On triangulable tensor products of B-pairs and trianguline representations

TL;DR

This paper proves that if the tensor product of two p-adic Galois representations is trianguline, then each representation is potentially trianguline, advancing understanding of the structure of p-adic representations.

Contribution

It establishes a new criterion linking the trianguline property of tensor products to the potential triangulinity of individual p-adic representations.

Findings

Tensor product of two representations being trianguline implies each is potentially trianguline.

Provides new insights into the structure of p-adic Galois representations.

Advances the theory of trianguline representations in p-adic Hodge theory.

Abstract

We show that if V and V' are two p-adic representations of Gal(Qp^alg/Qp) whose tensor product is trianguline, then V and V' are both potentially trianguline.

Equations

$$0\to E\to {\mathbf{B}}_{\mathrm{e},E}\to (E{\otimes }_{{\mathbf{Q}}_p}{\mathbf{B}}_{\mathrm{dR}})/(E{\otimes }_{{\mathbf{Q}}_p}{\mathbf{B}}_{\mathrm{dR}}^{+})\to 0.$$

$$W_{\mathrm{dR}}^{G_K}\to H^1(G_K,W)\to H^1(G_K,W_e)\oplus H^1(G_K,W_{\mathrm{dR}}^{+}).$$

$$W_{\mathrm{dR}}^{G_K}\to (W_{\mathrm{dR}}/W_{\mathrm{dR}}^{+}{)}^{G_K}\to H^1(G_K,W_{\mathrm{dR}}^{+})\to H^1(G_K,W_{\mathrm{dR}}),$$

Full text

1. 1 Reminders and complements
2. 2 The ring $\mathbf{B}_{\mathrm{e},E}$
3. 3 Triangulable representations
4. 4 Cohomology of $B$-pairs
5. 5 Proof of the main theorem

Introduction

The notion of a trianguline representation of $G_{\mathbf{Q}_{p}}=\operatorname{Gal}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})$ was introduced by Colmez [Col08] in the context of his work on the $p$-adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbf{Q}_{p})$. Examples of trianguline representations include the semi-stable representations of $G_{\mathbf{Q}_{p}}$ as well as the $p$-adic representations of $G_{\mathbf{Q}_{p}}$ attached to overconvergent cuspidal eigenforms of finite slope (theorem 6.3 of [Kis03] and proposition 4.3 of [Col08]). The category of all trianguline representations of $G_{\mathbf{Q}_{p}}$ is stable under extensions, tensor products, and duals. We refer the reader to the book [BC09] and the survey [Ber11] for a detailed discussion of trianguline representations. Let us at least mention the following analogue of the Fontaine-Mazur conjecture: if $V$ is an irreducible $2$-dimensional $p$-adic representation of $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ that is unramified at $\ell$ for almost all $\ell\neq p$, and whose restriction to a decomposition group at $p$ is trianguline, then $V$ is a twist of the Galois representation attached to an overconvergent cuspidal eigenform of finite slope. This conjecture is a theorem of Emerton (§1.2.2 of [Eme11]) under additional technical hypothesis on $V$. The trianguline property is in general a condition at $p$ reflecting (conjecturally at least) the fact that a $p$-adic representation comes from a $p$-adic automorphic form. This theme is pursued, for example, in [Han17], [Ber17] and [Con21].

If $K$ is a finite extension of $\mathbf{Q}_{p}$, we also have the notion of a trianguline representation of $G_{K}=\operatorname{Gal}(\overline{\mathbf{Q}}_{p}/K)$. We say that a representation $V$ of $G_{K}$ is potentially trianguline if there exists a finite extension $L/K$ such that the restriction of $V$ to $G_{L}$ is trianguline. The goal of this article is to prove the following theorem.

Theorem A**.**

If $V$ and $V^{\prime}$ are two non-zero $p$-adic representations of $G_{\mathbf{Q}_{p}}$ whose tensor product is trianguline, then $V$ and $V^{\prime}$ are both potentially trianguline.

We now give more details about the contents of this article. The definition of “trianguline” can be given either in terms of $(\varphi,\Gamma)$-modules over the Robba ring, or in terms of $B$-pairs. In this article, we use the theory of $B$-pairs, which was introduced in [Ber08]. We remark in passing that $B$-pairs are the same as $G_{K}$-equivariant bundles on the Fargues-Fontaine curve [FF18]. Let $K$ be a finite extension of $\mathbf{Q}_{p}$. Let $\mathbf{B}_{\mathrm{dR}}^{+}$, $\mathbf{B}_{\mathrm{dR}}$ and $\mathbf{B}_{\mathrm{e}}=(\mathbf{B}_{\mathrm{cris}})^{\varphi=1}$ be some of Fontaine’s rings of $p$-adic periods [Fon94]. A $B$-pair is a pair $W=(W_{e},W_{\mathrm{dR}}^{+})$ where $W_{e}$ is a free $\mathbf{B}_{\mathrm{e}}$-module of finite rank endowed with a continuous semi-linear action of $G_{K}$, and $W_{\mathrm{dR}}^{+}$ is a $G_{K}$-stable $\mathbf{B}_{\mathrm{dR}}^{+}$-lattice in $W_{\mathrm{dR}}=\mathbf{B}_{\mathrm{dR}}\otimes_{\mathbf{B}_{\mathrm{e}}}W_{e}$. If $V$ is a $p$-adic representation of $G_{K}$, then $W(V)=(\mathbf{B}_{\mathrm{e}}\otimes_{\mathbf{Q}_{p}}V,\mathbf{B}_{\mathrm{dR}}^{+}\otimes_{\mathbf{Q}_{p}}V)$ is a $B$-pair. If $E$ is a finite extension of $\mathbf{Q}_{p}$, the definition of $B$-pairs can be extended to $E$-linear objects, and we get objects called $\mathbf{B}^{\otimes E}_{\mid K}$-pairs in [BC10] or $E$-$B$-pairs of $G_{K}$ in [Nak09]. They are pairs $W=(W_{e},W_{\mathrm{dR}}^{+})$ where $W_{e}$ is a free $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$-module of finite rank endowed with a continuous semi-linear action of $G_{K}$, and $W_{\mathrm{dR}}^{+}$ is a $G_{K}$-stable $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$-lattice in $W_{\mathrm{dR}}=(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}})\otimes_{E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}}W_{e}$. Note that the action of $G_{K}$ is $E$-linear.

We say (definition 1.15 of [Nak09]) that a $\mathbf{B}^{\otimes E}_{\mid K}$-pair $W$ is split triangulable if $W$ is a successive extension of objects of rank $1$, triangulable if there exists a finite extension $F/E$ such that the $\mathbf{B}^{\otimes F}_{\mid K}$-pair $F\otimes_{E}W$ is split triangulable, and potentially triangulable if there exists a finite extension $L/K$ such that the $\mathbf{B}^{\otimes E}_{\mid L}$-pair $W|_{G_{L}}$ is triangulable. If $V$ is a $p$-adic representation of $G_{K}$, we say that $V$ is trianguline if $W(V)$ is triangulable.

Let $\Delta$ be a set of rank $1$ semi-linear $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$-representations of $G_{K}$. We say that a $\mathbf{B}^{\otimes E}_{\mid K}$-pair is split $\Delta$-triangulable if it is split triangulable, and the rank $1$ $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$-representations of $G_{K}$ that come from the triangulation are all in $\Delta$. Let $\Delta(\mathbf{Q}_{p})$ be the set of rank $1$ $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$-representations of $G_{K}$ that extend to $G_{\mathbf{Q}_{p}}$. Theorem A then results from the following more general result (theorem 5.4), applied to $K=\mathbf{Q}_{p}$.

Theorem B**.**

If $X$ and $Y$ are two non-zero $\mathbf{B}^{\otimes E}_{\mid K}$-pairs whose tensor product is $\Delta(\mathbf{Q}_{p})$-triangulable, then $X$ and $Y$ are both potentially triangulable.

The proof of theorem B relies on the study of $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$-representations of $G_{K}$ as well as on the study of the slopes, weights and cohomology of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs. The ring $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$ has many non-trivial units, which makes the study of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs more difficult than when $E=\mathbf{Q}_{p}$. Note finally that some of the results of this article already appear in [DM13].

1. Reminders and complements

If $K$ is a finite extension of $\mathbf{Q}_{p}$, let $G_{K}=\operatorname{Gal}(\overline{\mathbf{Q}}_{p}/K)$. Let $E$ be a finite Galois extension of $\mathbf{Q}_{p}$ such that $K\subset E$, and let $\Sigma=\operatorname{Gal}(E/\mathbf{Q}_{p})$. Let $E_{0}$ be the maximal unramified extension of $\mathbf{Q}_{p}$ inside $E$. Let $\mathbf{B}_{\mathrm{dR}}^{+}$, $\mathbf{B}_{\mathrm{dR}}$, $\mathbf{B}_{\mathrm{cris}}^{+}$ and $\mathbf{B}_{\mathrm{cris}}$ be Fontaine’s rings of $p$-adic periods (see for instance [Fon94]). They are all equipped with an action of $G_{\mathbf{Q}_{p}}$, and $\mathbf{B}_{\mathrm{cris}}^{+}$ and $\mathbf{B}_{\mathrm{cris}}$ have in addition a Frobenius map $\varphi$. Let $\mathbf{B}_{\mathrm{e}}=(\mathbf{B}_{\mathrm{cris}})^{\varphi=1}$ and $\mathbf{B}_{\mathrm{e},E}=E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$. The group $G_{\mathbf{Q}_{p}}$ acts $E$-linearly on $\mathbf{B}_{\mathrm{e},E}$.

Proposition 1.1**.**

The ring $\mathbf{B}_{\mathrm{e},E}$ is a principal ideal domain.

Proof.

The ring $\mathbf{B}_{\mathrm{e},E}$ is a Bézout domain; for $E=\mathbf{Q}_{p}$ this is shown in proposition 1.1.9 of [Ber08], and the same argument is used to show the general case in lemma 1.6 of [Nak09]. By theorem 6.5.2 of [FF18], the ring $\mathbf{B}_{\mathrm{e}}$ is a principal ideal domain, and therefore $\mathbf{B}_{\mathrm{e},E}$ is a principal ideal domain as well, since it is a quotient of the polynomial ring $\mathbf{B}_{\mathrm{e}}[X]$, and thus Noetherian. ∎

Recall that a $\mathbf{B}^{\otimes E}_{\mid K}$-pair is a pair $W=(W_{e},W_{\mathrm{dR}}^{+})$ where $W_{e}$ is a free $\mathbf{B}_{\mathrm{e},E}$-module of finite rank endowed with a continuous semi-linear action of $G_{K}$, and $W_{\mathrm{dR}}^{+}$ is a $G_{K}$-stable $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$-lattice in $W_{\mathrm{dR}}=(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}})\otimes_{\mathbf{B}_{\mathrm{e},E}}W_{e}$.

Proposition 1.2**.**

If $W_{e}$ is a $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$, then $(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}})\otimes_{\mathbf{B}_{\mathrm{e},E}}W_{e}$ admits an $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$-lattice stable under $G_{K}$.

Proof.

See §3.5 of [Fon04]. The same argument gives an $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$-lattice instead of a $\mathbf{B}_{\mathrm{dR}}^{+}$-lattice if one starts from an $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}$-representation. ∎

Recall that Nakamura has classified the $\mathbf{B}^{\otimes E}_{\mid K}$-pairs of rank $1$, under the assumption that $E$ contains the Galois closure of $K$. Given a character $\delta:K^{\times}\to E^{\times}$, he constructs in §1.4 of [Nak09] a rank $1$ $\mathbf{B}^{\otimes E}_{\mid K}$-pair $W(\delta)$, that we denote by $\mathbf{B}(\delta)$, and proves that every rank $1$ $\mathbf{B}^{\otimes E}_{\mid K}$-pair is of this form for a unique $\delta$. We have $\mathbf{B}(\delta_{1})\otimes\mathbf{B}(\delta_{2})=\mathbf{B}(\delta_{1}\delta_{2})$ (§1.4 of [Nak09]). We denote by $\mathbf{B}(\delta)_{e}$ the $\mathbf{B}_{\mathrm{e},E}$-component of $\mathbf{B}(\delta)$.

Recall (see for instance §2 of [BC10] or §1.3 of [Nak09]) that $\mathbf{B}^{\otimes E}_{\mid K}$-pairs have slopes. This comes from the equivalence of categories between $\mathbf{B}^{\otimes E}_{\mid K}$-pairs and $(\varphi,\Gamma)$-modules over the Robba ring, and Kedlaya’s constructions and results for $\varphi$-modules over the Robba ring (see [Ked04]). In particular, one can define the notion of isoclinic (pure of a certain slope) $\mathbf{B}^{\otimes E}_{\mid K}$-pairs. For example, if $V$ is an $E$-linear representation of $G_{K}$, then $W(V)=(\mathbf{B}_{\mathrm{e},E}\otimes_{E}V,(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+})\otimes_{E}V)$ is pure of slope [math], and every $\mathbf{B}^{\otimes E}_{\mid K}$-pair that is pure of slope [math] is of this form (proposition 2.2 of [BC10]).

We have the following slope filtration theorem (see theorem 2.1 of [BC10]).

Theorem 1.3**.**

If $W$ is a $\mathbf{B}^{\otimes E}_{\mid K}$-pair, there is a canonical filtration $\{0\}=W_{0}\subset W_{1}\subset\cdots\subset W_{\ell}=W$ by sub $\mathbf{B}^{\otimes E}_{\mid K}$-pairs such that

1. (1)

for every $1\leqslant i\leqslant\ell$, the quotient $W_{i}/W_{i-1}$ is isoclinic; 2. (2)

if $s_{i}$ is the slope of $W_{i}/W_{i-1}$, then $s_{1}<s_{2}<\cdots<s_{\ell}$.

The following proposition gathers the results that we need concerning slopes of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs. Recall that $\operatorname{Hom}(X,Y)=(\operatorname{Hom}_{E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}}(X_{e},Y_{e}),\operatorname{Hom}_{E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}}(X_{\mathrm{dR}}^{+},Y_{\mathrm{dR}}^{+}))$.

Proposition 1.4**.**

If $X$ is pure of slope $s$ and $Y$ is pure of slope $t$, then

1. (1)

$\operatorname{Hom}(X,Y)$* is pure of slope $t-s$ and $X\otimes Y$ is pure of slope $s+t$;* 2. (2)

if $X$ and $Y$ have the same rank and $X\subset Y$ and $s=t$, then $X=Y$; 3. (3)

if $Y$ is a direct summand of $X$, then $s=t$.

Proof.

For (1), see theorem 6.10 and proposition 5.13 of [Ked04]. For (2), we can take determinants and assume that $X$ and $Y$ are of rank $1$. The claim is then proposition 2.3 of [Ber08]. Item (3) follows from the fact that if $X=Y\oplus Z$, then the set of slopes of $X$ is the union of those of $Y$ and $Z$ (proposition 5.13 of [Ked04]). ∎

2. The ring $\mathbf{B}_{\mathrm{e},E}$

Recall that $\mathbf{B}_{\mathrm{e},E}=E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$. In this section, we determine the units of $\mathbf{B}_{\mathrm{e},E}$ and study the rank $1$ $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{E}$. Let $q=p^{h}$ be the cardinality of the residue field of $\mathcal{O}_{E}$, so that $E_{0}=\mathbf{Q}_{p^{h}}$. Let $\varphi_{E}:E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}\to E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}$ be the map $\operatorname{Id}\otimes\varphi^{h}$.

Proposition 2.1**.**

We have an exact sequence

[TABLE]

Proof.

This follows from tensoring by $E$ the usual fundamental exact sequence $0\to\mathbf{Q}_{p}\to\mathbf{B}_{\mathrm{e}}\to\mathbf{B}_{\mathrm{dR}}/\mathbf{B}_{\mathrm{dR}}^{+}\to 0$ (proposition 1.17 of [BK90]). ∎

Proposition 2.2**.**

The natural map $\mathbf{B}_{\mathrm{e},E}\to(E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}})^{\varphi_{E}=1}$ is an isomorphism.

Proof.

Since $\varphi_{E}$ is $E$-linear, we have $(E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}})^{\varphi_{E}=1}=E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{\varphi^{h}=1}$ and it is therefore enough to prove that $\mathbf{B}_{\mathrm{cris}}^{\varphi^{h}=1}=\mathbf{Q}_{p^{h}}\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{cris}}^{\varphi=1}$. The group $\operatorname{Gal}(\mathbf{Q}_{p^{h}}/\mathbf{Q}_{p})$ acts $\mathbf{Q}_{p^{h}}$-semi-linearly on $\mathbf{B}_{\mathrm{cris}}^{\varphi^{h}=1}$ via $\varphi$, and the claim follows from Galois descent (Speiser’s lemma). ∎

Remark 2.3**.**

The isomorphism of proposition 2.2 is $G_{E}$-equivariant.

In addition, if $g\in G_{\mathbf{Q}_{p}}$ acts by $\operatorname{Id}\otimes g$ on $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$, then it acts by $\operatorname{Id}\otimes g\varphi^{-n(g)}$ on $(E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}})^{\varphi_{E}=1}$ (where $n(g)$ is defined below).

Let $\pi$ be a uniformizer of $\mathcal{O}_{E}$, and let $\chi_{\pi}$ denote the Lubin-Tate character $\chi_{\pi}:G_{E}\to\mathcal{O}_{E}^{\times}$ attached to $\pi$. For each $\tau\in\Sigma=\operatorname{Gal}(E/\mathbf{Q}_{p})$, let $n(\tau)$ be the element of $\{0,\ldots,h-1\}$ such that $\tau=\varphi^{n(\tau)}$ on $E_{0}$. Let $t_{\tau}\in E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}$ denote the element constructed in §5 of [Ber16], where (in the notation of [Ber16]) we take $F=E$. We have $t_{\tau}=(\tau\otimes\varphi^{n(\tau)})(t_{\operatorname{Id}})$. The element $t_{\operatorname{Id}}$ is also denoted by $t_{\pi}$ in [Ber16], and it is the same as the element $t_{E}$ constructed in §9 of [Col02]. The usual $t$ of $p$-adic Hodge theory is $t=t_{\mathbf{Q}_{p}}$ for $\pi=p$.

For each $\sigma\in\Sigma$, we have a map $E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}\to\mathbf{B}_{\mathrm{dR}}^{+}$ given by $x\mapsto(\sigma\otimes\varphi^{n(\sigma)})(x)$, followed by the natural injection of $E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}$ in $\mathbf{B}_{\mathrm{dR}}^{+}$ (theorem 4.2.4 of [Fon94]). Finally, note that $E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}}=E\otimes_{E_{0}}\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}}$ is contained in $E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}$.

Proposition 2.4**.**

Let the notation be as above.

1. (1)

We have $\varphi_{E}(t_{\tau})=\tau(\pi)\cdot t_{\tau}$ and $g(t_{\tau})=\tau(\chi_{\pi}(g))\cdot t_{\tau}$ if $g\in G_{E}$; 2. (2)

the $t$-adic valuation of the $\sigma$-component of the image of $t_{\tau}$ via the map $E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}\to E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}=\prod_{\sigma\in\Sigma}\mathbf{B}_{\mathrm{dR}}$ given by $x\mapsto\{(\sigma\otimes\varphi^{n(\sigma)})(x)\}_{\sigma\in\Sigma}$ is $1$ if $\sigma=\tau^{-1}$ and [math] otherwise; 3. (3)

there exists $u\in(E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})^{\times}$ such that $\prod_{\tau\in\Sigma}t_{\tau}=u\cdot t$ in $E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}$.

Proof.

Since $t_{\tau}=(\tau\otimes\varphi^{n(\tau)})(t_{\operatorname{Id}})$, it is enough to check (1) for $\tau=\operatorname{Id}$. The corresponding statement is at the end of §3 of [Ber16] (page 3578). Likewise, (2) follows from the case $\tau=\operatorname{Id}$. That case now follows from (1) and the fact that the Hodge-Tate weight of $\chi_{\pi}$ is $1$ at $\sigma=\operatorname{Id}$ and [math] at $\sigma\neq\operatorname{Id}$. Finally, we have $\operatorname{N}_{E/\mathbf{Q}_{p}}(\chi_{\pi})=\chi_{\mathrm{cyc}}\eta$ where $\eta:G_{E}\to\mathbf{Q}_{p}^{\times}$ is unramified, and by (1), this implies (3). ∎

Note that $t_{\tau}^{-1}\in E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}$ since $t_{\tau}$ divides $t$ in $\mathbf{B}_{\mathrm{cris}}^{+}$ by (3) of proposition 2.4.

Proposition 2.5**.**

If $\mathbf{n}=\{n_{\tau}\}_{\tau\in\Sigma}$ is a tuple of integers whose sum is [math], then there exists $u_{\mathbf{n}}\in(E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})^{\times}$ such that $u=\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}}u_{\mathbf{n}}$ belongs to $\mathbf{B}_{\mathrm{e},E}$. The element $u$ is then a unit of $\mathbf{B}_{\mathrm{e},E}$ and every unit of $\mathbf{B}_{\mathrm{e},E}$ is of this form up to multiplication by $E^{\times}$.

Proof.

Let $w=\varphi_{E}(\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}})/\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}}=\prod_{\tau\in\Sigma}\tau(\pi)^{n_{\tau}}$ by (1) of proposition 2.4. Since $\sum_{\tau\in\Sigma}n_{\tau}=0$, we have $w\in\mathcal{O}_{E}^{\times}$. There exists $u_{\mathbf{n}}\in(E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})^{\times}$ such that $\varphi_{E}(u_{\mathbf{n}})/u_{\mathbf{n}}=w^{-1}$, and then $u=\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}}u_{\mathbf{n}}$ belongs to $\mathbf{B}_{\mathrm{e},E}$. The inverse of $u$ is $\prod_{\tau\in\Sigma}t_{\tau}^{-n_{\tau}}u_{\mathbf{n}}^{-1}$ which also belongs to $\mathbf{B}_{\mathrm{e},E}$, so that $u\in\mathbf{B}_{\mathrm{e},E}^{\times}$.

We now show that every $u\in\mathbf{B}_{\mathrm{e},E}^{\times}$ is of this form. Let $n_{\tau}$ be the $t$-adic valuation in $\mathbf{B}_{\mathrm{dR}}$ of the $\tau^{-1}$-component $u_{\tau^{-1}}=(\tau^{-1}\otimes\operatorname{Id})(u)$ of the image of $u\in E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{e}}$ in $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}=\prod_{\sigma\in\Sigma}\mathbf{B}_{\mathrm{dR}}$. Note that $u_{\sigma}\in\mathbf{B}_{\mathrm{e},E}^{\times}$ for all $\sigma\in\Sigma$ and that $\prod_{\sigma\in\Sigma}u_{\sigma}\in(\mathbf{B}_{\mathrm{e},E}^{\times})^{\Sigma}=\mathbf{B}_{\mathrm{e}}^{\times}$. We have $\mathbf{B}_{\mathrm{e}}^{\times}=\mathbf{Q}_{p}^{\times}$ by lemma 1.1.8 of [Ber08], so that $\sum_{\tau\in\Sigma}n_{\tau}=0$. By (2) of proposition 2.4, the element $u\cdot\prod_{\tau\in\Sigma}t_{\tau}^{-n_{\tau}}u_{\mathbf{n}}^{-1}$ belongs to $(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+})\cap\mathbf{B}_{\mathrm{e},E}^{\times}$, and $(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+})\cap\mathbf{B}_{\mathrm{e},E}^{\times}=E^{\times}$ by proposition 2.1. ∎

Recall that an $E$-linear representation is crystalline or de Rham if the underlying $\mathbf{Q}_{p}$-linear representation is crystalline or de Rham. We say that a character $\delta:G_{E}\to E^{\times}$ is $\mathbf{B}_{\mathrm{e},E}$-admissible if there exists $y\in\mathbf{B}_{\mathrm{e},E}\setminus\{0\}$ such that $\delta(g)=g(y)/y$. Such a character is then crystalline, hence also de Rham.

Proposition 2.6**.**

If $y\in\mathbf{B}_{\mathrm{e},E}\setminus\{0\}$ is such that $y\cdot\mathbf{B}_{\mathrm{e},E}$ is stable under $G_{E}$, then $y\in\mathbf{B}_{\mathrm{e},E}^{\times}$ and there exists $n_{\tau}\in\mathbf{Z}$ with $\sum_{\tau\in\Sigma}n_{\tau}=0$ and $y_{0}\in(E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})^{\times}$ such that $y=\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}}y_{0}$.

Proof.

If $y\cdot\mathbf{B}_{\mathrm{e},E}$ is stable under $G_{E}$, then $g(y)/y\in\mathbf{B}_{\mathrm{e},E}$ for all $g\in G_{E}$. Note that if $z\in\mathbf{B}_{\mathrm{dR}}^{\times}$, then $g(z)/z\in\mathbf{B}_{\mathrm{dR}}^{+}$. This implies that $g(y)/y\in\mathbf{B}_{\mathrm{e},E}\cap(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+})$. By proposition 2.1, $g(y)/y\in E^{\times}$. The map $\delta:G_{E}\to E^{\times}$ given by $\delta(g)=g(y)/y$ is a crystalline character of $G_{E}$, and hence of the form $\prod_{\tau\in\Sigma}\tau(\chi_{\pi})^{n_{\tau}}\eta_{0}$ where $n_{\tau}\in\mathbf{Z}$ and $\eta_{0}:G_{E}\to E^{\times}$ is unramified. This implies that there exists $y_{0}\in(E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})^{\times}$ such that $y=\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}}y_{0}$. If $y\in\mathbf{B}_{\mathrm{e},E}$, then $\varphi_{E}(y)=y$ so that $\sum_{\tau\in\Sigma}n_{\tau}=0$ by (1) of proposition 2.4, and hence $y\in\mathbf{B}_{\mathrm{e},E}^{\times}$. ∎

Corollary 2.7**.**

If $\delta:G_{E}\to E^{\times}$ is a $\mathbf{B}_{\mathrm{e},E}$-admissible character, then $\delta$ is de Rham and the sum of its weights at all $\tau\in\Sigma$ is [math]. Conversely, any character $\delta:G_{E}\to E^{\times}$ that is de Rham with the sum of its weights at all $\tau\in\Sigma$ equal to [math] is the product of a $\mathbf{B}_{\mathrm{e},E}$-admissible character by a potentially unramified character.

Proof.

The first assertion follows immediately from proposition 2.6. We now prove the second assertion. If $\delta:G_{E}\to E^{\times}$ is de Rham, it is of the form $\prod_{\tau\in\Sigma}\tau(\chi_{\pi})^{n_{\tau}}\eta_{0}$ where $n_{\tau}\in\mathbf{Z}$ and $\eta_{0}:G_{E}\to E^{\times}$ is potentially unramified. Let $\mathbf{n}=\{n_{\tau}\}_{\tau\in\Sigma}$ and $u$ be the corresponding unit (proposition 2.5). If $g\in G_{E}$, then $g(u)/u=\prod_{\tau\in\Sigma}\tau(\chi_{\pi}(g))^{n_{\tau}}\eta_{u}(g)$ where $\eta_{u}:G_{E}\to E^{\times}$ is unramified. The second assertion then follows from this. ∎

A $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$ is a free $\mathbf{B}_{\mathrm{e},E}$-module of finite rank with a semi-linear and continuous action of $G_{K}$ (recall that $G_{K}$ acts linearly on $E$). If $\delta\in H^{1}(G_{K},\mathbf{B}_{\mathrm{e},E}^{\times})$ (for example if $\delta:G_{K}\to E^{\times}$ is a character), we denote by $\mathbf{B}_{\mathrm{e},E}(\delta)$ the resulting rank $1$ $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$.

Proposition 2.8**.**

If $W_{e}$ is a $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$, and if $X_{e}$ is a sub $\mathbf{B}_{\mathrm{e},E}$-module of $W_{e}$ stable under $G_{K}$, then $X_{e}$ is a free $\mathbf{B}_{\mathrm{e},E}$-module, and it is saturated in $W_{e}$.

Proof.

See lemma 1.10 of [Nak09]. ∎

Proposition 2.9**.**

If $W$ is a rank $1$ $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{E}$, then there exists $\delta:G_{E}\to E^{\times}$ such that $W=\mathbf{B}_{\mathrm{e},E}(\delta)$.

Proof.

If we choose a basis $w$ of $W$, then $g(w)=\delta(g)w$ with $\delta(g)\in\mathbf{B}_{\mathrm{e},E}^{\times}$, so that $\delta(g)$ is of the form $\prod_{\tau\in\Sigma}t_{\tau}^{n_{\tau}(g)}u_{\mathbf{n}(g)}$ by proposition 2.5. Since $\delta(gh)=\delta(g)g(\delta(h))$, (1) of proposition 2.4 implies that the maps $n_{\tau}:G_{E}\to\mathbf{Z}$ are continuous homomorphisms. They are therefore trivial, and this implies that $\delta(g)\in E^{\times}$. ∎

Remark 2.10**.**

The character $\delta$ in proposition 2.9 is not unique, since it can be multiplied by any $\mathbf{B}_{\mathrm{e},E}$-admissible character of $G_{E}$.

Remark 2.11**.**

If $K\neq E$, it is not necessarily true that every rank $1$ $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$ is of the form $\mathbf{B}_{\mathrm{e},E}(\delta)$ for a character $\delta:G_{K}\to E^{\times}$.

Proof.

Take $E=\mathbf{Q}_{p}(\sqrt{p})$ and $K=\mathbf{Q}_{p}$ and $W=(E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{cris}})^{\varphi=\pi}=t_{\operatorname{Id}}\cdot\mathbf{B}_{\mathrm{e},E}$. The $E$-linear action of $G_{\mathbf{Q}_{p}}$ on $W$ is given by the map $\delta:g\mapsto g(t_{\operatorname{Id}})/t_{\operatorname{Id}}$. If $g\in G_{E}$, then $\delta(g)=\chi_{\pi}(g)$. If $u=t_{\operatorname{Id}}^{n}t_{\tau}^{-n}u_{n,-n}\in\mathbf{B}_{\mathrm{e},E}^{\times}$ as in proposition 2.5, and $g\notin G_{E}$, then $g(ut_{\operatorname{Id}})/ut_{\operatorname{Id}}=t_{\operatorname{Id}}^{-2n-1}t_{\tau}^{2n+1}v$ with $v\in(E\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})^{\times}$. Therefore, there is no character $\eta:G_{\mathbf{Q}_{p}}\to E^{\times}$ such that $W=\mathbf{B}_{\mathrm{e},E}(\eta)$.

Note that $W$ is the $\mathbf{B}_{\mathrm{e},E}$-component of the $\mathbf{B}^{\otimes E}_{\mid K}$-pair $W_{0}^{-1}$ of §1.4 of [Nak09]. ∎

Remark 2.12**.**

The results of this section provide a new proof of proposition 1.1.

Proof.

By theorem 6.5.2 of [FF18], the ring $(E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}[1/t_{\operatorname{Id}}])^{\varphi_{E}=1}$ is a PID. Since we have shown $\mathbf{B}_{\mathrm{e},E}$ is a localization of $(E\otimes_{E_{0}}\mathbf{B}_{\mathrm{cris}}^{+}[1/t_{\operatorname{Id}}])^{\varphi_{E}=1}$, it is itself a PID. ∎

Proposition 2.13**.**

We have $\operatorname{Frac}(\mathbf{B}_{\mathrm{e},E})^{G_{K}}=E$.

Proof.

Take $x/y\in\operatorname{Frac}(\mathbf{B}_{\mathrm{e},E})^{G_{K}}$ with $x$, $y\in\mathbf{B}_{\mathrm{e},E}$ coprime. If $g\in G_{K}$, then $g(x)y=xg(y)$ so that $x$ divides $g(x)$ and $y$ divides $g(y)$ in $\mathbf{B}_{\mathrm{e},E}$ (recall that $\mathbf{B}_{\mathrm{e},E}$ is a PID). By proposition 2.6, $x$ and $y$ belong to $\mathbf{B}_{\mathrm{e},E}^{\times}$. This implies that $x/y\in\mathbf{B}_{\mathrm{e},E}^{G_{K}}=E$. ∎

Corollary 2.14**.**

If $W_{e}$ is a $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$, then $\dim_{E}W_{e}^{G_{K}}\leqslant\operatorname{rk}W_{e}$.

Proof.

By a standard argument, proposition 2.13 implies that the map $\mathbf{B}_{\mathrm{e},E}\otimes_{E}W_{e}^{G_{K}}\to W_{e}$ is injective. This implies the corollary. ∎

3. Triangulable representations

In this section, we study triangulable $\mathbf{B}^{\otimes E}_{\mid K}$-pairs and $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$. We say that a $\mathbf{B}^{\otimes E}_{\mid K}$-pair is irreducible if it has no non-trivial saturated sub $\mathbf{B}^{\otimes E}_{\mid K}$-pair (see §2.1 of [Ber08]).

Proposition 3.1**.**

If $W=(W_{e},W_{\mathrm{dR}}^{+})$ is an irreducible $\mathbf{B}^{\otimes E}_{\mid K}$-pair, then $W_{e}$ is an irreducible $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$.

Proof.

Let $X_{e}$ be a sub-object of $W_{e}$. By proposition 2.8, it is a saturated and free submodule of $W_{e}$. The space $X_{\mathrm{dR}}^{+}=X_{\mathrm{dR}}\cap W_{\mathrm{dR}}^{+}$ is an $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$ lattice of $X_{\mathrm{dR}}$ stable under $G_{K}$. Hence $X=(X_{e},X_{\mathrm{dR}}^{+})$ is a saturated sub $\mathbf{B}^{\otimes E}_{\mid K}$-pair of $W$. ∎

Corollary 3.2**.**

If $W$ is a $\mathbf{B}^{\otimes E}_{\mid K}$-pair, then $W$ is split triangulable as a $\mathbf{B}^{\otimes E}_{\mid K}$-pair if and only if $W_{e}$ is split triangulable as a $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$.

Proof.

It is clear that if $W$ is split triangulable, then so is $W_{e}$. Conversely, the proof of proposition 3.1 shows how to construct a triangulation of $W$ from a triangulation of $W_{e}$. ∎

Let $\Delta$ be a set of rank $1$ semi-linear $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$. Recall that a $\mathbf{B}^{\otimes E}_{\mid K}$-pair is split $\Delta$-triangulable if it is split triangulable, and the rank $1$ $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$ that come from the triangulation are all in $\Delta$.

Proposition 3.3**.**

If $0\to W^{\prime}\to W\to W^{\prime\prime}\to 0$ is an exact sequence of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs, then $W$ is split $\Delta$-triangulable if and only if $W^{\prime}$ and $W^{\prime\prime}$ are split $\Delta$-triangulable.

Proof.

If $W^{\prime}$ and $W^{\prime\prime}$ are split $\Delta$-triangulable, then $W$ is obviously split $\Delta$-triangulable. We now prove the converse. If $W_{e}$ admits a triangulation, then so do $W_{e}^{\prime}$ and $W_{e}^{\prime\prime}$. By corollary 3.2, $W^{\prime}$ and $W^{\prime\prime}$ are therefore split triangulable. Proposition 2.8 implies that two different triangulations of $W_{e}$ give rise to two composition series of $W_{e}$ (seen as a $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$). The set of rank $1$ $\mathbf{B}_{\mathrm{e},E}$-representations attached to any triangulation of $W_{e}$ is therefore well-defined up to permutation by the Jordan-Hölder theorem. Hence if $W$ is split $\Delta$-triangulable, then so are $W^{\prime}$ and $W^{\prime\prime}$. ∎

Proposition 3.4**.**

If $W_{e}$ is an irreducible $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$, and $\delta\in H^{1}(G_{K},\mathbf{B}_{\mathrm{e},E}^{\times})$, then every surjective map $\pi:\operatorname{End}(W_{e})\to\mathbf{B}_{\mathrm{e},E}(\delta)$ of $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$ is split.

Proof.

Write $\mathbf{B}_{\mathrm{e},E}(\delta)=\mathbf{B}_{\mathrm{e},E}\cdot e_{\delta}$, where $g(e_{\delta})=\delta(g)e_{\delta}$ with $\delta(g)\in\mathbf{B}_{\mathrm{e},E}^{\times}$. Recall that if $A$ is a ring and $M$ is a free $A$-module, then $\operatorname{End}_{A}(M)$ is its own dual, for the pairing $(f,g)\mapsto\operatorname{Tr}(fg)$. The map $\pi$ is therefore of the form $f\mapsto\operatorname{Tr}(fh)\cdot e_{\delta}$ for some $h\in\operatorname{End}(W_{e})$. The map $h$ satisfies $g(h)=\delta(g)^{-1}h$, and therefore gives rise to a $G_{K}$-equivariant map $h:W_{e}\to W_{e}(\delta)$. Since $W_{e}$ is irreducible, $h$ is invertible. We can then write $\operatorname{End}(W_{e})=\ker(\pi)\oplus\mathbf{B}_{\mathrm{e},E}\cdot h^{-1}$, which shows that $\pi$ is split. ∎

Theorem 3.5**.**

If $W_{e}$ is an irreducible $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$ such that $\operatorname{End}(W_{e})$ is split triangulable, then the triangulation of $\operatorname{End}(W_{e})$ splits.

Proof.

Write $\{0\}=X_{0}\subset X_{1}\subset\cdots\subset X_{d}=\operatorname{End}(W_{e})$, and $X_{i}/X_{i-1}=\mathbf{B}_{\mathrm{e},E}(\delta_{i})$ for some $\delta_{i}\in H^{1}(G_{K},\mathbf{B}_{\mathrm{e},E}^{\times})$. By proposition 3.4, the exact sequence $0\to X_{d-1}\to\operatorname{End}(W_{e})\to\mathbf{B}_{\mathrm{e},E}(\delta_{d})\to 0$ is split, and therefore $\operatorname{End}(W_{e})=X_{d-1}\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{d})$.

Suppose that we have an isomorphism $\operatorname{End}(W_{e})=X_{j}\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{j+1})\oplus\cdots\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{d})$. Let $\pi_{j}$ denote the composition $\operatorname{End}(W_{e})\to X_{j}\to\mathbf{B}_{\mathrm{e},E}(\delta_{j})$. By proposition 3.4, $\operatorname{End}(W_{e})=\ker(\pi_{j})\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{j})$. We have $\ker(\pi_{j})=X_{j-1}\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{j+1})\oplus\cdots\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{d})$, so that $\operatorname{End}(W_{e})=X_{j-1}\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{j})\oplus\cdots\oplus\mathbf{B}_{\mathrm{e},E}(\delta_{d})$. The claim follows by induction. ∎

Remark 3.6**.**

Theorem 3.5 is reminiscent of the following result of Chevalley: if $G$ is any group and if $X$ and $Y$ are finite dimensional semi-simple characteristic [math] representations of $G$, then $X\otimes Y$ is also semi-simple. The same holds for semi-linear representations and, more generally, in any Tannakian category over a field of characteristic [math] [Del16].

4. Cohomology of $B$-pairs

The cohomology of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs is defined and studied in §2.1 of [Nak09]. We recall what we need. Let $W$ be a $\mathbf{B}^{\otimes E}_{\mid K}$-pair. Nakamura constructs an $E$-vector space $H^{1}(G_{K},W)$ that has the following properties

1. (1)

$H^{1}(G_{K},W)=\operatorname{Ext}^{1}(\mathbf{B},W)$ (i.e. it classifies the extensions of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs); 2. (2)

there is an exact sequence of $E$-vector spaces

[TABLE]

If $W$ is a rank $1$ $\mathbf{B}^{\otimes E}_{\mid K}$-pair with $W_{e}\in\Delta(\mathbf{Q}_{p})$, then $W_{\mathrm{dR}}^{G_{\mathbf{Q}_{p}}}$ is an $E$-vector space of dimension $1$ or [math], depending on whether $W_{e}$ (extended to $G_{\mathbf{Q}_{p}}$) is de Rham or not. Since $W_{\mathrm{dR}}^{G_{K}}=K\otimes_{\mathbf{Q}_{p}}W_{\mathrm{dR}}^{G_{\mathbf{Q}_{p}}}$, this implies that $W_{\mathrm{dR}}^{G_{K}}=\{0\}$ if $W$ is not de Rham. Note that if $W$ is a rank $1$ $\mathbf{B}^{\otimes E}_{\mid K}$-pair with $K\neq\mathbf{Q}_{p}$, then $W$ may be “partially de Rham” in the sense of [Din17], so that in general $W_{\mathrm{dR}}^{G_{K}}$ can be non-zero even if $W$ is not de Rham.

Proposition 4.1**.**

If $W_{\mathrm{dR}}^{+}$ is a free $E\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}$-representation of $G_{K}$ of rank $1$, the map $H^{1}(G_{K},W_{\mathrm{dR}}^{+})\to H^{1}(G_{K},W_{\mathrm{dR}})$ is injective.

Proof.

Since we have an exact sequence

[TABLE]

it is enough to show that $W_{\mathrm{dR}}^{G_{K}}\to(W_{\mathrm{dR}}/W_{\mathrm{dR}}^{+})^{G_{K}}$ is surjective. To prove this, we can replace $K$ by a finite extension $L$, and in particular we can assume that $L$ contains $E$. In this case, $W_{\mathrm{dR}}^{+}{\mid}_{G_{L}}$ is a direct sum of rank $1$ $\mathbf{B}_{\mathrm{dR}}^{+}$-representations of $G_{L}$.

Let $X_{\mathrm{dR}}^{+}$ be a rank $1$ $\mathbf{B}_{\mathrm{dR}}^{+}$-representation of $G_{L}$. The $L$-vector space $X_{\mathrm{dR}}^{G_{L}}$ is of dimension [math] or $1$. If $\dim_{L}X_{\mathrm{dR}}^{G_{L}}=1$, then $X_{\mathrm{dR}}$ is de Rham, and the map $X_{\mathrm{dR}}^{G_{L}}\to(X_{\mathrm{dR}}/X_{\mathrm{dR}}^{+})^{G_{L}}$ is surjective by the same argument as in lemma 3.8.1 of [BK90] (see lemma 2.6 of [Nak09]). If $\dim_{L}X_{\mathrm{dR}}^{G_{L}}=0$, then for every $i\in\mathbf{Z}$, we have $(t^{i}X_{\mathrm{dR}}^{+}/t^{i+1}X_{\mathrm{dR}}^{+})^{G_{L}}=0$ by proposition 3.21 of [Fon04]. This implies that $(X_{\mathrm{dR}}/X_{\mathrm{dR}}^{+})^{G_{L}}=0$, so that the map $X_{\mathrm{dR}}^{G_{L}}\to(X_{\mathrm{dR}}/X_{\mathrm{dR}}^{+})^{G_{L}}$ is also surjective. ∎

Corollary 4.2**.**

If $X$ is a direct sum of rank $1$ $\mathbf{B}^{\otimes E}_{\mid K}$-pairs, the map $H^{1}(G_{K},X_{\mathrm{dR}}^{+})\to H^{1}(G_{K},X_{\mathrm{dR}})$ is injective.

Recall that every rank $1$ $\mathbf{B}^{\otimes E}_{\mid K}$-pair is of the form $\mathbf{B}(\delta)$ for a unique $\delta:K^{\times}\to E^{\times}$.

Proposition 4.3**.**

If a $\mathbf{B}^{\otimes E}_{\mid K}$-pair $W$ is split $\Delta(\mathbf{Q}_{p})$-triangulable, with subquotients $\{\mathbf{B}(\delta_{i})\}_{i}$ such that $\mathbf{B}(\delta_{i}\delta_{j}^{-1})$ is not de Rham for any $i\neq j$, and if the corresponding triangulation of $W_{e}$ splits as a direct sum of $1$-dimensional $\mathbf{B}_{\mathrm{e},E}$-representations, then the triangulation of $W$ splits.

Proof.

Let $0=W_{0}\subset W_{1}\subset\cdots\subset W_{d}=W$ be the given triangulation of $W$. We prove by induction on $j$ that $W_{j}=\mathbf{B}(\delta_{1})\oplus\cdots\oplus\mathbf{B}(\delta_{j})$. This is true for $j=1$, assume it holds for $j-1$. Write $0\to W_{j-1}\to W_{j}\to\mathbf{B}(\delta_{j})\to 0$ and $W_{j-1}=\mathbf{B}(\delta_{1})\oplus\cdots\oplus\mathbf{B}(\delta_{j-1})$. Let $X=W_{j-1}(\delta_{j}^{-1})$ and $Y=W_{j}(\delta_{j}^{-1})$. The $\mathbf{B}^{\otimes E}_{\mid K}$-pair $Y$ corresponds to a class in $H^{1}(G_{K},X)$. The $\mathbf{B}_{\mathrm{e},E}$-representation $Y_{e}$ is split, and therefore so is $Y_{\mathrm{dR}}$. By corollary 4.2, so is $Y_{\mathrm{dR}}^{+}$. The class of $Y$ in $H^{1}(G_{K},X)$ is therefore in the kernel of $H^{1}(G_{K},X)\to H^{1}(G_{K},X_{e})\oplus H^{1}(G_{K},X_{\mathrm{dR}}^{+})$. Since $X_{\mathrm{dR}}^{G_{K}}=0$ by hypothesis, Nakamura’s exact sequence (2) above implies that the class of $Y$ is trivial and hence $W_{j}=W_{j-1}\oplus\mathbf{B}(\delta_{j})$. The proposition follows by induction. ∎

5. Proof of the main theorem

In this section, we prove theorem B. Let $F$ be a finite extension of $E$ of degree $\geqslant 2$, and write $F\otimes_{E}F=\oplus_{i}F_{i}$. There are at least two summands since $F$ itself is one of them.

Proposition 5.1**.**

Let $F/E$ be as above, and let $W$ be an $F$-linear representation of $G_{K}$. We have $F\otimes_{E}W=\oplus_{i}(F_{i}\otimes_{F}W)$ as $F$-linear representations of $G_{K}$.

Proof.

We have $F\otimes_{E}W=(F\otimes_{E}F)\otimes_{F}W=\oplus_{i}(F_{i}\otimes_{F}W)$. ∎

Corollary 5.2**.**

If $W$ is a $\mathbf{B}_{\mathrm{e},E}$-representation of $G_{K}$ that has an $F$-linear structure, then $W$ becomes reducible after extending scalars from $E$ to $F$.

Let us say that a $\mathbf{B}^{\otimes E}_{\mid K}$-pair $W$ is completely irreducible if $(F\otimes_{E}W)|_{G_{L}}$ is an irreducible $\mathbf{B}^{\otimes F}_{\mid L}$-pair for all finite extensions $F$ of $E$ and $L$ of $K$.

Proposition 5.3**.**

If $K=E$ and if $X$ and $Y$ are two completely irreducible $\mathbf{B}^{\otimes E}_{\mid K}$-pairs such that $\operatorname{Hom}(X,Y)$ is split $\Delta(\mathbf{Q}_{p})$-triangulable, then $X$ and $Y$ are of rank $1$.

Proof.

Let $\{\mathbf{B}(\delta_{i})\}_{i}$ be the rank $1$ subquotients of the triangulation of $\operatorname{Hom}(X,Y)$. We have an inclusion $\mathbf{B}(\delta_{1})\subset\operatorname{Hom}(X,Y)$. This gives rise to a non-zero map $X\to Y(\delta_{1}^{-1})$ of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs. Write $\mathbf{B}(\delta_{1})_{e}=\mathbf{B}_{\mathrm{e},E}(\mu_{1})$ for some $\mu_{1}:G_{K}\to\mathbf{B}_{\mathrm{e},E}^{\times}$ (recall that $K=E$). Since $X$ and $Y$ are irreducible, $X_{e}$ and $Y_{e}$ are irreducible $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$ (proposition 3.1), and the map $X_{e}\to Y_{e}(\mu_{1}^{-1})$ is therefore an isomorphism. This implies that $\operatorname{Hom}(X_{e},Y_{e})=\operatorname{End}(X_{e})(\mu_{1})$, so that $\operatorname{End}(X_{e})$ is split triangulable. By theorem 3.5, the triangulation of $\operatorname{End}(X_{e})$ splits. The triangulation of $\operatorname{Hom}(X_{e},Y_{e})=\operatorname{End}(X_{e})(\mu_{1})$ therefore also splits. Let $n$ be the common rank of $X$ and $Y$.

Suppose that none of the $\mathbf{B}(\delta_{i}\delta_{j}^{-1})$ are de Rham for any $i\neq j$. By proposition 4.3 applied to $W=\operatorname{Hom}(X,Y)$, the triangulation of $\operatorname{Hom}(X,Y)$ splits. We can therefore write $\operatorname{Hom}(X,Y)=\oplus_{i}\mathbf{B}(\delta_{i})$. Since $X$ and $Y$ are both irreducible, they are pure of some slopes $s$ and $t$ by theorem 1.3. The $\mathbf{B}^{\otimes E}_{\mid K}$-pair $\operatorname{Hom}(X,Y)$ is then pure of slope $t-s$ by (1) of proposition 1.4. By (3) of ibid, each of the $\mathbf{B}(\delta_{i})$ is also pure of slope $t-s$. Each $\mathbf{B}(\delta_{i})$ gives rise to a map $X\to Y(\delta_{i}^{-1})$, which is an isomorphism of $\mathbf{B}^{\otimes E}_{\mid K}$-pairs by (2) of ibid, since $X$ and $Y(\delta_{i}^{-1})$ are both pure of slope $s$. By taking determinants, we get $\delta_{i}^{n}=\det(Y)\det(X)^{-1}$ for every $i$. This implies that $(\delta_{i}\delta_{j}^{-1})^{n}=1$ so that $\delta_{i}\delta_{j}^{-1}$ is of finite order, and $\mathbf{B}(\delta_{i}\delta_{j}^{-1})$ is de Rham (lemma 4.1 of [Nak09]), contradicting our assumption.

Therefore, one of the $\mathbf{B}(\delta_{i}\delta_{j}^{-1})$ is de Rham for some $i\neq j$. Write $\mathbf{B}(\delta_{k})_{e}=\mathbf{B}_{\mathrm{e},E}(\mu_{k})$ where the $\mu_{k}$ are characters $G_{K}\to E^{\times}$ (recall that $K=E$), so that $\operatorname{End}(X_{e})(\mu_{1})=\oplus_{k}\mathbf{B}_{\mathrm{e},E}(\mu_{k})$ as $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$. The fact that $\mathbf{B}(\delta_{i}\delta_{j}^{-1})$ is de Rham implies that $\mu_{i}\mu_{j}^{-1}$ is de Rham. We then have $X_{e}=X_{e}(\mu_{1}\mu_{i}^{-1})=X_{e}(\mu_{1}\mu_{j}^{-1})$, so that $X_{e}=X_{e}(\mu_{i}\mu_{j}^{-1})$. By taking determinants, we find that $\mathbf{B}_{e,E}((\mu_{i}\mu_{j}^{-1})^{n})=\mathbf{B}_{e,E}$ and therefore by corollary 2.7, $(\mu_{i}\mu_{j}^{-1})^{n}:G_{K}\to E^{\times}$ is de Rham and the sum of its weights is [math]. This implies that the sum of the weights of $\mu_{i}\mu_{j}^{-1}:G_{K}\to E^{\times}$ is [math]. By corollary 2.7, $\mu_{i}\mu_{j}^{-1}=\chi\eta$ with $\chi:G_{K}\to E^{\times}$ a $\mathbf{B}_{e,E}$-admissible character and $\eta:G_{K}\to E^{\times}$ potentially unramified. Since $X_{e}(\chi\eta)=X_{e}$ and $X_{e}(\chi)=X_{e}$, we get $X_{e}(\eta)=X_{e}$. By taking determinants, we get that $\eta^{n}$ is $\mathbf{B}_{e,E}$-admissible. Since $\eta^{n}$ is also potentially unramified, and $\mathbf{B}_{\mathrm{e},E}\cap(\overline{\mathbf{Q}}_{p}\cdot\widehat{\mathbf{Q}}_{p}^{\mathrm{nr}})=E$, it is trivial. Hence $\eta$ is a character of finite order of $G_{K}$, and so there exists a finite extension $L$ of $K$ such that $\mu_{i}=\chi\mu_{j}$ on $G_{L}$.

The space $\operatorname{End}(X_{e})(\mu_{1})$ contains $\mathbf{B}_{\mathrm{e},E}(\mu_{j})\oplus\mathbf{B}_{\mathrm{e},E}(\mu_{i})$, which is isomorphic to $\mathbf{B}_{\mathrm{e},E}(\mu_{j})\oplus\mathbf{B}_{\mathrm{e},E}(\mu_{j})$ after restricting to $G_{L}$. Let $f$ and $g$ be the two resulting isomorphisms $X_{e}\to X_{e}(\mu_{1}\mu_{j}^{-1})$. The map $h=f^{-1}\circ g:X_{e}\to X_{e}$ is $G_{L}$-equivariant and is not in $E^{\times}\cdot\operatorname{Id}$ since $f$ and $g$ are $\mathbf{B}_{\mathrm{e},E}$-linearly independent. Therefore, $\operatorname{End}(X_{e})^{G_{L}}$ is strictly larger than $E$.

Since $X_{e}|_{G_{L}}$ is irreducible, Schur’s lemma and corollary 2.14 imply that $\operatorname{End}(X_{e})^{G_{L}}$ contains a field $F$ such that $[F:E]\geqslant 2$ (for example, $F=E[h]$). Hence $X_{e}|_{G_{L}}$ has an $F$-linear structure. Corollary 5.2 implies that $(F\otimes_{E}X_{e})|_{G_{L}}$ is reducible. By proposition 3.1, $X$ is not completely irreducible. This is a contradiction, so $X$ had to be of rank $1$. Since $X$ and $Y$ have the same rank, we are done. ∎

We now recall and prove theorem B. A strict sub-quotient of a $\mathbf{B}^{\otimes E}_{\mid K}$-pair is a quotient of a saturated sub $\mathbf{B}^{\otimes E}_{\mid K}$-pair.

Theorem 5.4**.**

If $X$ and $Y$ are two non-zero $\mathbf{B}^{\otimes E}_{\mid K}$-pairs whose tensor product is $\Delta(\mathbf{Q}_{p})$-triangulable, then $X$ and $Y$ are both potentially triangulable.

Proof.

We can replace $E$ and $K$ by finite extensions $F$ and $L$ if necessary, and write $X$ and $Y$ as successive extensions of completely irreducible $\mathbf{B}^{\otimes F}_{\mid L}$-pairs with $F=L$. If $X^{\prime}$ and $Y^{\prime}$ are two strict sub-quotients of $X$ and $Y$, then $X^{\prime}\otimes Y^{\prime}$ is a strict sub-quotient of $X\otimes Y$, and it is $\Delta(\mathbf{Q}_{p})$-triangulable by proposition 3.3. Proposition 5.3, applied to $(X^{\prime})^{*}$ and $Y^{\prime}$ so that $X^{\prime}\otimes Y^{\prime}=\operatorname{Hom}((X^{\prime})^{*},Y^{\prime})$, tells us that $X^{\prime}$ and $Y^{\prime}$ are of rank $1$.

Hence the $\mathbf{B}^{\otimes F}_{\mid L}$-pairs $(F\otimes_{E}X)|_{G_{L}}$ and $(F\otimes_{E}Y)|_{G_{L}}$ are split triangulable. ∎

Corollary 5.5**.**

If $X_{e}$ and $Y_{e}$ are two $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{K}$ whose tensor product is triangulable, with the rank $1$ sub-quotients extending to $\mathbf{B}_{\mathrm{e},E}$-representations of $G_{\mathbf{Q}_{p}}$, then $X_{e}$ and $Y_{e}$ are both potentially triangulable.

Proof.

By proposition 1.2, $X_{e}$ and $Y_{e}$ extend to $\mathbf{B}^{\otimes E}_{\mid K}$-pairs. The result follows from corollary 3.2 and theorem 5.4. ∎

We finish with an example of a representation $V$ such that $V\otimes_{E}V$ is trianguline, but $V$ itself is not trianguline. This shows that the “potentially” in the statement of theorem A cannot be avoided. Let $Q_{8}$ denote the quaternion group. If $p\equiv 3\bmod{4}$, there is a Galois extension $K/\mathbf{Q}_{p}$ such that $\operatorname{Gal}(K/\mathbf{Q}_{p})=Q_{8}$ (see II.3.6 of [JY88]). Choose such a $p$ and $K$, and let $E$ be a finite extension of $\mathbf{Q}_{p}$ containing $\sqrt{-1}$. The group $Q_{8}$ has a (unique) irreducible $2$-dimensional $E$-linear representation, which we inflate to a representation $V$ of $G_{\mathbf{Q}_{p}}$. One can check that $V\otimes_{E}V$ is a direct sum of characters, hence trianguline, and that the semi-linear representation $\operatorname{Frac}(\mathbf{B}_{\mathrm{e},E})\otimes_{E}V$ is irreducible. This holds for all $E$ as above, so that $V$ is not trianguline.

Acknowledgements: We thank Léo Poyeton and Sandra Rozensztajn for their comments, and David Hansen and Andrea Conti for their questions and discussions.