u}$ and $\varphi_{q}$ in a basis of ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)/{\mathrm{Fil}}_{d-1}({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V))\simeq E\otimes_{K}{\bf
B}_{{\mathrm{rig}},K}(\delta_{d})$ are respectively of the form
$(w(\delta_{d}))$ and $(\delta_{d}(\pi))$. Therefore, in an adapted basis for
the refinement of ${\bf D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)$
corresponding to $\delta$, the matrices of $\varphi_{q}$ and $\nabla_{u}$ are
as we wanted. ∎

In particular, as in the crystalline case, a refinement defines an ordering on
both the eigenvalues of $\varphi_{q}$ and on the set of Hodge-Tate weights of
$V$, and encodes the data of the Hodge-Tate weights of its parameter.

###  7.7 Crystabelian representations

Recall that to any $K$-analytic representation $V$ of ${\mathcal{G}}_{K}$, we
can attach a filtered $\varphi_{q}$-module over $K$ ${\bf
D}_{{\mathrm{crys}}}(V)$, which is a finite dimensional $K$-vector space
(whose dimension is $\leq\dim_{K}(V)$), endowed with a Frobenius map
$\varphi_{q}$ such that $\varphi_{q}^{*}D\simeq D$, and with a filtration by
$K$ subspaces, indexed by ${\bf Z}$, which is decreasing and separated.

Given the construction of our rings of periods, it is quite obvious that the
modules ${\bf D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ attached to
$K$-analytic $p$-adic representations of ${\mathcal{G}}_{K}$ contain its
crystalline periods:

######  .

Let $V$ be a $K$-analytic $p$-adic representation. Then

  1. 1.

${\bf D}_{{\mathrm{crys}}}^{+}(V)\subset{\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)^{\nabla_{u}=0}$;

  2. 2.

${\bf D}_{{\mathrm{crys}}}(V)\subset({\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)[1/x_{K}])^{\nabla_{u}=0}$.

###### Proof.

Item 1 just follows from the facts that $\nabla_{u}$ and the action of
${\mathcal{G}}_{K}$ commute, that ${\bf
D}_{{\mathrm{crys}}}^{+}(V)=({\widetilde{\bf{B}}}_{{\mathrm{rig}}}^{+}\otimes_{{{\bf
Q}_{p}}}V)^{{\mathcal{G}}_{K}}$ and that $({\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}})^{\nabla_{u}=0}={{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}$.

Item 2 follows from the fact that ${\bf
D}_{{\mathrm{crys}}}(V)=({\widetilde{\bf{B}}}_{{\mathrm{rig}}}^{+}[1/t]\otimes_{{{\bf
Q}_{p}}}V)^{{\mathcal{G}}_{K}}$ by lemma 3.8 of [Por20] and that $({\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}[1/x_{K}])^{\nabla_{u}=0}={{\widetilde{\bf{B}}}_{\mathrm{rig}}^{+}}[1/t_{\pi}]$.
∎

A crystabelian representation of ${\mathcal{G}}_{K}$ is a representation of
${\mathcal{G}}_{K}$ which becomes crystalline over an abelian extension of
$K$. In what follows, we will assume that $V$ is a crystabelian $K$-analytic
representation of ${\mathcal{G}}_{K}$. Since
$K^{\mathrm{ab}}=K^{\mathrm{unr}}\cdot K_{\infty}$, and by the same argument
as in the proof of proposition 7.2, it means that there exists $n\geq 0$ such
that $V_{|G_{K_{n}}}$ is crystalline. We still let ${\bf
D}_{{\mathrm{crys}}}(V)$ stand for the corresponding filtered
$\varphi_{q}$-module, even though the filtration is defined over $K_{n}$.

Let $k\geq 0$ be such that ${\bf D}_{{\mathrm{crys}}}(V(\chi_{\pi}^{-k}))={\bf
D}_{{\mathrm{crys}}}^{+}(V(\chi_{\pi}^{-k}))$. A quick computation shows that
${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V(\chi_{\pi}^{-k}))=e^{ku_{K}}{\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ and that ${\bf
D}_{{\mathrm{crys}}}(V(\chi_{\pi}^{-k}))=t_{\pi}^{k}{\bf
D}_{{\mathrm{crys}}}(V)$. In particular, this shows that

${\bf D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)=e^{-ku_{K}}{\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)\supset{\bf
D}_{{\mathrm{crys}}}^{+}(V(\chi_{\pi}^{-k}))=t_{\pi}^{k}{\bf
D}_{{\mathrm{crys}}}(V)$

so that $x_{K}^{k}{\bf D}_{{\mathrm{crys}}}(V)\subset{\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$. Since we have
$\varphi(x_{K})=\pi\cdot x_{K}$, this shows that ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ contains a copy of ${\bf
D}_{{\mathrm{crys}}}(V)$, but with the filtration and Frobenius shifted by
some power. Note that we can also recover the filtration on
$K_{\infty}\otimes_{K}{\bf D}_{{\mathrm{crys}}}^{+}(V)$ as it was done for
${\bf D}_{{\mathrm{dR}}}^{+}(V)$ in §4.

Assume for now that $V$ is a crystalline positive $p$-adic representation,
which means that ${\bf D}_{{\mathrm{crys}}}(V)={\bf
D}_{{\mathrm{crys}}}^{+}(V)$. Let
${\mathcal{F}}=({\mathcal{F}}_{i})_{i=0,\dots,d}$ be a full $\varphi$\- and
$\nabla_{u}$-stable $E{\langle\langle x_{K}\rangle\rangle}$-filtration of
${\bf D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)$. Since the filtration is
stable by $\nabla_{u}$, the filtration
${\mathcal{F}}^{\nabla_{u}=0}=({\mathcal{F}}_{i}^{\nabla_{u}=0})_{i=0,\dots,d}$
is also a full $\varphi$-stable filtration of ${\bf
D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)^{\nabla_{u}=0}={\bf
D}_{{\mathrm{crys}}}(V)$. In particular, refinements in our sense for
trianguline representations give rise to refinements of positive crystalline
representations. In the other direction, we can for example use proposition
7.1 in order to construct a filtration on ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$ starting from a $\varphi$-stable filtration
on ${\bf D}_{{\mathrm{crys}}}(V)$ and then use proposition 7.6 to recover the
filtration on ${\bf D}_{{\mathrm{tri}},K}^{{\mathrm{an}}}(V)$. By
compatibility of the constructions, we see that our definition of refinements
is an extension of the one of [BC09a, §2.4].

We now explain how to recover our module ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ from ${\bf
D}_{{\mathrm{crys}}}(V)$.

######  .

Let $V$ be a $K$-analytic crystabelian representation of ${\mathcal{G}}_{K}$.
Then

${\bf D}_{{\mathrm{crys}}}(V){\langle\langle
x_{K}\rangle\rangle}[1/x_{K}]={\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)[1/x_{K}].$

###### Proof.

By [KR09, Prop. 2.2.6], we have ${\bf D}_{{\mathrm{crys}}}(V)=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)[1/t_{\pi}])^{{\mathcal{G}}_{K_{n}}}$, and by
proposition 7.3, we have ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V){\\{\\{u_{K}\\}\\}}_{n})^{{\mathcal{G}}_{K_{n}}}$.
In particular, if we let $D=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)[1/t_{\pi}]{\\{\\{u_{K}\\}\\}}_{n})^{{\mathcal{G}}_{K_{n}}}$,
then $D$ contains both ${\bf D}_{{\mathrm{crys}}}(V)$ and ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$. Moreover, since
$x_{K}=t_{\pi}e^{-u_{K}}$ and since $e^{-u_{K}}$ is invertible in ${\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}$, we have that $D=({\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)[1/x_{K}]{\\{\\{u_{K}\\}\\}}_{n})^{{\mathcal{G}}_{K_{n}}}$.
Since $\frac{1}{x_{K}}$ is invariant under the action of
${\mathcal{G}}_{K_{n}}$, this implies that $D={\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)[1/x_{K}]$. But since ${\bf
D}_{{\mathrm{crys}}}(V)$ has the right dimension, we also know that $D=({\bf
D}_{{\mathrm{crys}}}(V)\otimes_{K}{\bf
B}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}[1/x_{K}])^{{\mathcal{G}}_{K_{n}}}={\bf
D}_{{\mathrm{crys}}}(V)\otimes_{K}K{\langle\langle
x_{K}\rangle\rangle}[1/x_{K}]$. ∎

It remains to see how one can recover ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)$ inside ${\bf
D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)[1/x_{K}]$. First note that we can
define a filtration on ${\bf D}_{{\mathrm{crys}}}(V){\langle\langle
x_{K}\rangle\rangle}[1/x_{K}]$ by

${\mathrm{Fil}}^{k}({\bf D}_{{\mathrm{crys}}}(V){\langle\langle
x_{K}\rangle\rangle}[1/x_{K}])=\bigcup_{i+j=k}x^{i}\left({\mathrm{Fil}}^{j}({\bf
D}_{{\mathrm{crys}}}(V)\right){\langle\langle x_{K}\rangle\rangle}.$

######  .

We have

${\bf D}_{{\mathrm{tri}},K}^{n,{\mathrm{an}}}(V)=\left\\{\begin{aligned}
z\in{\bf D}_{{\mathrm{crys}}}(V){\langle\langle
x_{K}\rangle\rangle}[\frac{1}{x_{K}}]\textrm{ such that
}\varphi_{q}^{-n}(z)\in{\mathrm{Fil}}^{0}({\bf
D}_{{\mathrm{crys}}}(V){\langle\langle
x_{K}\rangle\rangle}[\frac{1}{x_{K}}])\\\ \textrm{ for }n\gg
0\end{aligned}\right\\}.$

###### Proof.

This follows from the usual way one recovers ${\bf
D}_{{\mathrm{rig}}}^{\dagger}(V)$ from ${\bf D}_{{\mathrm{crys}}}(V)$ (see for
example [Ber08, §V.3]) and from the previous lemma. ∎

Unfortunately, our ring of periods ${\bf
B}_{{\mathrm{tri}},K}^{{\mathrm{an}}}$ is too small in general, because it
computes $({\bf D}_{{\mathrm{rig}},K}^{\dagger}(V))^{F-an}$ by proposition 3.
The fact that this is too small for all trianguline representations, even in
dimension $2$, follows from theorem 3.23 of [Col14] (even though the link
between our constructions and Colmez’s is not straightforward). We explain the
particular case when $F=K={{\bf Q}_{p}}$ but the general case should be
similar: basically, the ring ${\bf B}_{{\mathrm{tri}},{{\bf
Q}_{p}}}^{{\mathrm{an}}}$ contains the periods of all representations which
are crystalline, up to a twist by a character ${\bf
Q}_{p}^{\times}{\rightarrow}E^{\times}$, and all the direct sums of such
representations. We could adapt our construction to “add a log to our ring”,
which would cover the semistable periods, but we would still be missing the
“nongeometric” periods of trianguline representations, which appear in item
$(ii)$ of theorem 3.23 of [Col14].

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