---
abstract: 'Let $p$ be a prime number and $f$ an overconvergent $p$-adic automorphic form on a definite unitary group which is split at $p$. Assume that $f$ is of “classical weight” and that its Galois representation is crystalline at $p$, then $f$ is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimensions by making crucial use of the patched eigenvariety constructed in [@BHS].'
address:
- |
    Christophe Breuil\
    C.N.R.S.\
    Laboratoire de Mathématiques d’Orsay\
    Université Paris-Sud\
    Université Paris-Saclay\
    F-91405 Orsay\
    France\
    Christophe.Breuil@math.u-psud.fr
- |
    Eugen Hellmann\
    Mathematisches Institut\
    Universität Bonn\
    Endenicher Allee 60\
    D-53115 Bonn\
    Germany\
    hellmann@math.uni-bonn.de
- |
    Benjamin Schraen\
    C.N.R.S.\
    Laboratoire de Mathématiques\
    Université de Versailles St. Quentin\
    45 Av. des États-Unis\
    F-78035 Versailles\
    France\
    benjamin.schraen@uvsq.fr
author:
- 'Christophe Breuil, Eugen Hellmann and Benjamin Schraen'
title: Smoothness and classicality on Eigenvarieties
---

Introduction
============

Let $p$ be a prime number. In this paper we are concerned with classicality of $p$-adic automorphic forms on some unitary groups, i.e. we are looking for criteria that decide whether a given $p$-adic automorphic form is classical or not. More precisely we work with $p$-adic forms of finite slope, that is, in the context of *eigenvarieties*.

Let $F^+$ be a totally real number field and $F$ be an imaginary quadratic extension of $F^+$. We fix a unitary group $G$ in $n$ variables over $F^+$ which splits over $F$ and over all $p$-adic places of $F^+$, and which is compact at all infinite places of $F^+$. Associated to such a group $G$ (and the choice of a tame level, i.e. a compact open subgroup of $G({\mathbb{A}}_{F^+}^{p\infty})$) there is a nice Hecke eigenvariety which is an equidimensional rigid analytic space of dimension $n[F^+:{\mathbb{Q}}]$, see e.g. [@Cheeig], [@BelChe] or [@Emerton]. One may view a [*$p$-adic overconvergent eigenform of finite slope*]{}, or simply overconvergent form, as a point $x$ of such an eigenvariety and one can associate to each overconvergent form a continuous semi-simple representation $\rho_x:{\rm Gal}(\overline F/F)\rightarrow {\mathrm{GL}}_n(\overline {\mathbb{Q}}_p)$ which is unramified outside a finite set of places of $F$ and which is trianguline in the sense of [@Co] at all places of $F$ dividing $p$ ([@KPX]).

A natural expectation deduced from the Langlands and Fontaine-Mazur conjectures is that, if $\rho_x$ is de Rham (in the sense of Fontaine) at places of $F$ dividing $p$, then $x$ is a [*classical*]{} automorphic form (see Definition \[defclass\] and Proposition \[compaclas\] for the precise definition). However, the naive version of this statement fails for two reasons: (1) a classical automorphic form for $G({\mathbb{A}}_{F^+})$ can only give Galois representations which have distinct Hodge-Tate weights (in each direction $F\hookrightarrow \overline{\mathbb{Q}}_p$) and (2) the phenomenon of [*companion*]{} forms shows that there can exist classical and non-classical forms giving the same Galois representation. However, we can resolve (1) by requiring $\rho_x$ to have distinct Hodge-Tate weights and (2) by requiring $x$ to be of “classical” (or dominant) weight. In fact, since the Hodge-Tate weights of $\rho_x$ are related to the weight of $x$, requiring the latter automatically implies the former, once $\rho_x$ is assumed to be de Rham. As a conclusion, it seems reasonable to expect that any overconvergent form $x$ of classical weight such that $\rho_x$ is de Rham at places of $F$ dividing $p$ is a classical automorphic form (see Conjecture $\ref{classiconj}$ and Remark \[extra\]).

Such a classicality theorem is due to Kisin ([@Kisinoverconvergent]) in the context of Coleman-Mazur’s eigencurve, i.e. in the slightly different setting of ${\rm GL}_2/{\mathbb{Q}}$. Note that, at the time of [@Kisinoverconvergent], the notion of a trianguline representation was not available, and in fact [@Kisinoverconvergent] inspired Colmez to define trianguline representations ([@Co]).

In the present paper we prove new cases of this classicality conjecture (in the above unitary setting). In particular we are able to deal with cases where the overconvergent form $x$ is [*critical*]{}. Throughout, we assume that $\rho_x$ is crystalline at $p$-adic places. Essentially the same proof should work if $\rho_x$ is only assumed crystabelline, but the crystalline assumption significantly simplifies the notation.

To state our main results, we fix an overconvergent form $x$ of classical weight such that $\rho_x$ is crystalline at all places dividing $p$. Such an overconvergent form can be described by a pair $(\rho_x,\delta_x)$, where $\rho_x$ is as above and $\delta_x=(\delta_{x,v})_{v\in S_p}$ is a locally ${\mathbb{Q}}_p$-analytic character of the diagonal torus of $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)\cong \prod_{v\in S_p}{\mathrm{GL}}_n(F^+_v).$ Here $S_p$ denotes the set of places of $F^+$ dividing $p$. There are nontrivial relations between $\rho_{x,v}:=\rho\vert_{{\rm Gal}(\overline {F^+_v}/F^+_v)}$ and $\delta_{x,v}$, in particular the character $\delta_{x,v}$ defines an ordering of the eigenvalues of the crystalline Frobenius on $D_{\rm cris}(\rho_{x,v})$. If we assume that these Frobenius eigenvalues are pairwise distinct, then this ordering defines a Frobenius stable flag in $D_{\rm cris}(\rho_{x,v})$. We can therefore associate to $x$ for each $v\in S_p$ a permutation $w_{x,v}$ that gives the relative position of this flag with respect to the Hodge filtration on $D_{\rm cris}(\rho_{x,v})$, see §\[weyl\] (where we rather use another equivalent definition of $w_{x,v}$ in terms of triangulations). Following [@BelChe §2.4.3] we say that $x$ is [*noncritical*]{} if, for each $v$, the permutation $w_{x,v}$ is trivial. The invariant $(w_{x,v})_{v\in S_p}$ can thus be seen as “measuring” the criticality of $x$.

In the statement of our main theorem, we need to assume a certain number of Working Hypotheses (basically the combined hypotheses of all the papers we use). We denote by ${\overline{\rho}}_x$ the mod $p$ semi-simplification of $\rho_x$. These Working Hypotheses are:

1.  The field $F$ is unramified over $F^+$ and $G$ is quasi-split at all finite places of $F^+$;

2.  the tame level of $x$ is hyperspecial at all finite places of $F^+$ inert in $F$;

3.  ${\overline{\rho}}_x({\rm Gal}(\overline F/F(\zeta_p))$ is adequate ([@Thorne]) and $\zeta_p\notin \overline F^{\ker({\rm ad}{\overline{\rho}}_x)}$;

4.  the eigenvalues of $\varphi$ on $D_{\rm cris}(\rho_{x,v})$ are sufficiently generic for any $v\in S_p$ (Definition \[veryreg\]).

Our main theorem is:

\[mainintro\] Let $p>2$ and assume that the group $G$ and the tame level satisfy (i) and (ii). Let $x$ be an overconvergent form of classical weight such that $\rho_x$ is crystalline and satisfies (iii) and (iv). If $w_{x,v}$ is a product of distinct simple reflections for all places $v$ of $F^+$ dividing $p$, then $x$ is classical.

Note that the assumption on the $w_{x,v}$ in Theorem \[mainintro\] is empty when $n=2$, and already this $n=2$ case was not previously known (to the knowledge of the authors). The noncritical case of Theorem \[mainintro\], i.e. the special case where all the $w_{x,v}$ are trivial, is already known and due to Chenevier ([@Chenevierfern Prop.4.2]). Thus the main novelty, and difficulty, in Theorem \[mainintro\] is that it deals with possibly critical (though not too critical) points.

In fact we give a more general classicality criterion and prove that it is satisfied under the assumptions of Theorem $\ref{mainintro}$. This criterion is formulated in terms of the rigid analytic space of trianguline representations $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ defined in [@HellmannFS] and [@BHS §2.2]. For every $v\in S_p$ there is a canonical morphism from the eigenvariety to $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$.

\[mainintro2\] Let $p>2$ and assume that the group $G$ and the tame level satisfy (i) and (ii). Let $x$ be an overconvergent form of classical weight such that $\rho_x$ is crystalline and satisfies (iii) and (iv). If for any $v\in S_p$ the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is contained in a unique irreducible component of $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$, then $x$ is classical.

According to this theorem we need to understand the local geometry of the space $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ at $x_v$. It turns out that much of this local geometry is controlled by the Weyl group element $w_{x,v}$ associated to $x$ which only depends on the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$. For $v\in S_p$ denote by $\lg(w_{x,v})$ the length of the permutation $w_{x,v}$ and by $d_{x,v}$ the rank of the ${\mathbb{Z}}$-module generated by $w_{x,v}(\alpha)-\alpha$, as $\alpha$ ranges over the roots of $({\mathrm{Res}}_{F_v^+/{\mathbb{Q}}_p}{\mathrm{GL}}_n)\times_{{\mathbb{Q}}_p} \overline {\mathbb{Q}}_p\cong \prod_{\tau:\, F_v^+\hookrightarrow \overline{\mathbb{Q}}_p}{\mathrm{GL}}_n$. Then $d_{x,v}\leq \lg(w_{x,v})$, with equality if and only if $w_{x,v}$ is a product of distinct simple reflections (Lemma \[coxeter\]).

\[mainlocal\] Let $v\in S_p$ and let $X\subseteq X_{\rm tri}^\square(\overline \rho_{x,v})$ be a union of irreducible components that contain $x_v$ and satisfy the [*accumulation property*]{} of Definition \[accu\] at $x_v$. Then $$\dim T_{X,x_v}\leq \dim X+ \lg(w_{x,v})-d_{x,v}=\dim X_{\rm tri}^\square(\overline \rho_{x,v}) + \lg(w_{x,v})-d_{x,v},$$ where $T_{X,x_v}$ is the tangent space to $X$ at $x_v$. In particular $X$ is smooth at $x_v$ when $w_{x,v}$ is a product of distinct simple reflections.

The accumulation condition in Theorem \[mainlocal\] actually prevents us from directly applying it to $X=X_{\rm tri}^\square(\overline \rho_{x,v})$ and thus directly deducing Theorem \[mainintro\] from Theorem \[mainintro2\]. Hence we have to sharpen Theorem \[mainintro2\], see Theorem \[classicalitycrit\].

Assuming the classical modularity lifting conjectures for ${\overline{\rho}}_x$ (in all weights with trivial inertial type), there is a certain union $\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ such that $\prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is (essentially) described by the patched eigenvariety $X_p({\overline{\rho}}_x)$ defined in [@BHS] (see Remark \[conjvariant\]). In the last section of the paper (§\[modularity\]), we prove (assuming modularity lifting conjectures) that the inequality in Theorem \[mainlocal\] for $X=\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is an [*equality*]{} for all $v\in S_p$, $$\label{mainequality}
\dim T_{\widetilde X_{\rm tri}^\square({\overline{\rho}}_{x,v}),x_v}=\dim X_{\rm tri}^\square(\overline \rho_{x,v}) + \lg(w_{x,v})-d_{x,v}\ \ \ \ \ \ {\rm (assuming\ modularity),}$$ see Corollary \[bonnedim\]. The precise computation (\[mainequality\]) of the dimension of the tangent space is intimately related to (and uses in its proof) the existence of many *companion points* on the patched eigenvariety $X_p({\overline{\rho}}_x)$. These companion points are provided by the following unconditional theorem, which is of independent interest.

\[companionptsintro\] Let $y=((\rho_v)_{v\in S_p},\epsilon)$ be a point on $X_p({\overline{\rho}}_x)$. Let $T$ be the diagonal torus in ${\mathrm{GL}}_n$ and let $\delta$ be a locally ${\mathbb{Q}}_p$-analytic character of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ such that $\epsilon\delta^{-1}$ is an algebraic character of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ and such that $\epsilon$ is strongly linked to $\delta$ in the sense of [@HumBGG §5.1] (as modules over the Lie algebra of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$). Then $((\rho_v)_{v\in S_p},\delta)$ is also a point on $X_p({\overline{\rho}}_x)$.

We also prove that the equality (\[mainequality\]) for all $v\in S_p$ (and thus the modularity lifting conjectures) imply that the initial Hecke eigenvariety is itself [*singular*]{} at $x$ as soon as the Weyl element $w_{x,v}$ is [*not*]{} a product of distinct simple reflections for some $v\in S_p$, see Corollary \[singhecke\].

Let us now outline the strategy of the proofs of Theorems \[mainintro2\] and \[mainlocal\].

The proof of Theorem \[mainlocal\] crucially uses results of Bergdall ([@Bergdall]), together with a fine analysis of the various conditions on the infinitesimal deformations of $\rho_{x,v}$ carried by vectors in $T_{X,x_v}$, see §\[wedgesection\] and §\[endofproof\]. Very recently, Bergdall proved an analogous bound for the dimension of the tangent space of the initial Hecke eigenvariety at $x$ assuming standard vanishing conjectures on certain Selmer groups ([@Bergdraft]).

The proof of Theorem \[mainintro2\] makes use of the patched eigenvariety $X_p({\overline{\rho}}_x)$ constructed in [@BHS] by applying Emerton’s construction of eigenvarieties [@Emerton] to the locally analytic vectors of the patched Banach $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$-representation $\Pi_\infty$ of [@CEGGPS]. As usual with the patching philosophy, the space $X_p({\overline{\rho}}_x)$ can be related to another geometric object which has a much more local flavour, namely the space $X_{\rm tri}^\square(\overline \rho_{x,p}):=\prod_{v\in S_p}X_{\rm tri}^\square(\overline \rho_{x,v})$ of trianguline representations. More precisely, by [@BHS Th.3.20] there is a Zariski closed embedding: $$\label{maininclusion}
X_p(\overline\rho_x)\hookrightarrow {\mathfrak{X}}_{{\overline{\rho}}_x^p}\times {\mathbb{U}}^g \times X_{\rm tri}^\square(\overline \rho_{x,p}),$$ identifying the source with a union of irreducible components of the target. Here ${\mathbb{U}}^g$ is an open polydisc (related to the patching variables) and ${\mathfrak{X}}_{{\overline{\rho}}_x^p}$ is the rigid analytic generic fiber of the framed deformation space of ${\overline{\rho}}_x$ at all the “bad” places prime to $p$. Moreover the Hecke eigenvariety containing $x$ can be embedded into the patched eigenvariety $X_p(\overline\rho_x)$ (see [@BHS Th.4.2]). As previously, we denote by $x_v$ the image of $x$ in $X_{\rm tri}^\square(\overline \rho_{x,v})$ via (\[maininclusion\]).

For $v\in S_p$ let us write ${\bf k}_{v}$ for the set of labelled Hodge-Tate weights of $\rho_{x,v}$, and $R_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ for the quotient defined in [@Kisindef] of the framed deformation ring of $\overline \rho_{x,v}$ parametrizing crystalline deformations of $\overline \rho_{x,v}$ of Hodge-Tate weight ${\bf k}_v$, and ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ for the rigid space $({\mathrm{Spf}}\, R_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}})^{\rm rig}$. We relate ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ to $X_{\rm tri}^\square(\overline \rho_{x,v})$ by introducing a third rigid analytic space $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ finite over ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ parametrizing crystalline deformations $\rho_v$ of $\overline \rho_{x,v}$ of Hodge-Tate weights ${\bf k}_v$ [*together with an ordering*]{} of the Frobenius eigenvalues on $D_{\rm cris}(\rho_{v})$, see §\[variant\] for a precise definition. The space $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ naturally embeds into $X_{\rm tri}^\square(\overline \rho_{x,v})$ and contains the point $x_v$ (and is smooth at $x_v$). We prove that there is a unique irreducible component $Z_{{\rm tri}}(x_v)$ of $X_{\rm tri}^\square(\overline \rho_{x,v})$ containing the unique irreducible component of $\widetilde {\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ passing through $x_v$ (Corollary \[defnZtri(x)\]). Let $Z_{\rm tri}(x):=\prod_{v\in S_p}Z_{\rm tri}(x_v)$, which is thus an irreducible component of $X_{\rm tri}^\square(\overline \rho_{x,p})$ containing $x$. Then Theorem \[mainintro2\] easily follows from the following theorem (see (i) of Remark \[uniqueirredcompo\]):

\[mainclasscrit\] Assume that ${\mathfrak{X}}_{{\overline{\rho}}_x^p}\times{\mathbb{U}}^g\times Z_{\rm tri}(x)\subseteq X_p(\overline\rho_x)$ via (\[maininclusion\]). Then the point $x$ is classical.

Let us finally sketch the proof of Theorem \[mainclasscrit\] (in fact, for the same reason as above, we have to sharpen Theorem \[mainclasscrit\], see Theorem \[classicalitycrit\]). Let $R_\infty$ be the usual patched deformation ring of ${\overline{\rho}}_x$, there is a canonical morphism of rigid spaces $X_p(\overline\rho_x)\longrightarrow {\mathfrak X}_\infty:=({\mathrm{Spf}}\, R_\infty)^{\rm rig}$. Let $L(\lambda)$ be the finite dimensional algebraic representation of $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ associated (via the usual shift) to the Hodge-Tate weights $({\bf k}_v)_{v\in S_p}$. Proving classicality of $x$ turns out to be equivalent to proving that the image of $x$ in ${\mathfrak X}_\infty$ is in the support of the $R_\infty$-module $\Pi_\infty(\lambda)'$ which is the continuous dual of: $$\Pi_\infty(\lambda):={\mathrm{Hom}}_{{\prod}_{v\in S_p}{\mathrm{GL}}_n(\mathcal{O}_{F_{\tilde{v}}})}\big(L(\lambda),\Pi_\infty\big).$$ By [@CEGGPS Lem.4.17], the $R_\infty$-module $\Pi_\infty(\lambda)'$ is essentially a Taylor-Wiles-Kisin “usual” patched module for the trivial inertial type and the Hodge-Tate weights $({\bf k}_v)_{v\in S_p}$. Forgetting the factors ${\mathfrak{X}}_{{\overline{\rho}}_x^p}$ and ${\mathbb{U}}^g$ which appear in ${\mathfrak X}_\infty$, its support is a union of irreducible components of the smooth rigid space $\prod_{v\in S_p}{\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$. It is thus enough to prove that the unique irreducible component $Z_{\rm cris}(\rho_x)$ of $\prod_{v\in S_p}{\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ passing through $(\rho_{x,v})_{v\in S_p}$ contains a point which is in the support of $\Pi_\infty(\lambda)'$. But it is easy to find a point $y$ in $Z_{\rm tri}(x)$ sufficiently close to $x$ such that $(\rho_{y,v})_{v\in S_p}\in Z_{\rm cris}(\rho_x)$ (in particular $\rho_{y,v}$ is crystalline of the same Hodge-Tate weights as $\rho_{x,v}$) [*and*]{} moreover $\rho_{y,v}$ is [*generic*]{} in the sense of [@BHS Def.2.8] for all $v\in S_p$. The assumption in Theorem \[mainclasscrit\] implies $y\in X_p(\overline\rho_x)$ and it is now not difficult to prove that such a generic crystalline point of $X_p(\overline\rho_x)$ is always classical, i.e. is in the support of $\Pi_\infty(\lambda)'$.

We end this introduction with the main notation of the paper.

If $K$ is a finite extension of ${\mathbb{Q}}_p$ we denote by $\mathcal{G}_K$ the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ and by $\Gamma_K$ the Galois group $\mathrm{Gal}(K(\zeta_{p^n},n\geq 1)/K)$ where $(\zeta_{p^n})_{n\geq 1}$ is a compatible system of primitive $p^n$-th roots of $1$ in $\overline K$. We normalize the reciprocity map ${\mathrm{rec}}_K:\, K^\times\rightarrow \mathcal{G}_K^{\rm ab}$ of local class field theory so that the image of a uniformizer of $K$ is a geometric Frobenius element. We denote by $\varepsilon$ the $p$-adic cyclotomic character and recall that its Hodge-Tate weight is $1$.

For $a\in L^\times$ (where $L$ is any finite extension of $K$) we denote by ${\rm unr}(a)$ the unramified character of $\mathcal{G}_K$, or equivalently of $\mathcal{G}_K^{\rm ab}$ or $K^\times$, sending to $a$ (the image by ${\mathrm{rec}}_K$ of) a uniformizer of $K$. For $z\in L$, we let $\vert z\vert_K:=p^{-[K:{\mathbb{Q}}_p]{{\mathrm{val}}}(z)}$ where ${{\mathrm{val}}}(p)=1$. We let $K_0\subseteq K$ be the maximal unramified subfield (we thus have $(\vert \ \vert_K)\vert_{K^\times}={\rm unr}(p^{-[K_0:{\mathbb{Q}}_p]})={\rm unr}(q^{-1})$ where $q$ is the cardinality of the residue field of $K$).

If $X={\mathrm{Sp}}\,A$ is an affinoid space, we write $\mathcal{R}_{A,K}$ for the Robba ring associated to $K$ with $A$-coefficients (see [@KPX Def.6.2.1] though our notation is slightly different). Given a continuous character $\delta: K^\times \rightarrow A^\times$ we write ${\mathcal{R}}_{A,K}(\delta)$ for the rank one $(\varphi,\Gamma_K)$-module on ${\mathrm{Sp}}\,A$ defined by $\delta$, see [@KPX Construction 6.2.4]. If $X$ is a rigid analytic space over $L$ (a finite extension of ${\mathbb{Q}}_p$) and $x$ is a point on $X$, we denote by $k(x)$ the residue field of $x$ (a finite extension of $L$), so that we have $x\in X(k(x))$. If $X$ and $Y$ are two rigid analytic spaces over $L$, we often write $X\times Y$ instead of $X\times_{{\mathrm{Sp}}\,L}Y$.

If $X$ is a “geometric object over ${\mathbb{Q}}_p$” (i.e. a rigid space, a scheme, an algebraic group, etc.), we denote by $X_K$ its base change to $K$ (for instance if $X$ is the algebraic group ${\mathrm{GL}}_n$ we write ${\mathrm{GL}}_{n,K}$). If $H$ is an abelian $p$-adic Lie group, we let $\widehat{H}$ be the rigid analytic space over ${\mathbb{Q}}_p$ which represents the functor mapping an affinoid space $X={\mathrm{Sp}}\,A$ to the group ${\mathrm{Hom}}_{\rm cont}(H,A^\times)$ of continous group homomorphisms (or equivalently locally ${\mathbb{Q}}_p$-analytic group homomorphisms) $H\rightarrow A^\times$. Finally, if $M$ is an $R$-module and $I\subseteq R$ an ideal, we denote by $M[I]\subseteq M$ the submodule of elements killed by $I$, and if $S$ is any finite set, we denote by $\vert S\vert$ its cardinality.

Crystalline points on the trianguline variety {#localpart1}
=============================================

We give several important definitions and results, including the key local statement bounding the dimension of some tangent spaces on the trianguline variety (Theorem \[upperbound\]).

Recollections {#begin}
-------------

We review some notation and definitions related to the trianguline variety.

We fix two finite extensions $K$ and $L$ of ${\mathbb{Q}}_p$ such that: $$\vert{\mathrm{Hom}}(K,L)\vert=[K:{\mathbb{Q}}_p]$$ and denote by $\mathcal{O}_K$, ${\mathcal O}_L$ their respective rings of integers. We fix a uniformizer $\varpi_K\in \mathcal{O}_K$ and denote by $k_L$ the residue field of ${\mathcal O}_L$. We let $\mathcal{T}:=\widehat{K^\times}$ and $\mathcal{W}:=\widehat{\mathcal{O}_K^\times}$. The restriction of characters to $\mathcal{O}_K^\times$ induces projections $\mathcal{T}\twoheadrightarrow \mathcal{W}$ and $\mathcal{T}_L\twoheadrightarrow \mathcal{W}_L$. If ${\bf k}:=(k_\tau)_{\tau:\, K\hookrightarrow L}\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$, we denote by $z^{\bf k}\in \mathcal{T}(L)$ the character: $$\begin{aligned}
\label{charc}
z\longmapsto\prod_{\tau\in{\mathrm{Hom}}(K,L)}\tau(z)^{k_\tau}\end{aligned}$$ where $z \in K^\times$. For ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$, we denote by $\delta_{\bf k}\in \mathcal{T}^n(L)$ the character: $$(z_1,\dots,z_n)\longmapsto\prod_{\stackrel{1\leq i\leq n}{\tau:\, K\hookrightarrow L}}\tau(z_i)^{k_{\tau,i}}$$ where $(z_1,\dots,z_n)\in (K^\times)^n$. We also denote by $\delta_{\bf k}$ its image in $\mathcal{W}^n(L)$ (i.e. its restriction to $({\mathcal O}_K^\times)^n$). We say that a point $\delta\in \mathcal{W}^n_L$ is [*algebraic*]{} if $\delta=\delta_{\bf k}$ for some ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. We say that an algebraic $\delta=\delta_{\bf k}$ is [*dominant*]{} (resp. [*strictly dominant*]{}) if moreover $k_{\tau,i}\geq k_{\tau,i+1}$ (resp. $k_{\tau,i}> k_{\tau,i+1}$) for $i\in \{1,\dots,n-1\}$ and $\tau\in {\mathrm{Hom}}(K,L)$.

We write ${\mathcal{T}}_{\rm reg}\subset {\mathcal{T}}_L$ for the Zariski-open complement of the $L$-valued points $z^{-\bf k}, \vert z\vert_K z^{{\bf k}+{\bf 1}}$, with ${\bf k}=(k_\tau)_{\tau:K\hookrightarrow L}\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$. We write ${\mathcal{T}}_{\rm reg}^n$ for the Zariski-open subset of characters $(\delta_1,\dots,\delta_n)$ such that $\delta_i/\delta_j\in {\mathcal{T}}_{\rm reg}$ for $i\neq j$.

We fix a continuous representation ${\overline{r}}:\mathcal{G}_K\rightarrow {\mathrm{GL}}_n(k_L)$ and let $R_{{\overline{r}}}^{\square}$ be the framed local deformation ring of ${\overline{r}}$ (a local complete noetherian $\mathcal{O}_L$-algebra of residue field $k_L$). We write $\mathfrak{X}_{{\overline{r}}}^\square:=({\mathrm{Spf}}\, R_{{\overline{r}}}^\square)^{{\mathrm{rig}}}$ for the rigid analytic space over $L$ associated to the formal scheme ${\mathrm{Spf}}\, R_{{\overline{r}}}^\square$. Recall that a representation $r$ of ${\mathcal{G}}_K$ on a finite dimension $L$-vector space is called *trianguline of parameter* $\delta=(\delta_1,\dots,\delta_n)$ if the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ over ${\mathcal{R}}_{L,K}$ associated to $r$ admits an increasing filtration ${\mathrm{Fil}}_\bullet$ by sub-$(\varphi,\Gamma_K)$-modules over ${\mathcal{R}}_{L,K}$ such that the graded piece ${\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}$ is isomorphic to ${\mathcal{R}}_{L,K}(\delta_i)$. We let $X_{\rm tri}^\square({\overline{r}})$ be the associated framed trianguline variety, see [@BHS §2.2] and [@HellmannFS]. Recall that $X_{\rm tri}^\square({\overline{r}})$ is the reduced rigid analytic space over $L$ which is the Zariski closure in $\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ of: $$\label{ucris}
U_{\rm tri}^\square({\overline{r}}):=\{{\rm points}\ (r,\delta)\ {\rm in}\ {\mathfrak{X}}^\square_{{\overline{r}}}\times{\mathcal{T}}^n_{\rm reg}\ {\rm such\ that}\ r\ \text{is trianguline of parameter}\ \delta\}$$ (the space $U_{\rm tri}^\square({\overline{r}})$ is denoted $U_{\rm tri}^\square({\overline{r}})^{\rm reg}$ in [@BHS §2.2]). The rigid space $X_{\rm tri}^\square({\overline{r}})$ is reduced equidimensional of dimension $n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}$ and its subset $U_{\rm tri}^\square({\overline{r}})\subset X_{\rm tri}^\square({\overline{r}})$ turns out to be Zariski-open, see [@BHS Th.2.6]. Moreover by [*loc. cit.*]{} the rigid variety $U_{\rm tri}^\square({\overline{r}})$ is smooth over $L$ and equidimensional, hence there is a bijection between the set of connected components of $U_{\rm tri}^\square({\overline{r}})$ and the set of irreducible components of $X_{\rm tri}^\square({\overline{r}})$.

We denote by $\omega:X_{\rm tri}^\square({\overline{r}})\rightarrow \mathcal{W}^n_L$ the composition $X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\twoheadrightarrow \mathcal{T}^n_L \twoheadrightarrow \mathcal{W}^n_L$. If $x$ is a point of $X_{\rm tri}^\square({\overline{r}})$, we write $x=(r,\delta)$ where $r\in \mathfrak{X}_{{\overline{r}}}^\square$ and $\delta=(\delta_1,\dots,\delta_n)\in \mathcal{T}_L^n$. We say that a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is *crystalline* if $r$ is a crystalline representation of $\mathcal{G}_K$.

\[paramofcrystpt\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline point. Then for $i\in \{1,\dots,n\}$ there exist ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ and $\varphi_i\in k(x)^\times$ such that: $$\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i).$$ Moreover the $(k_{\tau,i})_{i,\tau}$ are the labelled Hodge-Tate weights of $r$ and the $\varphi_i$ are the eigenvalues of the geometric Frobenius on the (unramified) Weil-Deligne representation ${\rm WD}(r)$ associated to $r$ (cf. [@Fo]).

The fact that the $(k_{\tau,i})_{i,\tau}$ are the Hodge-Tate weights of $r$ follows for instance from [@BHS Prop.2.9]. By [@KPX Th.6.3.13] there exist for each $i$ a continuous character $\delta'_i:K^\times \rightarrow k(x)^\times$ such that $r$ is trianguline of parameter $\delta':=(\delta'_1,\dots,\delta'_n)$ and such that $\delta_i/\delta'_i$ is an algebraic character of $K^\times$ (i.e. of the form $z^{{\bf k}}$ for some ${\bf k}\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$). It thus suffices to prove that each $\delta'_i$ is of the form $z^{{\bf k}'_i}{\rm unr}(\varphi_i)$ for some ${\bf k}'_i\in{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$ where the $\varphi_i\in k(x)^\times$ are the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$, or equivalently (using the definition of ${\rm WD}(r)$) are the eigenvalues of the linearized Frobenius $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on the $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-module $D_{\rm cris}(r):=(B_{\rm cris}\otimes_{{\mathbb{Q}}_p}r)^{\mathcal{G}_K}$. By [@Berger Th.3.6] there is an isomorphism (recall $t$ is “Fontaine’s $2i\pi$”): $$\begin{aligned}
\label{berger}
D_{\rm cris}(r)\cong D_{{\mathrm{rig}}}(r)[\tfrac{1}{t}]^{\Gamma_K},\end{aligned}$$ and a triangulation ${\mathrm{Fil}}_\bullet$ of $D_{{\mathrm{rig}}}(r)$ with graded pieces giving the parameter $\delta'$ induces a complete $\varphi$-stable filtration ${\mathcal{F}}_\bullet$ on $D_{\rm cris}(r)$ such that ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ is the filtered $\varphi$-module associated to ${\mathcal{R}}_{L,K}(\delta'_i)={\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}$ by the same recipee as (\[berger\]). It follows from this and from [@KPX Example 6.2.6(3)] that $\delta'_i$ is of the form $z^{{\bf k}'_i}{\rm unr}(a)$ where $a\in k(x)^\times$ is the unique element such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on the underlying $\varphi$-module of ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $1\otimes a\in K_0\otimes_{{\mathbb{Q}}_p} k(x)$. This finishes the proof.

Note that Lemma \[paramofcrystpt\] implies that if $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is a crystalline point, then $\omega(x)$ is algebraic ($=\delta_{\bf k}$ for ${\bf k}:=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ where the $k_{\tau,i}$ are as in Lemma \[paramofcrystpt\]). We say that a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ such that $\omega(x)$ is algebraic is [*dominant*]{} (resp. [*strictly dominant*]{}) if $\omega(x)$ is dominant (resp. strictly dominant).

A variant of the crystalline deformation space {#variant}
----------------------------------------------

We define a certain irreducible component $Z_{{\rm tri},U}(x)$ of a sufficiently small open neighbouhood $U\subseteq X_{\rm tri}^\square({\overline{r}})$ containing a crystalline strictly dominant point $x$ (Corollary \[defnZtri(x)\]).

We fix ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in ({\mathbb{Z}}^n)^{{\mathrm{Hom}}(K,L)}$ such that $k_{\tau,i}> k_{\tau,i+1}$ for all $i,\tau$ and write $R_{\overline r}^{\square,{\bf k}{\rm -cr}}$ for the crystalline deformation ring of $\overline r$ with Hodge-Tate weights ${\bf k}$, i.e. the reduced and ${\mathbb{Z}}_p$-flat quotient of $R^\square_{\overline r}$ such that, for any finite extension $L'$ of $L$, a morphism $x:{\mathrm{Spec}}\, L'\rightarrow {\mathrm{Spec}}\, R_{{\overline{r}}}^\square$ factors through ${\mathrm{Spec}}\, R_{\overline r}^{\square, {\bf k}{\rm -cr}}$ if and only if the representation ${\mathcal{G}}_K\rightarrow {\mathrm{GL}}_n(L')$ defined by $x$ is crystalline with labelled Hodge-Tate weights $(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$. That this ring exists is the main result of [@Kisindef]. We write ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ for the rigid analytic space associated to ${\mathrm{Spf}}\, R_{\overline r}^{\square,{\bf k}{\rm -cr}}$. By [@Kisindef], it is smooth over $L$.

Let $\tilde r:{\mathcal G}_K\rightarrow {\mathrm{GL}}_n(R_{\overline r}^{\square,{\bf k}{\rm -cr}})$ be the corresponding universal deformation. By [@Kisindef Th.2.5.5] or [@BergerColmez Cor.6.3.3] there is a coherent $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$-module ${\mathcal{D}}$ that is locally on ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ free over $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$ together with a $\varphi\otimes {\mathrm{id}}$-linear automorphism $\Phi_{\rm cris}$ such that: $$({\mathcal{D}},\Phi_{\rm cris})\otimes_{{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}} k(x)\cong D_{\rm cris}\big(\tilde r\otimes_{R_{\overline r}^{\square,{\bf k}{\rm -cr}}} k(x)\big)$$ for all $x\in {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$. Fixing an embedding $\tau_0:K_0\hookrightarrow L$ we can define the associated family of Weil-Deligne representations: $$({\rm WD}(\tilde r),\Phi):=\big({\mathcal{D}}\otimes_{K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}},\tau_0\otimes{\mathrm{id}}}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}},\Phi_{\rm cris}^{[K_0:{\mathbb{Q}}_p]}\otimes{\mathrm{id}}\big)$$ on ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ whose isomorphism class does not depend on the choice of the embedding $\tau_0$.

Let $T^{\mathrm{rig}}\cong ({\mathbb G}_{{\rm m}}^{\mathrm{rig}})^{n}$ be the rigid analytic space over ${\mathbb{Q}}_p$ associated to the diagonal torus $T\subset {\mathrm{GL}}_{n}$ and let ${\mathcal{S}}_n$ be the Weyl group of $({\mathrm{GL}}_{n},T)$ acting on $T$, and thus on $T^{\mathrm{rig}}$, in the usual way. Recall that the map: $${\mathrm{diag}}(\varphi_1,\varphi_2,\dots,\varphi_n)\mapsto {\rm coefficients\ of}\ (X-\varphi_1)(X-\varphi_2)\dots (X-\varphi_n)$$ induces an isomorphism of schemes over ${\mathbb{Q}}_p$: $$T/{\mathcal{S}}_n\buildrel\sim\over\longrightarrow {\mathbb G}_{{\rm a}}^{n-1}\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p} {\mathbb G}_{{\rm m}}$$ and also of the associated rigid analytic spaces. We deduce that the coefficients of the characteristic polynomial of the Frobenius $\Phi$ on ${\rm WD}(\tilde r)$ determine a morphism of rigid analytic spaces over $L$: $${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}\longrightarrow T_L^{{\mathrm{rig}}}/{\mathcal{S}}_n.$$ Let us define: $$\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}:={\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}.$$ Concretely $\widetilde{\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm-cr}$ parametrizes crystalline framed ${\mathcal{G}}_K$-deformations $r$ of $\overline r$ of labelled Hodge-Tate weights ${\bf k}$ together with an ordering $(\varphi_1,\dots,\varphi_n)$ of the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$.

\[reduced\] The rigid analytic space $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is reduced.

It is sufficient to prove this result locally. Let $\mathrm{Sp}\,C$ be an admissible irreducible affinoid open subspace of $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ whose image in $T_L^{\rm rig}/\mathcal{S}_n$ is contained in an admissible affinoid open irreducible subspace $\mathrm{Sp}\,A$ of $T_L^{\rm rig}/\mathcal{S}_n$. As both $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ and $T_L^{\rm rig}/\mathcal{S}_n$ are smooth over $L$ we can find an admissible open affinoid covering of $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$ by such $\mathrm{Sp}\,C$. The map $T_L^{\rm rig}\rightarrow T_L^{\rm rig}/\mathcal{S}_n$ is finite flat being the rigidification of a map of affine schemes $T_L\rightarrow T_L/\mathcal{S}_n$ which is finite flat. Consequently the inverse image of $\mathrm{Sp}\,A$ in $T_L^{\rm rig}$ is an admissible affinoid open subspace $\mathrm{Sp}\,B$ with $B$ an affinoid algebra which is finite flat over $A$. As $B$ is a finite $A$-algebra, we have an isomorphism $C\widehat{\otimes}_AB\simeq C\otimes_AB$. It follows, by definition of the fiber product of rigid analytic spaces, that the rigid analytic spaces of the form $\mathrm{Sp}(C\otimes_AB)$ form an admissible open covering of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}=\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\times_{T_L^{\rm rig}/\mathcal{S}_n}T_L^{\rm rig}$. It is sufficient to prove that rings $C\otimes_AB$ as above are reduced. From Lemma \[red\] below it is sufficient to prove that $C\otimes_AB$ is a finite flat generically étale $C$-algebra. As $B$ is finite flat over $A$, the $C$-algebra $C\otimes_AB$ is clearly finite flat. It is sufficient to prove that it is a generically étale $C$-algebra. As $B$ is generically étale over $A$, it is sufficient to prove that the map $\mathrm{Spec}\,C\rightarrow\mathrm{Spec}\,A$ is dominant. It is thus sufficient to prove that the map of rigid analytic spaces $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\rightarrow T_L^{\rm rig}/\mathcal{S}_n$ is open. This follows from the fact that it has, locally on $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$, a factorization: $$\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\longrightarrow ({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0} \times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p} {\rm Flag})^{\mathrm{rig}}\times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p}{\mathrm{Sp}}\,L\longrightarrow T_L^{\rm rig}/\mathcal{S}_n$$ where the first map is the smooth map in the proof of Lemma \[genericopensinXcris\] below, and the second is the projection on $(\mathrm{Res}_{K_0/\mathbb{Q}_p}\mathrm{GL}_{n,K_0})_L^{\rm rig}$ followed by the base change to $L$ of the rigidification of the morphism $\mathrm{Res}_{K_0/\mathbb{Q}_p}\mathrm{GL}_{n,K_0}\rightarrow T/\mathcal{S}_n$ defined in [@HartlHellmann (9.1)]. The first map being smooth is flat and thus open by [@Bo Cor.9.4.2], and the last two are easily seen to be open.

The following (well-known) lemma was used in the proof of Lemma \[reduced\].

\[red\] Let $A$ a commutative noetherian domain and $B$ a finite flat $A$-algebra. Then the ring $B$ has no embedded component, i.e. all its associated ideals are minimal prime ideals. Moreover if $B$ is generically étale over $A$, i.e. $\mathrm{Frac}(A)\otimes_A B$ is a finite étale $\mathrm{Frac}(A)$-algebra, then the ring $B$ is reduced.

As $B$ is flat over $A$, the map $\mathrm{Spec}\,B\rightarrow\mathrm{Spec}\,A$ has an open image, and $A$ being a domain it contains the unique generic point of $\mathrm{Spec}\,A$, which implies that the natural map $A\rightarrow B$ is injective. Moreover $B$ being finite over $A$, the image of $\mathrm{Spec}\,B\rightarrow\mathrm{Spec}\,A$ is closed, hence it is $\mathrm{Spec}\,A$ since $\mathrm{Spec}\,A$ is connected. In particular $B$ is a faithfully flat $A$-algebra. As $B$ is a flat $A$-module, it follows from [@BourbakiAC §IV.2.6 Lem.1] applied with $E=A$ and $F=B$ that $\mathfrak{p}\in\mathrm{Ass}(B)$ implies $\mathfrak{p}\cap A=0$ ($A$ is a domain, so $\mathrm{Ass}(A)=\{0\}$). It then follows from [@BourbakiAC §V.2.1 Cor.1] that if $\mathfrak{p}\in\mathrm{Ass}(B)$, then $\mathfrak{p}$ is a minimal prime of $B$. Indeed, $A$ being noetherian and $B$ a finite $A$-module, $B$ is an integral extension of $A$. We can apply [*loc. cit.*]{} to the inclusion $\mathfrak{q}\subseteq\mathfrak{p}$ where $\mathfrak{q}$ is a minimal prime ideal of $B$ (both ideals $\mathfrak{q}$ and $\mathfrak{p}$ being above the prime ideal $(0)$ of $A$ since $\mathfrak{p}\cap A=\mathfrak{q}\cap A=0$).

Let $\mathfrak{d}_{B/A}$ be the discriminant of $B/A$ (its existence comes from the fact that $B$ is a finite faithfully flat $A$-algebra, hence a finite projective $A$-module). As the extension is generically étale, we can find $f\in\mathfrak{d}_{B/A}$ such that $B_f$ is étale over $A_f$. As $A_f$ is a domain, $B_f$ is then reduced. Thus the nilradical $\mathfrak{n}$ of $B_f$ is killed by some power of $f$. Replacing $f$ by this power, we can assume that the vanishing ideal of $\mathfrak{n}$ contains $f$. Assume that $\mathfrak{n}$ is nonzero and let $\mathfrak{p}$ be a prime ideal of $B$ minimal among prime ideals containing $\mathrm{Ann}_B(\mathfrak{n})$. It follows from [@BourbakiAC §IV.1.3 Cor.1] that $\mathfrak{p}$ is an associated prime of the $B$-module $\mathfrak{n}$ and consequently of $B$. But we have $f\in\mathfrak{p}$ which contradicts the fact that $\mathfrak{p}\cap A=0$.

We now embed this “refined” crystalline deformation space $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ into the space $X_{\rm tri}^\square({\overline{r}})$ as follows. We define a morphism of rigid spaces over $L$: $$\begin{aligned}
\label{cristotriang}
{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{{\mathrm{Sp}}\,L} T_L^{\mathrm{rig}}&\longrightarrow &{\mathfrak{X}}_{\overline r}^\square\times_{{\mathrm{Sp}}\,L}{\mathcal{T}}^n_L\\
\nonumber (r,\varphi_1,\dots,\varphi_n)&\longmapsto & (r, z^{{\bf k}_1}{\rm unr}(\varphi_1),\dots, z^{{\bf k}_n}{\rm unr}(\varphi_n)).\end{aligned}$$ This morphism is a closed embedding of reduced rigid spaces as both maps $r\mapsto r$ and $(\varphi_1,\dots,\varphi_n)\mapsto (z^{{\bf k}_1}{\rm unr}(\varphi_1),\dots, z^{{\bf k}_n}{\rm unr}(\varphi_n))$ respectively define closed embeddings ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm-cr}\!\!\hookrightarrow {\mathfrak{X}}_{\overline r}^\square$ and $T_L^{\mathrm{rig}}\hookrightarrow {\mathcal{T}}^n_L$. We claim that the restriction of the morphism $(\ref{cristotriang})$ to: $$\label{cristotriangres}
\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\times_{{\mathrm{Sp}}\,L}T_L^{\mathrm{rig}}$$ factors through $X_{\rm tri}^\square(\overline r)\subset {\mathfrak{X}}_{\overline r}^\square\times_{{\mathrm{Sp}}\,L}{\mathcal{T}}^n_L$. As the source of this restriction is reduced by Lemma \[reduced\], it is enough to check it on a Zariski-dense set of points of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$.

Let $r$ be an $n$-dimensional crystalline representation of ${\mathcal G}_K$ over a finite extension $L'$ of $L$ of Hodge-Tate weights ${\bf k}$ and let $\varphi_1,\dots,\varphi_n$ be an ordering of the eigenvalues of a geometric Frobenius on ${\rm WD}(r)$, equivalently of the eigenvalues of $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on $D_{\rm cris}(r)$ (that are assumed to be in $L'^\times$). Assuming moreover that the $\varphi_i$ are pairwise distinct, this datum gives rise to a unique complete $\varphi$-stable flag of free $K_0\otimes_{{\mathbb{Q}}_p}L'$-modules: $$0={\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \dots\subset {\mathcal{F}}_n=D_{\rm cris}(r)$$ on $D_{\rm cris}(r)$ such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $\varphi_i$ (this is a *refinement* in the sense of [@BelChe Def.2.4.1]). By the same argument as in the proof of Lemma \[paramofcrystpt\] using Berger’s dictionary between crystalline $(\varphi,\Gamma_K)$-modules and filtered $\varphi$-modules (see e.g. (\[berger\])), the filtration ${\mathcal{F}}_\bullet$ induces a triangulation ${\mathrm{Fil}}_{\bullet}$ on $D_{{\mathrm{rig}}}(r)$. If we assume that ${\mathcal{F}}_\bullet$ is [*noncritical*]{} in the sense of [@BelChe Def.2.4.5], i.e. the filtration ${\mathcal{F}}_\bullet$ is in general position with respect to the Hodge filtration ${\mathrm{Fil}}^\bullet D_{\rm dR}(r)$ on $D_{\rm dR}(r)$, that is, for all embeddings $\tau:K\hookrightarrow L$ and all $i=1,\dots,n-1$ we have: $$\begin{gathered}
\label{generalpos}
\big({\mathcal{F}}_{i}\otimes_{K_0\otimes_{{\mathbb{Q}}_p}L',\tau\otimes {\mathrm{id}}}L'\big)\oplus \big({\mathrm{Fil}}^{-k_{\tau,i+1}}D_{\rm dR}(r)\otimes_{K\otimes_{{\mathbb{Q}}_p}L',\tau\otimes {\mathrm{id}}}L'\big)=D_{\rm cris}(r)\otimes_{K_0\otimes_{{\mathbb{Q}}_p}L,\tau\otimes {\mathrm{id}}}L'\\
=D_{\rm dR}(r)\otimes_{K\otimes_{{\mathbb{Q}}_p}L',\tau\otimes {\mathrm{id}}}L',\end{gathered}$$ then ${\mathcal{F}}_{i}/{\mathcal{F}}_{i-1}$ is a filtered $\varphi$-module of Hodge-Tate weights ${\bf k}_i$, or equivalently ${\mathrm{Fil}}_i/{\mathrm{Fil}}_{i-1}\cong {\mathcal{R}}_{L',K}(\delta_i)$ with $\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i)$.

\[genericopensinXcris\] There are smooth (over $L$) Zariski-open and Zariski-dense subsets in $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$: $$\widetilde V_{\overline r}^{\square,{\bf k}\rm -cr} \subset \widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}\subset\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$$ such that:

1.  a point $(r,\varphi_1,\dots,\varphi_n)\in \widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ lies in $\widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}$ if and only if the $\varphi_i$ are pairwise distinct;\

2.  a point $(r,\varphi_1,\dots,\varphi_n)\in \widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}$ lies in $\widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}$ if and only if it satisfies assumption $(\ref{generalpos})$ above and $z^{{\bf k}_i-{\bf k}_j}{\rm unr}(\varphi_i\varphi_j^{-1})\in{\mathcal{T}}_{\rm reg}$ for $i\ne j$.

Moreover the image of $\widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}$ via (\[cristotriang\]) composed with (\[cristotriangres\]) lies in $U_{\rm tri}^\square(\overline r)$.

The idea of the proof is the same as that of [@Chenevier Lem.4.4]. It is enough to show that all the statements are true locally on $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Let us (locally) fix a basis of the coherent locally free $K_0\otimes_{{\mathbb{Q}}_p}{\mathcal{O}}_{\!{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}}$-module ${\mathcal{D}}$ on ${\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm -cr}$. By the choice of such a basis, the matrix of the crystalline Frobenius $\Phi_{\rm cris}$ and the Hodge filtration define (locally) a morphism: $${\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm -cr}\longrightarrow ({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0} \times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p} {\rm Flag})^{\mathrm{rig}}\times_{{\mathrm{Sp}}\,{\mathbb{Q}}_p}{\mathrm{Sp}}\,L$$ where ${\rm Flag}:=({\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K})/({\mathrm{Res}}_{K/{\mathbb{Q}}_p} B)$ (compare [@HartlHellmann §8]). By [@HartlHellmann Prop.8.12] and the discussion preceding it, it follows that this morphism is smooth, hence so is the morphism: $$\label{morphismbasechange}
\widetilde {\mathfrak{X}}_{{\overline{r}}}^{\square,{\bf k}\rm -cr}\longrightarrow \big(({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0})^{\mathrm{rig}}_L\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}\big)\times_{{\mathrm{Sp}}\,L} {\rm Flag}_L^{\mathrm{rig}}$$ where ${\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0}\rightarrow T/{\mathcal{S}}_n$ is the morphism defined in [@HartlHellmann (9.1)]. On the other hand, using that the morphism $T\rightarrow T/{\mathcal{S}}_n$ is obviously smooth in the neighbourhood of a point $(\varphi_1,\dots,\varphi_n)\in T$ where the $\varphi_i$ are pairwise distinct, we see that the conditions of (i), resp. (ii), in the statement cut out smooth (over $L$) Zariski-open and Zariski-dense subspaces of: $$\label{RHS}
\big(({\mathrm{Res}}_{K_0/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K_0})^{\mathrm{rig}}_L\times_{T_L^{\mathrm{rig}}/{\mathcal{S}}_n}T_L^{\mathrm{rig}}\big)\times_{{\mathrm{Sp}}\,L} {\rm Flag}_L^{\mathrm{rig}}.$$ Their inverse images in $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ via (\[morphismbasechange\]) are thus smooth over $L$ and Zariski-open in $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Let us prove that these inverse images are also Zariski-dense in $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. It is enough to prove that they intersect nontrivially every irreducible component of $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Let ${\mathrm{Sp}}\,A$ be any affinoid open subset of $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$, it follows from [@Bo Cor.9.4.2] that the image of ${\mathrm{Sp}}\,A$ by the smooth, hence flat, morphism (\[morphismbasechange\]) is admissible open in (\[RHS\]). In particular its intersection with one of the above Zariski-open and Zariski-dense subspaces of (\[RHS\]) can’t be empty, which proves the statement. The final claim of the lemma follows from (ii), the discussion preceding Lemma \[genericopensinXcris\] and the definition (\[ucris\]) of $U_{\rm tri}^\square(\overline r)$ .

Note that $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is equidimensional of the same dimension as ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$. Indeed, by Lemma \[genericopensinXcris\] it is enough to prove the same statement for $\widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}$. But this is clear since the map $\widetilde U_{\overline r}^{\square,{\bf k}\rm -cr}\rightarrow {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is smooth of relative dimension $0$, hence étale, and since ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}$ is equidimensional ([@Kisindef]). Lemma \[genericopensinXcris\] also implies that $(\ref{cristotriang})$ induces (as claimed above) a morphism: $$\label{embeddingcristri}
\iota_{\bf k}:\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\longrightarrow X_{\rm tri}^\square(\overline r)$$ which is obviously a closed immersion as $(\ref{cristotriang})$ is.

\[defnZtri(x)\] Let $x=(r,\delta)\in X_{\rm tri}^\square(\overline r)$ be a crystalline strictly dominant point such that $\omega(x)=\delta_{\bf k}$ and the Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ (cf. Lemma \[paramofcrystpt\]) are pairwise distinct and let $U$ be an open subset of $X_{\rm tri}^\square(\overline r)$ containing $x$.\
(i) The point $x$ belongs to $\iota_{\bf k}(\widetilde U_{\overline r}^{\square,{\bf k}{\rm -cr}})$ and there is a unique irreducible component $\widetilde Z_{\rm cris}(x)$ of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ containing $\iota_{\bf k}^{-1}(x)$.\
(ii) If $U$ is small enough there is a unique irreducible component $Z_{{\rm tri},U}(x)$ of $U$ containing $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$, and it is such that $Z_{{\rm tri},U}(x)\cap U'=Z_{{\rm tri},U'}(x)$ for any open $U'\subseteq U$ containing $x$.

\(i) The assumptions and Lemma \[paramofcrystpt\] imply that $x$ is in the image of the map $\iota_{\bf k}$ in (\[embeddingcristri\]) and the fact that the $\varphi_i$ are pairwise distinct implies that $x\in \iota_{\bf k}(\widetilde U_{\overline r}^{\square,{\bf k}{\rm -cr}})$. In particular $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ is smooth at $\iota_{\bf k}^{-1}(x)$ by Lemma \[genericopensinXcris\] and thus $\iota_{\bf k}^{-1}(x)$ belongs to a unique irreducible component $\widetilde Z_{\rm cris}(x)$ of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$.

\(ii) We have that $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$ is a Zariski-closed subset of $U$, and it is easy to see that it is still irreducible if $U$ is small enough since $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$ is smooth at $x$. Hence there exists at least one irreducible component of $U$ containing the irreducible Zariski-closed subset $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$. If there are two such irreducible components, then in particular [*any*]{} point of $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U$ is singular in $U$, hence in $X_{\rm tri}^\square(\overline r)$. But Lemma \[genericopensinXcris\] implies $\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}\ne \emptyset$ is Zariski-open and Zariski-dense in $\widetilde Z_{\rm cris}(x)$, hence: $$\big(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr}\big)\cap \big(\widetilde Z_{\rm cris}(x)\cap \iota_{\bf k}^{-1}(U)\big) \ne \emptyset$$ from which we get $\iota_{\bf k}(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr})\cap U\ne \emptyset$. The last statement of Lemma \[genericopensinXcris\] also implies $\iota_{\bf k}(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}\rm -cr})\cap U\subseteq U_{\rm tri}^\square(\overline r)$, which is then a contradiction since $U_{\rm tri}^\square(\overline r)$ is smooth over $L$.

Finally, shrinking $U$ again if necessary, we can assume that, for any open subset $U'\subseteq U$ containing $x$, the map $Z\mapsto Z\cap U'$ induces a bijection between the irreducible components of $U$ containing $x$ and the irreducible components of $U'$ containing $x$. It then follows from the definition of $Z_{{\rm tri},U}(x)$ that $Z_{{\rm tri},U}(x)\cap U'=Z_{{\rm tri},U'}(x)$.

\[down\] [(i) Since the map $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}\rightarrow {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ is finite, hence closed, and since $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$, ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ are both equidimensional (of the same dimension), the image of any irreducible component of $\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ is an irreducible component of ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$. In particular the image of $\widetilde Z_{\rm cris}(x)$ in (i) of Corollary \[defnZtri(x)\] is the unique irreducible component of ${\mathfrak{X}}_{\overline r}^{\square,{\bf k}{\rm -cr}}$ containing $r$.\
(ii) Either by the same proof as that for (ii) of Corollary \[defnZtri(x)\] or as a consequence of (ii) of Corollary \[defnZtri(x)\], we see that there is also a unique irreducible component $Z_{{\rm tri}}(x)$ of the whole $X_{\rm tri}^\square(\overline r)$ which contains the irreducible closed subset $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$.]{}

The Weyl group element associated to a crystalline point {#weyl}
--------------------------------------------------------

We review the definition of the Weyl group element associated to certain crystalline points on $X_{\rm tri}^\square(\overline r)$ (measuring their “criticality”) and state our main local results (Theorem \[upperbound\], Corollary \[Xtrismooth\]).

We keep the notation of §\[variant\]. We let $W\cong \prod_{\tau:\, K\hookrightarrow L}{\mathcal S}_n$ be the Weyl group of the algebraic group: $$({\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K})\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L\cong \prod_{\tau:\, K\hookrightarrow L}{{\mathrm{GL}}_{n,L}}$$ and $X^\ast(({\mathrm{Res}}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L)\cong \prod_{\tau:\, K\hookrightarrow L}X^\ast(T_L)$ be the ${\mathbb{Z}}$-module of algebraic characters of $({\mathrm{Res}}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L$ (recall $T$ is the diagonal torus in ${\mathrm{GL}}_n$ and $T_K$, $T_L$ its base change to $K$, $L$). We write ${\rm lg}(w)$ for the length of $w$ in the Coxeter group $W$ (for the set of simple reflections associated to the simple roots of the upper triangular matrices).

Let $x=(r,\delta)=(r,\delta_1,\dots,\delta_n)$ be a crystalline strictly dominant point on $X_{\rm tri}^\square({\overline{r}})$. Then by Lemma $\ref{paramofcrystpt}$ the characters $\delta_i$ are of the form $\delta_i=z^{{\bf k}_i}{\rm unr} (\varphi_i)$ where ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}$ and the $\varphi_i\in k(x)^\times$ are the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$. Assume that the $\varphi_i$ are pairwise distinct, then as in §\[variant\] the ordering $(\varphi_1,\dots,\varphi_n)$ defines a complete $\varphi$-stable flag of free $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-modules $0={\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \dots\subset {\mathcal{F}}_n=D_{\rm cris}(r)$ on $D_{\rm cris}(r)$ such that $\varphi^{[K_0:{\mathbb{Q}}_p]}$ acts on ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$ by multiplication by $\varphi_i$. We view ${\mathcal{F}}_i$ as a filtered $\varphi$-module with the induced Hodge filtration. If we write $(k'_{\tau,i})_{\tau:K\hookrightarrow L}$ for the Hodge-Tate weights of ${\mathcal{F}}_i/{\mathcal{F}}_{i-1}$, we find that there is a unique $w_x=(w_{x,\tau})_{\tau:\, K\hookrightarrow L}\in W=\prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$ such that: $$\label{wxdef}
k'_{\tau, i}=k_{\tau, w_{x,\tau}^{-1}(i)}$$ for all $i\in \{1,\dots, n\}$ and each $\tau: K\hookrightarrow L$. We call $w_x$ the *Weyl group element associated to* $x$. Note that ${\mathcal{F}}_\bullet$ is noncritical (see §\[variant\]) if and only if $w_{x,\tau}=1$ for all $\tau:K\hookrightarrow L$, in which case we say that the crystalline strictly dominant point $x=(r,\delta)$ is [*noncritical*]{}.

For $w\in W$ we denote by $d_w\in {\mathbb{Z}}_{\geq 0}$ the rank of the ${\mathbb{Z}}$-submodule of $X^\ast(({\mathrm{Res}}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L)$ generated by the $w(\alpha)-\alpha$ where $\alpha$ runs among the roots of $({\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,K})\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L$.

\[coxeter\] With the above notations we have: $$d_w\leq \lg(w)\leq [K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}$$ and $\lg(w)= d_w$ if and only if $w$ is a product of distinct simple reflections.

Note first that the right hand side inequality is obvious. Let us write in this proof $X:=X^\ast(({\rm Res}_{K/{\mathbb{Q}}_p}T_K)\times_{{\mathrm{Spec}}\,{\mathbb{Q}}_p}{\mathrm{Spec}}\,L)$, $X_{{\mathbb{Q}}}:=X\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$, and let us denote by $S$ the subset of simple reflections in $W$ (thus $\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})=[K:{\mathbb{Q}}_p]n$ and $\vert S\vert=[K:{\mathbb{Q}}_p](n-1)$). The rank of the subgroup of $X$ generated by the $w(\alpha)-\alpha$ for $\alpha$ as above, or equivalently by the $w(\alpha)-\alpha$ for $\alpha\in X$, is equal to the dimension of the ${\mathbb{Q}}$-vector space $(w-{\mathrm{id}})X_{{\mathbb{Q}}}$ which, by the rank formula, is equal to $\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})-\dim_{{\mathbb{Q}}}(\ker(w-{\mathrm{id}}))$. Let $I$ be the set of simple reflections appearing in $w$, we have $|I|\leq \lg(w)$ and $|I|= \lg(w)$ if and only if $w$ is a product of distinct simple reflections. It is thus enough to prove $\dim_{{\mathbb{Q}}}(\ker(w-{\mathrm{id}}))\geq \dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})-|I|$ with equality when $w$ is a product of distinct simple reflections. Note that $\ker(w-{\mathrm{id}})$ obviously contains the ${\mathbb{Q}}$-subvector space of $X_{\mathbb{Q}}$ of fixed points by the subgroup $W_I$ of $W$ generated by the elements of $I$, and it follows from [@Hum Th.1.12(c)] that, when $w$ is a product of distinct simple reflections, then $\ker(w-{\mathrm{id}})$ is exactly this ${\mathbb{Q}}$-subvector space. It is thus enough to prove that this ${\mathbb{Q}}$-subvector space of $X_{\mathbb{Q}}$, which is just the intersection of the hyperplanes $\ker(s-{\mathrm{id}})$ for $s\in I$, has dimension $\dim_{{\mathbb{Q}}}(X_{\mathbb{Q}})-|I|$. However we know that for any ${\mathbb{Q}}$-subvector space $V\subset X_{{\mathbb{Q}}}$ and any reflection $s$ of $X_{{\mathbb{Q}}}$, we have $\dim_{{\mathbb{Q}}}(V\cap\ker(s-{\mathrm{id}}))\geq \dim_{{\mathbb{Q}}}(V)-1$ and thus by induction: $$\dim_{{\mathbb{Q}}}\big(V\bigcap\big(\bigcap_{s\in S}\ker(s-{\mathrm{id}})\big)\big)\geq \dim_{{\mathbb{Q}}}(V)-|S|$$ with equality if and only if $\dim_{{\mathbb{Q}}}\big(V\bigcap\big(\bigcap_{s\in J}\ker(s-{\mathrm{id}})\big)\big)= \dim_{{\mathbb{Q}}}(V)-|J|$ for all $J\subseteq S$. As the ${\mathbb{Q}}$-subvector space $X_{{\mathbb{Q}}}^W$ of fixed points by $W$ has dimension $[K:{\mathbb{Q}}_p]$ (it is generated by the characters $\tau\circ\det$ for $\tau:K\hookrightarrow L$), we have: $$\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}}^W)=\bigcap_{s\in S}\ker(s-{\mathrm{id}})=[K:{\mathbb{Q}}_p]=\dim_{{\mathbb{Q}}}(X_{{\mathbb{Q}}})-|S|.$$ Consequently we deduce (taking $V=X_{\mathbb{Q}}$): $$\dim_{{\mathbb{Q}}}\big(\bigcap_{s\in I}\ker(s-{\mathrm{id}})\big)= \dim_{{\mathbb{Q}}}(X_{\mathbb{Q}})-|I|$$ which is the desired formula.

Recall that, if $X$ is a rigid analytic variety over $L$ and $x\in X$, the tangent space to $X$ at $x$ is the $k(x)$-vector space: $$\begin{aligned}
\label{tangent}
T_{X,x}:={\mathrm{Hom}}_{k(x)}\big({\mathfrak m}_{X,x}/{\mathfrak m}^2_{X,x},k(x)\big)={\mathrm{Hom}}_{k(x)-{\rm alg}}\big({\mathcal O}_{X,x},k(x)[\varepsilon]/(\varepsilon^2)\big)\end{aligned}$$ where ${\mathfrak m}_{X,x}$ is the maximal ideal of the local ring ${\mathcal O}_{X,x}$ at $x$ to $X$. If $X$ is equidimensional, recall also that $\dim_{k(x)}T_{X,x}\geq \dim X$ and that $X$ is smooth at $x$ if and only if $\dim_{k(x)}T_{X,x}= \dim X$.

We let $\widetilde X_{\rm tri}^\square({\overline{r}})\subseteq X_{\rm tri}^\square({\overline{r}})$ be the union of the irreducible components $C$ of $X_{\rm tri}^\square({\overline{r}})$ such that $C\cap U_{\rm tri}^\square({\overline{r}})$ contains a crystalline point. For instance it follows from Lemma \[genericopensinXcris\] that all the closed embeddings (\[embeddingcristri\]) factor as closed embeddings $\iota_{\bf k}:\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow \widetilde X_{\rm tri}^\square(\overline r)\subseteq X_{\rm tri}^\square(\overline r)$. In particular any point $x\in X_{\rm tri}^\square({\overline{r}})$ which is crystalline strictly dominant is in $\widetilde X_{\rm tri}^\square({\overline{r}})$.

The following statement is our main local conjecture.

\[mainconj\] Let $x\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline strictly dominant point such that the Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ (cf. Lemma \[paramofcrystpt\]) are pairwise distinct, let $w_x$ be the Weyl group element associated to $x$ (cf. (\[wxdef\])) and let $d_x:=d_{w_x}$. Then: $$\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}=\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}})=\lg(w_x)-d_{x}+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}.$$

In particular, since $\dim \widetilde X_{\rm tri}^\square({\overline{r}})=\dim X_{\rm tri}^\square({\overline{r}})=n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}$, we see by Lemma \[coxeter\] that $\widetilde X_{\rm tri}^\square({\overline{r}})$ should be smooth at $x$ if and only if $w_x$ is a product of distinct simple reflections.

[The reader can wonder why we don’t state Conjecture \[mainconj\] with $X_{\rm tri}^\square({\overline{r}})$ instead of $\widetilde X_{\rm tri}^\square({\overline{r}})$. The reason is that Conjecture \[mainconj\] with $\widetilde X_{\rm tri}^\square({\overline{r}})$ is actually implied by other conjectures, see §\[modularity\], and we don’t know if this is the case with $X_{\rm tri}^\square({\overline{r}})$.]{}

In order to state our main result in the direction of (a weaker version of) Conjecture \[mainconj\], we need the following two definitions.

\[veryreg\] A crystalline strictly dominant point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ is very regular if it satisfies the following conditions (where the $(\varphi_i)_{1\leq i\leq n}$ are the geometric Frobenius eigenvalues on ${\rm WD}(r)$):

1.  $\varphi_i\varphi_j^{-1}\notin \{1,q\}$ for $1\leq i\ne j\leq n$;\

2.  $\varphi_1\varphi_2\dots\varphi_i$ is a simple eigenvalue of the geometric Frobenius acting on $\bigwedge_{k(x)}^i\!\!{\rm WD}(r)$ for $1\leq i\leq n$.

\[varphi\] [If $x=(r,\delta)$ is crystalline strictly dominant, it easily follows from the dominance property that (i) of Definition \[veryreg\] is equivalent to $\delta_i\delta_j^{-1}\notin \{z^{-\bf h},\vert z\vert_K z^{\bf h},\ \vert z\vert_K^{-1} z^{\bf h},\ {\bf h}\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}\}$ for $1\leq i\ne j\leq n$. In particular it implies $\delta\in {\mathcal{T}}_{\rm reg}^n$, whence the terminology (compare also [@Bergdall §6.1]).]{}

\[accu\] Let $X$ be a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ (over $L$) and let $x\in X_{\rm tri}^\square({\overline{r}})$ such that $\omega(x)$ is algebraic. Then $X$ satisfies the [accumulation property at]{} $x$ if $x\in X$ and if, for any positive real number $C>0$, the set of crystalline strictly dominant points $x'=(r',\delta')$ such that:

1.  the eigenvalues of $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on $D_{\rm cris}(r')$ are pairwise distinct;\

2.  $x'$ is noncritical;\

3.  $\omega(x')=\delta'\vert_{({\mathcal O}_K^\times)^n}=\delta_{\bf k'}$ with $k'_{\tau,i}-k'_{\tau,i+1}>C$ for $i\in \{1,\dots,n-1\}$, $\tau\in {\mathrm{Hom}}(K,L);$

accumulate at $x$ in $X$ in the sense of [@BelChe §3.3.1].

It easily follows from Definition \[accu\] that $X$ satisfies the accumulation property at $x$ if and only if each irreducible component of $X$ containing $x$ satisfies the accumulation property at $x$. In particular, if $x$ belongs to each irreducible component of $X$, we see that for every $C>0$ the set of points $x'$ in the statement of Definition \[accu\] is also Zariski-dense in $X$. Since $U_{\rm tri}^\square({\overline{r}})\cap X$ is Zariski-open and Zariski-dense in $X$, we also see that each irreducible component of $X$ containing $x$ also contains points $x'$ as in Definition \[accu\] which are in $U_{\rm tri}^\square({\overline{r}})$, hence each irreducible component of $X$ containing $x$ is in $\widetilde X_{\rm tri}^\square({\overline{r}})$.

In §\[phiGammacohomology\] below we will prove the theorem that follows.

\[upperbound\] Let $x\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline strictly dominant very regular point and let $X\subseteq X_{\rm tri}^\square({\overline{r}})$ be a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ such that $X$ satisfies the accumulation property at $x$. Then we have: $$\dim_{k(x)}T_{X,x}\leq \lg(w_x)-d_x+\dim X_{\rm tri}^\square({\overline{r}})=\lg(w_x)-d_x+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}.$$

By Lemma \[coxeter\] we thus deduce the following important corollary.

\[Xtrismooth\] Let $x\in X_{\rm tri}^\square({\overline{r}})$ be a crystalline strictly dominant very regular point and let $X\subseteq X_{\rm tri}^\square({\overline{r}})$ be a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ such that $X$ satisfies the accumulation property at $x$. Assume that $w_x$ is a product of distinct simple reflections. Then $X$ is smooth at $x$.

[Note that for $X$, $x$ as above we only have $\dim_{k(x)}T_{X,x}\leq \dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$, thus Theorem \[upperbound\] doesn’t give an upper bound on $\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$ (but Conjecture \[mainconj\] implies Theorem \[upperbound\]). However Theorem \[upperbound\] and Corollary \[Xtrismooth\] will be enough for our purpose.]{}

Crystalline points on the patched eigenvariety {#globalpart}
==============================================

We state the classicality conjecture (Conjecture \[classiconj\]) and prove new cases of it (Corollary \[mainclassic\]).

The classicality conjecture {#classic}
---------------------------

We review the definition of classicality (Definition \[class\], Proposition \[compaclas\]) and state the classicality conjecture (Conjecture \[classiconj\]).

We first recall the global setting, basically the same as [@BHS §2.4]. We fix a totally real field $F^+$, we write $q_v$ for the cardinality of the residue field of $F^+$ at a finite place $v$ and we denote by $S_p$ the set of places of $F^+$ dividing $p$ . We fix a totally imaginary quadratic extension $F$ of $F^+$ that splits at all places of $S_p$ and let ${\mathcal{G}}_F:={\rm Gal}(\overline F/F)$. We fix a unitary group $G$ in $n$ variables over $F^+$ (with $n\geq 2$) such that $G\times_{F^+}F\cong {\mathrm{GL}}_{n,F}$ and $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$ is compact. We fix an isomorphism $i:G\times_{F^+}F\buildrel \sim\over\rightarrow {\mathrm{GL}}_{n,F}$ and, for each $v\in S_p$, a place $\tilde v$ of $F$ dividing $v$. The isomorphisms $F_v^+\buildrel\sim\over\rightarrow F_{\tilde v}$ and $i$ induce an isomorphism $i_{\tilde v}:\,G(F_v^+)\xrightarrow{\sim}{\mathrm{GL}}_n(F_{\tilde v})$ for $v\in S_p$. We let $G_v:=G(F^+_v)\cong {\mathrm{GL}}_n(F_{\tilde v})$ and $G_p:=\prod_{v\in S_p}G(F^+_v)\cong \prod_{v\in S_p}{\mathrm{GL}}_n(F_{\tilde v})$. We denote by $K_v$ (resp. $B_v$, resp. $\overline B_v$, resp. $T_v$) the inverse image of ${\mathrm{GL}}_n(\mathcal{O}_{F_{\tilde{v}}})$ (resp. of the subgroup of upper triangular matrices of ${\mathrm{GL}}_n(F_{\tilde v})$, resp. of the subgroup of lower triangular matrices of ${\mathrm{GL}}_n(F_{\tilde v})$, resp. of the subgroup of diagonal matrices of ${\mathrm{GL}}_n(F_{\tilde v})$) in $G_v$ under $i_{\tilde v}$ and we let $K_p:=\prod_{v\in S_p}K_v$ (resp. $B_p:=\prod_{v\in S_p}B_v$, resp. $\overline B_p:=\prod_{v\in S_p}\overline B_v$, resp. $T_p:=\prod_{v\in S_p}T_v$). We let $T_p^0:=T_p\cap K_p=\prod_{v\in S_p}(T_v\cap K_v)$.

We fix a finite extension $L$ of ${\mathbb{Q}}_p$ that is assumed to be large enough so that $|{\mathrm{Hom}}(F^+_v,L)|=[F^+_v:{\mathbb{Q}}_p]$ for $v\in S_p$. We let $\widehat{T}_{p,\rm reg}\subset \widehat{T}_{p,L}$ the open subspace of characters $\delta=(\delta_v)_{v\in S_p}=(\delta_{v,1},\dots,\delta_{v,n})_{v\in S_p}$ such that $\delta_{v,i}/\delta_{v,j}\in {\mathcal T}_{v,\rm reg}$ for all $v\in S_p$ and all $i\ne j$, where ${\mathcal T}_{v,\rm reg}$ is defined as ${\mathcal T}_{\rm reg}$ of §\[begin\] but with $F^+_v=F_{\tilde v}$ instead of $K$.

We fix a tame level $U^p=\prod_{v}U_v\subset G({\mathbb{A}}_{F^+}^{p\infty})$ where $U_v$ is a compact open subgroup of $G(F_v^+)$ and we denote by $\widehat S(U^p,L)$ the associated space of $p$-adic automorphic forms on $G({\mathbb{A}}_{F^+})$ of tame level $U^p$ with coefficients in $L$, that is, the $L$-vector space of continuous functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p\longrightarrow L$. Since $G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p$ is compact, it is a $p$-adic Banach space (for the sup norm) endowed with the linear continuous unitary action of $G_p$ by right translation on functions. In particular a unit ball is given by the ${\mathcal{O}}_{L}$-submodule $\widehat S(U^p,{\mathcal{O}}_{L})$ of continuous functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p\longrightarrow {\mathcal{O}}_{L}$ and the corresponding residual representation is the $k_{L}$-vector space $S(U^p,k_{L})$ of locally constant functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^p\longrightarrow k_{L}$ (a smooth admissible representation of $G_p$). Note that $S(U^p,k_L)=\varinjlim_{U_p}S(U^pU_p,k_L)$ where the inductive limit is taken over compact open subgroups $U_p$ of $G_p$ and where $S(U^pU_p,k_L)$ is the finite dimensional $k_L$-vector space of functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})/U^pU_p\longrightarrow k_L$. We also denote by $\widehat S(U^p,L)^{\rm an}\subset \widehat S(U^p,L)$ the $L$-subvector space of locally ${\mathbb{Q}}_p$-analytic vectors for the action of $G_p$ ([@STdist §7]). This is a strongly admissible locally ${\mathbb{Q}}_p$-analytic representation of $G_p$.

We fix $S$ a finite set of finite places of $F^+$ that split in $F$ containing $S_p$ and the set of finite places $v\nmid p$ (that split in $F$) such that $U_v$ is not maximal. We consider the commutative spherical Hecke algebra: $$\mathbb{T}^S:=\varinjlim_I\big(\bigotimes_{v\in I}{\mathcal{O}}_L[U_v\backslash G(F^+_v)/U_v]\big),$$ the inductive limit being taken over finite sets $I$ of finite places of $F^+$ that split in $F$ such that $I\cap S=\emptyset$. This Hecke algebra naturally acts on the spaces $\widehat S(U^p,L)$, $\widehat S(U^p,L)^{\rm an}$, $\widehat S(U^p,{\mathcal{O}}_L)$, $S(U^p,k_L)$ and $S(U^pU_p,k_L)$ (for any compact open subgroup $U_p$). If $C$ is a field, $\theta:\mathbb{T}^S\rightarrow C$ a ring homomorphism and $\rho: {\mathcal{G}}_F\rightarrow {\mathrm{GL}}_n(C)$ a group homomorphism which is unramified at each finite place of $F$ above a place of $F^+$ which splits in $F$ and is not in $S$, we refer to [@BHS §2.4] for what it means for $\rho$ to be [*associated to*]{} $\theta$.

Though we could state a more general classicality conjecture, it is convenient for us to assume right now the following two extra hypothesis: $p>2$ and $G$ quasi-split at each finite place of $F^+$ (these assumptions will be needed anyway for our partial results, note however that they imply that $4$ divides $n[F^+:{\mathbb{Q}}]$ which rules out the case $n=2$, $F^+={\mathbb{Q}}$). We fix $\mathfrak{m}^S$ a maximal ideal of $\mathbb{T}^S$ of residue field $k_L$ (increasing $L$ if necessary) such that $\widehat S(U^p,L)_{\mathfrak{m}^S}\neq 0$, or equivalently $\widehat S(U^p,{\mathcal{O}}_L)_{\mathfrak{m}^S}\neq 0$, or $S(U^p,k_L)_{\mathfrak{m}^S}=\varinjlim_{U_p}S(U^pU_p,k_L)_{\mathfrak{m}^S}\neq 0$, or $S(U^pU_p,k_L)_{\mathfrak{m}^S}\ne 0$ for some $U_p$ (note that $\widehat S(U^p,L)_{\mathfrak{m}^S}$ is then a closed subspace of $\widehat S(U^p,L)$ preserved by $G_p$). We denote by ${\overline{\rho}}={\overline{\rho}}_{\mathfrak{m}^S}:{\mathcal{G}}_F\rightarrow {\mathrm{GL}}_n(k_L)$ the unique absolutely semi-simple Galois representation associated to $\mathfrak{m}^S$ (see [@Thorne Prop.6.6] and note that the running assumption $F/F^+$ unramified in [*loc. cit.*]{} is useless at this point). We assume $\mathfrak{m}^S$ [*non-Eisenstein*]{}, that is, ${\overline{\rho}}$ absolutely irreducible. Then it follows from [@Thorne Prop.6.7] (with the same remark as above) that the spaces $\widehat S(U^p,L)_{\mathfrak{m}^S}$, $\widehat S(U^p,{\mathcal{O}}_L)_{\mathfrak{m}^S}$ and $S(U^p,k_L)_{\mathfrak{m}^S}$ become modules over $R_{{\overline{\rho}},S}$, the complete local noetherian ${\mathcal{O}}_L$-algebra of residue field $k_L$ pro-representing the functor of deformations $\rho$ of ${\overline{\rho}}$ that are unramified outside $S$ and such that $\rho'\circ c\cong \rho\otimes \varepsilon^{n-1}$ (where $\rho'$ is the dual of $\rho$ and $c\in {\rm Gal}(F/F^+)$ is the complex conjugation).

The continuous dual $(\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an})'$ of $\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}:=(\widehat S(U^p,L)_{\mathfrak{m}^S})^{\rm an}=(\widehat S(U^p,L)^{\rm an})_{\mathfrak{m}^S}$ becomes a module over the global sections $\Gamma({\mathfrak{X}}_{{\overline{\rho}},S},{\mathcal{O}}_{{\mathfrak{X}}_{{\overline{\rho}},S}})$ where ${\mathfrak{X}}_{{\overline{\rho}},S}:=({\mathrm{Spf}}\, R_{{\overline{\rho}},S})^{{\mathrm{rig}}}$ (see for instance [@BHS §3.1]). We denote by $Y(U^p,{\overline{\rho}})$ the *eigenvariety of tame level $U^p$* (over $L$) defined in [@Emerton] (see also [@BHS §4.1]) associated to $\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}$, that is, the support of the coherent ${\mathcal{O}}_{{\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}}$-module $(J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}))'$ on ${\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}$ where $J_{B_p}$ is Emerton’s locally ${\mathbb{Q}}_p$-analytic Jacquet functor with respect to the Borel $B_p$ and $(\cdot)'$ means the continuous dual. This is a reduced closed analytic subset of ${\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}$ of dimension $n[F^+:{\mathbb{Q}}]$ whose points are: $$\label{points}
\left\{x=(\rho,\delta)\in {\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}\ {\rm such\ that}\ {\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\ne 0\right\}$$ where $\mathfrak{p}_\rho\subset R_{{\overline{\rho}},S}$ denotes the prime ideal corresponding to the point $\rho\in {\mathfrak{X}}_{{\overline{\rho}},S}$ under the identification of the sets underlying ${\mathfrak{X}}_{{\overline{\rho}},S}=({\mathrm{Spf}}\, R_{{\overline{\rho}},S})^{{\mathrm{rig}}}$ and ${\rm Spm}\,R_{{\overline{\rho}},S}[1/p]$ ([@deJong Lem.7.1.9]) and where $k(\mathfrak{p}_\rho)$ is its residue field. We denote by $\omega:Y(U^p,{\overline{\rho}})\rightarrow \widehat T^0_{p,L}$ the composition $Y(U^p,{\overline{\rho}})\hookrightarrow {\mathfrak{X}}_{{\overline{\rho}},S}\times \widehat T_{p,L}\twoheadrightarrow \widehat T_{p,L} \twoheadrightarrow \widehat T^0_{p,L}$.

\[changeu\] [If ${U'}^p\subseteq {U}^p$ (and $S$ contains $S_p$ and the set of finite places $v\nmid p$ that split in $F$ such that $U'_v$ is not maximal), then a point $x=(\rho,\delta)$ of $Y(U^p,{\overline{\rho}})$ is also in $Y({U'}^p,{\overline{\rho}})$ since $\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\subseteq \widehat S({U'}^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]$ and $J_{B_p}$ is left exact.]{}

We let $X_{\rm tri}^\square({\overline{\rho}}_p)$ be the product rigid analytic variety $\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde{v}})$ (over $L$) where ${\overline{\rho}}_{\tilde v}$ is the restriction of $\rho$ to the decomposition subgroup of ${\mathcal G}_F$ at $\tilde v$ (that we identify with ${\mathcal{G}}_{F_{\tilde v}}={\rm Gal}(\overline F_{\tilde v}/F_{\tilde v})$) and where $X_{\rm tri}^\square({\overline{\rho}}_{\tilde{v}})$ is as in §\[begin\]. This is a reduced closed analytic subvariety of $({\mathrm{Spf}}\, R^\square_{{\overline{\rho}}_p})^{{\mathrm{rig}}}\times\widehat{T}_{p,L}$ where $R^\square_{{\overline{\rho}}_p}:=\widehat{\bigotimes}_{v\in S_p}R^\square_{{\overline{\rho}}_{\tilde{v}}}$. Identifying $B_v$ (resp. $T_v$) with the upper triangular (resp. diagonal) matrices of ${\mathrm{GL}}_n(F_{\tilde v})$ via $i_{\tilde v}$, we let $\delta_{B_v}:=|\cdot|_{F_{\tilde v}}^{n-1}\otimes |\cdot|_{F_{\tilde v}}^{n-3}\otimes \cdots \otimes |\cdot|_{F_{\tilde v}}^{1-n}$ be the modulus character of $B_v$ and define as in [@BHS §2.3] an automorphism $\imath_v:\widehat T_v\buildrel\sim\over\rightarrow \widehat T_v$ by: $$\imath_v(\delta_1,\dots,\delta_n):=\delta_{B_v}\cdot(\delta_1,\dots,\delta_i\cdot(\varepsilon\circ{\mathrm{rec}}_{F_{\tilde v}})^{i-1},\dots,\delta_n\cdot(\varepsilon\circ{\mathrm{rec}}_{F_{\tilde v}})^{n-1})$$ (the twist by $\delta_{B_v}$ here ultimately comes from the same twist appearing in the definition of $J_{B_v}$). It then follows from [@BHS Th.4.2] that the morphism of rigid spaces: $$\begin{aligned}
({\mathrm{Spf}}\,R_{{\overline{\rho}},S})^{{\mathrm{rig}}}\times \widehat T_{p,L}&\longrightarrow &({\mathrm{Spf}}\, R^\square_{{\overline{\rho}}_p})^{{\mathrm{rig}}}\times\widehat{T}_{p,L}\\
\nonumber \big(\rho,(\delta_v)_{v\in S_p}\big)=\big(\rho,(\delta_{v,1},\dots,\delta_{v,n})_{v\in S_p}\big)&\longmapsto & \big((\rho|_{{\mathcal{G}}_{F_{\tilde v}}})_{v\in S_p},(\imath_v^{-1}(\delta_{v,1},\dots,\delta_{v,n}))_{v\in S_p}\big)\end{aligned}$$ induces a morphism of reduced rigid spaces over $L$: $$\label{eigenvartotrianguline}
Y(U^p,{\overline{\rho}})\longrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$$ (note that (\[eigenvartotrianguline\]) is thus [*not*]{} compatible with the weight maps $\omega$ on both sides). We say that a point $x=(\rho,\delta)=(\rho,(\delta_v)_{v\in S_p})\in Y(U^p,{\overline{\rho}})$ is crystalline (resp. dominant, resp. strictly dominant, resp. crystalline strictly dominant very regular etc.) if for each $v\in S_p$ its image in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ via (\[eigenvartotrianguline\]) is (see §\[begin\] and Definition \[veryreg\]). Due to the twist $\imath_v$ beware that $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ is strictly dominant if and only if $\delta\vert_{T_v\cap K_v}$ is (algebraic) dominant for each $v\in S_p$.

Let $\delta\in \widehat T_{p,L}$ be any locally algebraic character. Then we can write $\delta=\delta_{\lambda}\delta_{\rm sm}$ in $\widehat T_{p,L}$ where $\lambda=(\lambda_v)_{v\in S_p}\in \prod_{v\in S_p}({\mathbb{Z}}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$, $\delta_\lambda:=\prod_{v\in S_p}\delta_{\lambda_v}$ (see §\[begin\] for $\delta_{\lambda_v}\in \widehat T_{v,L}$) and $\delta_{\rm sm}$ is a smooth character of $T_{p}$ with values in $k(\delta)$ (the residue field of the point $\delta\in \widehat T_{p,L}$). Using the theory of Orlik and Strauch ([@OrlikStrauch]), we define as in [@BHS (3.7)] the following strongly admissible locally ${\mathbb{Q}}_p$-analytic representation of $G_p$ over $k(\delta)$: $${\mathcal{F}}_{\overline B_p}^{G_p}(\delta):={\mathcal{F}}_{\overline B_p}^{G_p}\big((U(\mathfrak{g}_L)\otimes_{U({\overline {\mathfrak b}}_L)}(-\lambda))^\vee, \delta_{\rm sm}\delta_{B_p}^{-1}\big)$$ where $\delta_{B_p}:=\prod_{v\in S_p}\delta_{B_v}$ and where we refer to [@BHS §3.5] for the details and notation. Recall that ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)$ has the same constituents as the locally ${\mathbb{Q}}_p$-analytic principal series $({\rm Ind}_{\overline B_p}^{G_p}\delta_{\lambda}\delta_{\rm sm}\delta_{B_p}^{-1})^{\rm an}=({\rm Ind}_{\overline B_p}^{G_p}\delta\delta_{B_p}^{-1})^{\rm an}$ but in the “reverse order” (at least generically). If $\lambda$ is dominant (that is $\lambda_v$ is dominant for each $v$ in the sense of §\[begin\]), we denote by ${\rm LA}(\delta)$ the locally algebraic representation: $$\begin{gathered}
\label{ladelta}
{\rm LA}(\delta):={\mathcal{F}}_{\overline B_p}^{G_p}(L(\lambda)',\delta_{\rm sm}\delta_{B_p}^{-1})={\mathcal{F}}_{G_p}^{G_p}\big(L(\lambda)',({\rm Ind}_{\overline B_p}^{G_p}\delta_{\rm sm}\delta_{B_p}^{-1})^{\rm sm}\big)\\
=L(\lambda)\otimes_L \big({\rm Ind}_{\overline B_p}^{G_p}\delta_{\rm sm}\delta_{B_p}^{-1}\big)^{\rm sm}\end{gathered}$$ where $L(\lambda)$ is the simple $U(\mathfrak{g}_L)$-module over $L$ of highest weight $\lambda$ relative to the Lie algebra of $B_p$ (which is finite dimensional over $L$ since $\lambda$ is dominant) that we see as an irreducible algebraic representation of $G_p$ over $L$, where $L(\lambda)'$ is its dual, and where $(-)^{\rm sm}$ denotes the smooth Borel induction over $k(\delta)$ (the second equality in (\[ladelta\]) following from [@OrlikStrauch Prop.4.9(b)]). Arguing as in [@OrlikStrauch §6] (note that $L(\lambda)'$ is the unique irreducible subobject of $(U(\mathfrak{g}_L)\otimes_{U({\overline {\mathfrak b}}_L)}(-\lambda))^\vee$), it easily follows from [@OrlikStrauch Th.5.8] (see also [@BreuilAnalytiqueI Th.2.3(iii)]) and [@HumBGG §5.1] that ${\rm LA}(\delta)$ is identified with the maximal locally ${\mathbb{Q}}_p$-algebraic quotient of ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)$ (or the maximal locally algebraic subobject of $({\rm Ind}_{\overline B_p}^{G_p}\delta\delta_{B_p}^{-1})^{\rm an}$).

It follows from (\[points\]) together with [@BreuilAnalytiqueII Th.4.3] that a point $x=(\rho,\delta)\in  {\mathfrak{X}}_{{\overline{\rho}},S}\times\widehat T_{p,L}$ lies in $Y(U^p,{\overline{\rho}})$ if and only if: $$\label{adj}
{\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\cong{\mathrm{Hom}}_{G_p}\big({\mathcal{F}}_{\overline B_p}^{G_p}(\delta),\widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x)\big)\ne 0.$$

\[defclass\] A point $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ is called *classical* if there exists a nonzero continuous $G_p$-equivariant morphism: $${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)\longrightarrow \widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x)$$ that factors through the locally ${\mathbb{Q}}_p$-algebraic quotient ${\rm LA}(\delta)$ of ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)$ (equivalently $(\rho,\delta)$ is classical if ${\mathrm{Hom}}_{G_p}({\rm LA}(\delta),\widehat S(U^p,L)_{\mathfrak{m}^S}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x))\ne 0$).

[(i) This definition is [@BHS Def.3.14].\
(ii) It seems reasonnable to expect that if $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ is classical, then in fact [*any*]{} continuous $G_p$-equivariant morphism ${\mathcal{F}}_{\overline B_p}^{G_p}(\delta)\longrightarrow \widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_\rho)}k(x)$ factors through ${\rm LA}(\delta)$. See the last statement of Corollary \[mainclassic\] below for a partial result in that direction.]{}

We fix an algebraic closure $\overline{\mathbb{Q}}_p$ of $L$ and embeddings $j_\infty:\overline{\mathbb{Q}}\hookrightarrow {\mathbb{C}}$, $j_p:\overline{\mathbb{Q}}\hookrightarrow \overline{\mathbb{Q}}_p$. Recall that, if $\pi=\pi_{\infty}\otimes_{{\mathbb{C}}}\pi_f$ is an automorphic representation of $G({\mathbb{A}}_{F^+})$ over ${\mathbb{C}}$ where $\pi_{\infty}$ (resp. $\pi_f$) is a representation of $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$ (resp. of $G({\mathbb{A}}_{F^+}^\infty)$), then due to the compactness of $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$, we have that $\pi_\infty$ is a finite dimensional irreducible representation that comes from an algebraic representation of ${\mathrm{Res}}_{F^+/{\mathbb{Q}}}G$ over ${\mathbb{C}}$ (argue as in [@BelChe §§6.2.3,6.7]). Moreover, arguing again as in [*loc. cit.*]{}, $\pi_\infty$ (resp. $\pi_f$) has a ${\overline{\mathbb{Q}}}$-structure given by $j_\infty$ which is stable under the action of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)({\overline{\mathbb{Q}}})$ (resp. of $G({\mathbb{A}}_{F^+}^\infty)$). Hence the scalar extension of the ${\overline{\mathbb{Q}}}$-structure of $\pi_\infty$ to $\overline {\mathbb{Q}}_p$ via $j_p$ is endowed with an action of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)(\overline {\mathbb{Q}}_p)$, thus in particular of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)({\mathbb{Q}}_p)=G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)=G_p$. This latter representation of $G_p$ is easily seen to be defined over $L$ and of the form $L(\lambda)$ for a dominant $\lambda$ as above. We say that $\pi_\infty$ [*is of weight $\lambda$*]{} if the resulting representation of $G_p$ is $L(\lambda)$.

For the sake of completeness, we recall the following proposition showing that Definition \[defclass\] coincides with the usual classicality definition.

\[compaclas\] A strictly dominant point $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$, that is such that $\omega(x)=\delta_{\lambda}$ for some dominant $\lambda\in \prod_{v\in S_p}({\mathbb{Z}}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$, is classical if and only if there exists an automorphic representation $\pi=\pi_{\infty}\otimes_{{\mathbb{C}}}\pi_f^p\otimes_{{\mathbb{C}}}\pi_p$ of $G({\mathbb{A}}_{F^+})$ over ${\mathbb{C}}$ such that the following conditions hold:

1.  the $G(F^+\otimes_{\mathbb{Q}}\mathbb{R})$-representation $\pi_\infty$ is of weight $\lambda$ in the above sense;\

2.  the ${\mathcal{G}}_F$-representation $\rho$ is the Galois representation associated to $\pi$ (see proof below);\

3.  the invariant subspace $(\pi_f^p)^{U^p}$ is nonzero;\

4.  the $G_p$-representation $\pi_p$ is a quotient of $\big({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1}\big)^{\rm sm}\otimes_{k(\delta)}\overline {\mathbb{Q}}_p$.

If moreover $F$ is unramified over $F^+$ and $U_v$ is hyperspecial when $v$ is inert in $F$, then such a $\pi$ is unique and appears with multiplicity $1$ in $L^2(G(F^+)\backslash G({\mathbb{A}}_{F^+}),{\mathbb{C}})$.

Let $W$ be any linear representation of $G_p$ over an $L$-vector space and $U$ any compact open subgroup of $G({\mathbb{A}}_{F^+}^{\infty})$, we define $S(U,W)$ to be the $L$-vector space of functions $f:G(F^+)\backslash G({\mathbb{A}}_{F^+}^{\infty})\longrightarrow W$ such that $f(gu)=u_p^{-1}(f(g))$ for $g\in G({\mathbb{A}}_{F^+}^{\infty})$ and $u\in U$, where $u_p$ is the projection of $u$ in $G_p$. Fixing $U^p$ as previously, we define $S(U^p,W):=\varinjlim_{U_p}S(U^pU_p,W)$ (inductive limit taken over compact open subgroups $U_p$ of $G_p$) endowed with the linear left action of $G_p$ given by $(h_p\cdot f)(g):=h_p(f(gh_p))$ ($h_p\in G_p$, $g\in G({\mathbb{A}}_{F^+}^{\infty})$) where the second $h_p$ is seen in $G({\mathbb{A}}_{F^+}^{\infty})$ in the obvious way. Note that $\mathbb{T}^S$ also naturally acts on $S(U^p,W)$ (the representation $W$ here playing no role since this action is “outside $p$”). Then it follows from [@EGH §7.1.4] that there is an isomorphism of smooth representations of $G_p$ over $\overline{\mathbb{Q}}_p$: $$\label{class}
S(U^p,L(\lambda)')\otimes_{L}\overline {\mathbb{Q}}_p\cong \bigoplus_{\pi}\big[\big((\pi_f^p)^{U^p}\otimes_{\overline {\mathbb{Q}}}\pi_p\big)\otimes_{\overline{\mathbb{Q}},j_p}\overline {\mathbb{Q}}_p\big]^{\oplus m(\pi)}$$ where the direct sum is over the automorphic representations $\pi=\pi_\infty\otimes_{{\mathbb{C}}}\pi_p$ of $G({\mathbb{A}}_{F^+})$ such that $\pi_\infty$ is of weight $\lambda$ and $(\pi_f^p)^{U^p}\ne 0$ (we take the ${\overline{\mathbb{Q}}}$-structures) and where $m(\pi)$ is the multiplicity of $\pi$ in $L^2(G(F^+)\backslash G({\mathbb{A}}_{F^+}),{\mathbb{C}})$. We then say that a point $\rho\in {\mathfrak{X}}_{{\overline{\rho}},S}$ is the Galois representation associated to $\pi$ (with $\pi_\infty$ of weight $\lambda$) if we have: $$\big[\big((\pi_f^p)^{U^p}\otimes_{\overline {\mathbb{Q}}}\pi_p\big)\otimes_{\overline{\mathbb{Q}},j_p}\overline {\mathbb{Q}}_p\big]^{\oplus m(\pi)}\subseteq S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}\overline {\mathbb{Q}}_p$$ where $\mathfrak{p}_\rho$ is as in (\[points\]) (and $R_{{\overline{\rho}},S}$ acts on $S(U^p,L(\lambda)')_{\mathfrak{m}^S}$ again using [@Thorne Prop.6.7]). Note that $S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}\overline {\mathbb{Q}}_p\ne 0$ (equivalently $S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\ne 0$) if and only if there exists an automorphic representation $\pi$ such that $\pi_\infty$ is of weight $\lambda$, $(\pi_f^p)^{U^p}\ne 0$ and $\rho$ is the Galois representation associated to $\pi$.

Let $\widehat S(U^p,L)^{\lambda-\rm la}\subset \widehat S(U^p,L)^{\rm an}$ be the closed $G_p$-subrepresentation of locally $L(\lambda)$-algebraic vectors, that is the $L$-subvector space of $\widehat S(U^p,L)^{\rm an}$ (or equivalently of $\widehat S(U^p,L)$) of vectors $v$ such that there exists a compact open subgroup $U_p$ of $G_p$ such that the $U_p$-subrepresentation generated by $v$ in $\widehat S(U^p,L)\vert_{U_p}$ is isomorphic to $(L(\lambda)\vert_{U_p})^{\oplus d}$ for some positive integer $d$. Note that the subspace $\widehat S(U^p,L)^{\lambda-\rm la}$ is preserved under the action of $\mathbb{T}^S$ (since the latter commutes with $G_p$). Then it follows from [@Emerton Prop.3.2.4] and its proof that there is an isomorphism of locally ${\mathbb{Q}}_p$-algebraic representations of $G_p$ over $L$ which is $\mathbb{T}^S$-equivariant (with the action of $\mathbb{T}^S$ on the right hand side given by its action on $S(U^p,L(\lambda)')$): $$\widehat S(U^p,L)^{\lambda-\rm la}\cong L(\lambda)\otimes_L S(U^p,L(\lambda)').$$ We then deduce a $G_p$-equivariant isomorphism of $R_{{\overline{\rho}},S}$-modules: $$\label{lambdaalgm}
\widehat S(U^p,L)^{\lambda-\rm la}_{\mathfrak{m}^S}\cong L(\lambda)\otimes_L S(U^p,L(\lambda)')_{\mathfrak{m}^S}$$ where $\widehat S(U^p,L)^{\lambda-\rm la}_{\mathfrak{m}^S}:=(\widehat S(U^p,L)^{\lambda-\rm la})_{\mathfrak{m}^S}=(\widehat S(U^p,L)_{\mathfrak{m}^S})^{\lambda-\rm la}$.

Now let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ with $\omega(x)=\delta_{\lambda}$ for $\lambda$ dominant and define $\mathfrak{p}_\rho$ as in (\[points\]). From Definition \[defclass\] and the definition of $\widehat S(U^p,L)^{\lambda-\rm la}$, we get that the point $x$ is classical if and only if there exists a nonzero $G_p$-equivariant morphism: $$L(\lambda)\otimes_L \big({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1}\big)^{\rm sm} \longrightarrow \widehat S(U^p,L)_{\mathfrak{m}^S}^{\lambda-\rm la}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x)$$ if and only if by (\[lambdaalgm\]) there exists a nonzero $G_p$-equivariant morphism: $$\big({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1}\big)^{\rm sm}\longrightarrow S(U^p,L(\lambda)')_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x)$$ if and only if by (\[class\]) there exists an automorphic representation $\pi=\pi_\infty\otimes_{{\mathbb{C}}}\pi_f^p\otimes_{{\mathbb{C}}}\pi_p$ of $G({\mathbb{A}}_{F^+})$ such that $\pi_\infty$ is of weight $\lambda$, $(\pi_f^p)^{U^p}\ne 0$, $\rho$ is the Galois representation associated to $\pi$ and $\pi_p$ is a quotient of $({\rm Ind}_{\overline B_p}^{G_p} \delta\delta_{\lambda}^{-1}\delta_{B_p}^{-1})^{\rm sm}\otimes_{k(\delta)}\overline {\mathbb{Q}}_p$.

Now let us prove the last assertion. According to [@Labesse Cor.5.3], there exists an isobaric representation $\Pi=\Pi_1\boxplus\cdots\boxplus\Pi_r$ where $n=m_1+\cdots+m_r$ and $\Pi_i$ nonzero automorphic representations of $\mathrm{GL}_{m_i}(\mathbb{A}_F)$ occuring in the discrete spectrum such that $\Pi$ is a weak base change of $\pi$ in the sense of [@Labesse §4.10]. Since $\overline{\rho}$, hence $\rho$, are absolutely irreducible, we have $r=1$ and $\Pi=\Pi_1$ cuspidal. The equality $m(\pi)=1$ then follows from [@Labesse Th.5.4] (which uses the extra assumption $F/F^+$ unramified). The uniqueness of $\pi$ is a consequence of the strong base change theorem [@Labesse Th.5.9] together with the fact that $\pi_v$ is unramified at finite places $v$ of $F^+$ which are inert in $F$ (which uses the extra assumption $U_v$ hyperspecial for $v$ inert) and the fact that the $L$-packets at finite places of $F^+$ which are split in $F$ are singletons.

[With the notation of Proposition \[compaclas\], write $\delta\delta_{\lambda}^{-1}=(\delta_{{\rm sm},v,1},\dots,\delta_{{\rm sm},v,n})_{v\in S_p}$, if moreover $\delta_{{\rm sm},v,i}/\delta_{{\rm sm},v,j}\notin \{1,\vert\cdot\vert_{F_{\tilde v}}^{-2}\}$ for $1\leq i\ne j\leq n$ and $v\in S_p$, then we see from (iv) of Proposition \[compaclas\] that $\pi_p\cong ({\rm Ind}_{B_p}^{G_p} \delta\delta_{\lambda}^{-1})^{\rm sm}\otimes_{k(\delta)}\overline {\mathbb{Q}}_p$.]{}

We then have the following conjecture, which by Proposition \[compaclas\] is essentially a consequence of the Fontaine-Mazur conjecture and the Langlands philosophy, and which is the natural generalization in the context of definite unitary groups of the main result of [@Kisinoverconvergent] for ${\mathrm{GL}}_2/{\mathbb{Q}}$ (in the crystalline case).

\[classiconj\] Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant point. Then $x$ is classical.

\[extra\] [We didn’t seek to state the most general classicality conjecture. Obviously, the assumptions that $p>2$ and $G$ is quasi-split at each finite place of $F^+$ shouldn’t be crucial, and one could replace crystalline by de Rham.]{}

Proof of the main classicality result {#firstclassical}
-------------------------------------

We prove a criterion for classicality (Theorem \[classicalitycrit\]) in terms of the patched eigenvariety of [@BHS], which itself builds on the construction in [@CEGGPS] of a “big” patching module $M_\infty$. We use it to prove our main classicality result (Corollary \[mainclassic\]).

We keep the notation of §\[classic\] and make the following extra assumptions (which are required for the construction of $M_\infty$): $F$ is unramified over $F^+$, $U_v$ is hyperspecial if $v$ is inert in $F$, ${\overline{\rho}}(\mathcal{G}_{F(\zeta_p)})$ is adequate in the sense of [@Thorne Def.2.3] and $\zeta_p\notin \overline F^{\ker({\overline{\rho}}\otimes {\overline{\rho}}')}$. For instance if $p>2n+1$ and ${\overline{\rho}}\vert_{\mathcal{G}_{F(\zeta_p)}}$ is (still) absolutely irreducible, then ${\overline{\rho}}(\mathcal{G}_{F(\zeta_p)})$ is automatically adequate ([@GHTT Th.9]). We first briefly recall some notation, definitions and statements and refer to [@BHS §3.2] for more details on what follows.

We let $R_{{\overline{\rho}}_{\tilde v}}^{\overline\square}$ be the maximal reduced and ${\mathbb{Z}}_p$-flat quotient of the framed local deformation ring $R_{{\overline{\rho}}_{\tilde v}}^\square$ and set: $$R^{\rm loc}:=\widehat{\bigotimes}_{v\in S} R_{{\overline{\rho}}_{\tilde v}}^{\overline\square},\ \ R_{{\overline{\rho}}^p}:=\widehat{\bigotimes}_{v\in S\backslash S_p}R_{{\overline{\rho}}_{\tilde v}}^{\overline\square},\ \ R_{{\overline{\rho}}_p}:=\widehat{\bigotimes}_{v\in S_p}R_{{\overline{\rho}}_{\tilde v}}^{\overline\square},\ \ R_\infty:=R^{\rm loc}{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}x_1\dots,x_g{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}$$ where $g\geq 1$ is some integer which will be fixed below. We let ${\mathfrak{X}}_{{\overline{\rho}}^p}:=({\mathrm{Spf}}\, R_{{\overline{\rho}}^p})^{{\mathrm{rig}}}$, ${\mathfrak{X}}_{{\overline{\rho}}_p}:=({\mathrm{Spf}}R_{{\overline{\rho}}_p})^{{\mathrm{rig}}}$ and ${\mathfrak{X}}_\infty:=({\mathrm{Spf}}\, R_\infty)^{{\mathrm{rig}}}$ so that: $$\label{product}
{\mathfrak{X}}_\infty={\mathfrak{X}}_{{\overline{\rho}}^p}\times {\mathfrak{X}}_{{\overline{\rho}}_p}\times {\mathbb{U}}^g$$ where ${\mathbb{U}}:=({\mathrm{Spf}}\,{\mathcal{O}}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}y {{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}})^{{\mathrm{rig}}}$ is the open unit disc over $L$. We also define $S_\infty:={\mathcal{O}}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}y_1,\dots,y_t{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}$ where $t:=g+[F^+:{\mathbb{Q}}]\tfrac{n(n-1)}{2}+|S|n^2$ and $\mathfrak{a}:=(y_1,\dots,y_t)$ (an ideal of $S_\infty$).

Thanks to Remark \[changeu\] and Proposition \[compaclas\] we can (and do) assume that the tame level $U^p$ is small enough so that we have: $$\label{net}
G(F)\cap (hU^pK_ph^{-1})=\{1\}\ \ {\rm for\ all\ }h\in G({\mathbb{A}}^\infty_{F^+})$$ (indeed, let $w\nmid p$ be a finite place of $F^+$ that splits in $F$ such that $U_w$ is maximal, replace $U^p$ by $U'^p:=U'_w\prod_{v\ne w}U_v$ where $U'_w$ is small enough so that $U'^p$ satisfies (\[net\]), and use Proposition \[compaclas\] and local-global compatibility at $w$ to deduce classicality in level $U^p$ from classicality in level $U'^p$). Then there is a quotient $R_{{\overline{\rho}},S}\twoheadrightarrow R_{{\overline{\rho}}, {\mathcal{S}}}$ such that the action of $R_{{\overline{\rho}},S}$ on $\widehat S(U^p,L)_{\mathfrak{m}^S}$ factors through $R_{{\overline{\rho}},{\mathcal{S}}}$, an integer $g\geq 1$ and:

1.  a continuous $R_\infty$-admissible (see [@BHS Def.3.1]) unitary representation $\Pi_\infty$ of $G_p$ over $L$ together with a $G_p$-stable and $R_\infty$-stable unit ball $\Pi_\infty^\circ\subset \Pi_\infty$;\

2.  a morphism of local ${\mathcal{O}}_L$-algebras $S_\infty\rightarrow R_\infty$ such that $M_\infty:={\mathrm{Hom}}_{{\mathcal{O}}_L}(\Pi_\infty^\circ,{\mathcal{O}}_L)$ is finite projective as an $S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                                  {\mbox{\rm [\hspace{-0.15em}[}}
                                  {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                                  {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                                  {\mbox{\rm ]\hspace{-0.15em}]}}
                                  {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                                  {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}$-module;\

3.  compatible isomorphisms $R_\infty/\mathfrak{a}R_\infty\cong R_{{\overline{\rho}},{\mathcal{S}}}$ and $\Pi_\infty[\mathfrak{a}]\cong \widehat S(U^p,L)_{\mathfrak{m}^S}$ where the latter is $G_p$-equivariant.

We then define the *patched eigenvariety* $X_p({\overline{\rho}})$ as the support of the coherent ${\mathcal{O}}_{{\mathfrak{X}}_\infty\times \widehat T_{p,L}}$-module $\mathcal{M}_\infty=(J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}))'$ on ${\mathfrak{X}}_\infty\times \widehat T_{p,L}$ (see [@BHS Def.3.2] for $\Pi_\infty^{R_\infty-{\rm an}}$; strictly speaking $(J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}))'$ is the global sections of the sheaf $\mathcal{M}_\infty$). This is a reduced closed analytic subset of ${\mathfrak{X}}_\infty\times \widehat T_{p,L}$ ([@BHS Cor.3.19]) whose points are ([@BHS Prop3.7]): $$\label{pointpatching}
\left\{x=(y,\delta)\in {\mathfrak{X}}_\infty\times \widehat T_{p,L}\ {\rm such\ that}\ {\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x))\big)\ne 0\right\}$$ where $\mathfrak{p}_y\subset R_\infty$ denotes the prime ideal corresponding to the point $y\in {\mathfrak{X}}_\infty$ (under the identification of the sets underlying ${\mathfrak{X}}_\infty$ and ${\rm Spm}\,R_\infty[1/p]$) and ${k(\mathfrak{p}_y)}$ is the residue field of $\mathfrak{p}_y$. It follows from the proof of [@BHS Th.4.2] that we can recover the eigenvariety $Y(U^p,{\overline{\rho}})$ as the reduced Zariski-closed subspace of $X_p({\overline{\rho}})$ underlying the vanishing locus of the ideal $\mathfrak{a}\Gamma({\mathfrak{X}}_\infty,{\mathcal{O}}_{{\mathfrak{X}}_\infty})$.

\[CM\] The coherent sheaf $\mathcal{M}_\infty$ is Cohen-Macaulay over $X_p(\overline{\rho})$.

From the proof of [@BHS Prop.3.10] (to which we refer the reader for more details) we deduce that there exists an admissible affinoid covering $(U_i)_i$ of $X_p(\overline{\rho})$ such that $\Gamma(U_i,\mathcal{M}_\infty)$ is a finite projective module over a ring $\mathcal{O}_{\mathcal{W}_\infty}(W_i)$ whose action on $\Gamma(U_i,\mathcal{M}_\infty)$ factors through a ring homomorphism $\mathcal{O}_{\mathcal{W}_\infty}(W_i)\rightarrow\mathcal{O}_{X_p(\overline{\rho})}(U_i)$. Consequently we can deduce from [@EGAIV1 Prop.16.5.3] that $\Gamma(U_i,\mathcal{M}_\infty)$ is a Cohen-Macaulay $\mathcal{O}_{X_p(\overline{\rho})}(U_i)$-module.

It follows from [@BHS Th.3.20] that the isomorphism of rigid spaces: $$\begin{aligned}
{\mathfrak{X}}_\infty\times \widehat T_{p,L}&\buildrel\sim\over\longrightarrow &{\mathfrak{X}}_\infty\times\widehat{T}_{p,L}\\
\nonumber \big(x,(\delta_v)_{v\in S_p}\big)=\big(x,(\delta_{v,1},\dots,\delta_{v,n})_{v\in S_p}\big)&\longmapsto & \big(x,(\imath_v^{-1}(\delta_{v,1},\dots,\delta_{v,n}))_{v\in S_p}\big)\end{aligned}$$ induces via (\[product\]) a morphism of reduced rigid spaces over $L$: $$\label{patchedeigenvartoXtri}
X_p({\overline{\rho}})\longrightarrow {\mathfrak{X}}_{{\overline{\rho}}^p}\times X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g$$ which identifies the source with a union of irreducible components of the target. Note that the composition: $$Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})\buildrel{(\ref{patchedeigenvartoXtri})}\over\longrightarrow{\mathfrak{X}}_{{\overline{\rho}}^p}\times X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g\twoheadrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)$$ is the map (\[eigenvartotrianguline\]). An irreducible component of the right hand side of (\[patchedeigenvartoXtri\]) is of the form ${\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g$ where ${\mathfrak{X}}^p$ (resp. $Z$) is an irreducible component of ${\mathfrak{X}}^p$ (resp. $X_{\rm tri}^\square({\overline{\rho}}_p)$). Given an irreducible component ${\mathfrak{X}}^p\subseteq {\mathfrak{X}}_{{\overline{\rho}}^p}$, we denote by $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\subseteq X_{\rm tri}^\square({\overline{\rho}}_p)$ the union (possibly empty) of those irreducible components $Z\subseteq X_{\rm tri}^\square({\overline{\rho}}_p)$ such that ${\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g$ is an irreducible component of $X_p({\overline{\rho}})$ via (\[patchedeigenvartoXtri\]). The morphism (\[patchedeigenvartoXtri\]) thus induces an isomorphism: $$\label{union}
X_p({\overline{\rho}})\buildrel\sim\over\longrightarrow \bigcup_{{\mathfrak{X}}^p} \big({\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\big)$$ the union (inside ${\mathfrak{X}}_{{\overline{\rho}}^p}\times X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g$) being over the irreducible components ${\mathfrak{X}}^p$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}$.

We now state and prove the main result of this section, which gives a criterion for classicality on $Y(U^p,{\overline{\rho}})$. Recall that, given a crystalline strictly dominant point $x_v=(r_v,\delta_v)\in X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ such that the geometric Frobenius eigenvalues on ${\rm WD}(r_v)$ are pairwise distinct and $V_v\subseteq X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ a sufficiently small open neighbourhood of $x_v$, we have constructed in Corollary $\ref{defnZtri(x)}$ an irreducible component $Z_{{\rm tri},V_v}(x_v)$ of $V_v$ containing $x_v$.

\[classicalitycrit\] Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant point such that the eigenvalues $\varphi_{\tilde v,1},\dots,\varphi_{\tilde v,n}$ of the geometric Frobenius on the (unramified) Weil-Deligne representation ${\rm WD}(\rho\vert_{{\mathcal{G}}_{F_{\tilde v}}})$ satisfy $\varphi_{\tilde v,i}\varphi_{\tilde v,j}^{-1}\notin \{1,q_v\}$ for all $i\neq j$ and all $v\in S_p$. Let ${\mathfrak{X}}^p\subset {\mathfrak{X}}_{{\overline{\rho}}^p}$ be an irreducible component such that $x\in {\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\subseteq X_p({\overline{\rho}})$ via (\[union\]), let $x_v\in X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (for $v\in S_p$) be the image of $x$ via: $${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\twoheadrightarrow X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\hookrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)\twoheadrightarrow X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$$ and let $V_v\subseteq X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (for $v\in S_p$) be a sufficiently small open neighbourhood of $x_v$ so that $Z_{{\rm tri},V_v}(x_v)\subseteq V_v$ is defined. If we have: $$\prod_{v\in S_p} Z_{{\rm tri},V_v}(x_v)\subseteq X_{\rm tri}^{{\mathfrak{X}}^p\rm-aut}({\overline{\rho}}_p)$$ then the point $x$ is classical.

Let us write $\mathfrak{p}_y\subset R_\infty$ for the prime ideal corresponding to the image $y$ of $x$ in ${\mathfrak{X}}_\infty$ via $Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})\hookrightarrow {\mathfrak{X}}_\infty\times \widehat{T}_{p,L}\twoheadrightarrow {\mathfrak{X}}_\infty$ and $\mathfrak{p}_\rho\subset R_{{\overline{\rho}},S}$ for the prime ideal corresponding to the global representation $\rho$. Then it follows from property (iii) above that we have $\mathfrak{a}R_\infty\subseteq \mathfrak{p}_{\rho}$ and $\widehat S(U^p,L)_{\mathfrak{m}^S}[\mathfrak{p}_{\rho}]=\Pi_{\infty}[\mathfrak{p}_y]$. From Definition \[defclass\] we thus need to show that ${\mathrm{Hom}}_{G_p}({\rm LA}(\delta),\Pi_{\infty}[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x))\ne 0$.

As in §\[classic\] let us write $\delta=\delta_{\lambda}\delta_{\rm sm}$ with $\lambda=(\lambda_v)_{v\in S_p}$ and: $$\lambda_v:=(\lambda_{v,\tau,i})_{1\leq i\leq n,\tau\in{\mathrm{Hom}}(F_{\tilde v},L)}\in {\mathbb{Z}}^{{\mathrm{Hom}}(F_{\tilde v},L)}$$ (recall that each $\lambda_v$ is dominant with respect to $B_v$). Consider the usual induction with compact support ${\rm ind}_{K_p}^{G_p}(L(\lambda)\vert_{K_p})$ (resp. ${\rm ind}_{K_v}^{G_v}(L(\lambda_v)\vert_{K_v})$) where $L(\lambda_p)$ (resp. $L(\lambda_v)$) is the irreducible algebraic representation of $G_p$ (resp. $G_v$) over $L$ of highest weight $\lambda$ (resp. $\lambda_v$) with respect to $B_p$ (resp. $B_v$). Let $\mathcal{H}(\lambda):={\mathrm{End}}_{G_p}({\rm ind}_{K_p}^{G_p}L(\lambda))$ and $\mathcal{H}(\lambda_v):={\mathrm{End}}_{G_v}({\rm ind}_{K_v}^{G_v}L(\lambda_v))$ be the respective convolution algebras (which are commutative $L$-algebras), we have $\mathcal{H}(\lambda)\cong \prod_{v\in S_p}\mathcal{H}(\lambda_v)$. Moreover by Frobenius reciprocity: $$\Pi_\infty(\lambda):={\mathrm{Hom}}_{K_p}(L(\lambda),\Pi_\infty)\cong {\mathrm{Hom}}_{G_p}\big({\rm ind}_{K_p}^{G_p}L(\lambda),\Pi_\infty\big)$$ carries an action of $\mathcal{H}(\lambda)$. By a slight extension of [@CEGGPS Lem.4.16(1)] (see the proof of [@BHS Prop.3.15]), the action of $R_{{\overline{\rho}}_{\tilde v}}^\square$ on $\Pi_\infty(\lambda)$ via $R_{{\overline{\rho}}_{\tilde v}}^\square\rightarrow R^{\rm loc}\hookrightarrow R_\infty$ factors through its quotient $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ where, for $v\in S_p$, ${\bf k}_v:=(k_{v,\tau,i})_{1\leq i\leq n,\tau\in{\mathrm{Hom}}(F_{\tilde v},L)}$ with $k_{v,\tau,i}:=\lambda_{v,\tau,i}-(i-1)$ (note that $\omega(x_v)=\delta_{{\bf k}_v}$ and that $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ is also a quotient of $R_{{\overline{\rho}}_{\tilde v}}^{\overline\square}$).

These two actions of $\mathcal{H}(\lambda_v)$ and $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ on the $L$-vector space $\Pi_\infty(\lambda)$ are related. By [@CEGGPS Th.4.1] and a slight extension of [@CEGGPS Lem.4.16(2)] (see the proof of [@BHS Prop.3.15]), there is a unique $L$-algebra homomorphism $\eta_v:\mathcal{H}(\lambda_v)\rightarrow R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}[1/p]$ which interpolates the local Langlands correspondence (in a sense given in [@CEGGPS Th.4.1]) and such that the above action of $\mathcal{H}(\lambda_v)$ on $\Pi_\infty(\lambda)$ agrees with the action induced by that of $R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}[1/p]$ composed with the morphism $\eta_v$.

In order to show that ${\rm LA}(\delta)$ admits a nonzero $G_p$-equivariant morphism to $\Pi_{\infty}[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x)$, we claim it is enough to show that $\Pi_\infty(\lambda)[\mathfrak{p}_y]\cong {\mathrm{Hom}}_{G_p}({\rm ind}_{K_p}^{G_p}L(\lambda),\Pi_\infty[\mathfrak{p}_y])$ is nonzero. Indeed, by what we just saw, any nonzero $G_p$-equivariant morphism ${\rm ind}_{K_p}^{G_p}L(\lambda)\rightarrow \Pi_\infty[\mathfrak{p}_y]$ induces a nonzero $G_p$-equivariant morphism: $${\rm ind}_{K_p}^{G_p}L(\lambda)\otimes_Lk(x)\longrightarrow \Pi_\infty[\mathfrak{p}_y]\otimes_{k(\mathfrak{p}_y)}k(x)$$ which factors through ${\rm ind}_{K_p}^{G_p}L(\lambda)\otimes_{\mathcal{H}(\lambda)}\theta_{\mathfrak{p}_y}$ where $\theta_{\mathfrak{p}_y}$ is the character: $$\xymatrix{
\theta_{\mathfrak{p}_y}:\mathcal{H}(\lambda)\ar[r]^<<<<{\otimes_{v\in S_p}\eta_v}&\widehat\bigotimes_{v\in S_p} R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v{\rm -cr}}[1/p]\ar[r] & k(\mathfrak{p}_y)\subseteq k(x),
}$$ the last morphism being the canonical projection to the residue field $k(\mathfrak{p}_y)$ at $\mathfrak{p}_y$ (the map $R_{{\overline{\rho}}_p}\hookrightarrow R_\infty\twoheadrightarrow R_\infty/\mathfrak{p}_y$ factoring through $\widehat\bigotimes_{v\in S_p} R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v{\rm -cr}}$ by the assumption on $\rho$). But by the compatibility with the local Langlands correspondence in [@CEGGPS Th.4.1] together with the assumption $\varphi_{\tilde v,i}/\varphi_{\tilde v,j}\neq q_v$ for $1\leq i,j\leq n$ and $v\in S_p$, we have ${\rm ind}_{K_p}^{G_p}L(\lambda)\otimes_{\mathcal{H}(\lambda)}\theta_{\mathfrak{p}_y}\cong {\rm LA}(\delta)\otimes_{k(\delta)}k(x)$.

By the same proof as that of [@CEGGPS Lem.4.17(2)], the $R_\infty \otimes_{R_{{\overline{\rho}}_p}}\widehat\bigotimes_{v\in S_p} R_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v{\rm -cr}}$-module $\Pi_{\infty}(\lambda)'$ is supported on a union of irreducible components of: $${\mathfrak{X}}_{{\overline{\rho}}^p}\times \prod_{v\in S_p} {\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}\times{\mathbb{U}}^g$$ and we have to prove that $y$ is a point on one of these irreducible components. Since $y\in {\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$ where $Z_{\rm cris}(\rho_{\tilde v})$ is the unique irreducible component of ${\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ containing $\rho_{\tilde v}:=\rho\vert_{{\mathcal{G}}_{F_{\tilde v}}}$ (recall ${\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$ is smooth over $L$ by [@Kisindef]), it is enough to prove that ${\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$ is one of the irreducible components in the support of $\Pi_{\infty}(\lambda)'$, or equivalently that ${\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$ contains at least one point which is in the support of $\Pi_{\infty}(\lambda)'$.

For each $v\in S_p$ let $x'_v=(r'_v,\delta'_v)$ be any point in $\iota_{{\bf k}_v}(\widetilde Z_{\rm cris}(x_v))\cap V_v\subseteq V_v\subseteq X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ where $\widetilde Z_{\rm cris}(x_v)$ is as in (i) of Corollary \[defnZtri(x)\] (so in particular $x'_v$ is crystalline strictly dominant of Hodge-Tate weights ${\bf k}_v$ and $r'_v$ lies on $Z_{\rm cris}(\rho_{\tilde v})$ by (i) of Remark \[down\]). Then we have $x'_v\in Z_{{\rm tri},V_v}(x_v)$ for $v\in S_p$ by (ii) of Corollary \[defnZtri(x)\]. From the assumption: $$\prod_{v\in S_p}Z_{{\rm tri},V_v}(x_v)\subset X_{\rm tri}^{{\mathfrak{X}}^p\rm -aut}({\overline{\rho}}_p)$$ it then follows that there exists: $$x'=(y',\epsilon')\in {\mathfrak{X}}^p\times\prod_{v\in S_p}Z_{{\rm tri},V_v}(x_v)\times {\mathbb{U}}^g\subseteq {\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\buildrel (\ref{union})\over \subseteq X_p({\overline{\rho}})\subset {\mathfrak{X}}_\infty\times \widehat T_{p,L}$$ (with $y'\in {\mathfrak{X}}_\infty$, $\epsilon'\in \widehat T_{p,L}$) mapping to $x'_v$ via ${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g\twoheadrightarrow X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\hookrightarrow X_{\rm tri}^\square({\overline{\rho}}_p)\twoheadrightarrow X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (so $\epsilon'_v=\imath_v^{-1}(\delta'_v)$) and where $y'$ still belongs to ${\mathfrak{X}}^p\times \prod_{v\in S_p} Z_{\rm cris}(\rho_{\tilde v})\times{\mathbb{U}}^g$. It is thus enough to prove that $y'$ is in the support of $\Pi_{\infty}(\lambda)'$, i.e. that $\Pi_\infty(\lambda)[\mathfrak{p}_{y'}]\cong {\mathrm{Hom}}_{K_p}(L(\lambda),\Pi_\infty[\mathfrak{p}_{y'}])$ is nonzero.

We conclude by a similar argument as in the proof of [@BHS Prop.3.27]. By (the proof of) [@Chenevier Lem.4.4] and the same argument as at the end of the proof of Lemma \[genericopensinXcris\] (using the smoothness, hence flatness, of $\widetilde U_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}\rightarrow {\mathfrak{X}}_{{\overline{\rho}}_{\tilde v}}^{\square,{\bf k}_v\rm -cr}$), we may choose $x_v'\in \iota_{{\bf k}_v}(\widetilde Z_{\rm cris}(x_v))\cap V_v$ such that the crystalline Galois representation $r'_v$ is generic in the sense of [@BHS Def.2.8]. Then we claim that the nonzero $G_p$-equivariant morphism ${\mathcal{F}}_{\overline B_p}^{G_p}(\epsilon')\rightarrow \Pi_\infty^{R_\infty-\rm an}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x')$ corresponding by [@BreuilAnalytiqueII Th.4.3] to the nonzero $T_p$-equivariant morphism $\epsilon'\rightarrow J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x'))$ given by the point $x'$ factors through its locally ${\mathbb{Q}}_p$-algebraic quotient ${\rm LA}(\epsilon')$ (which provides a nonzero $K_p$-equivariant morphism $L(\lambda)\rightarrow \Pi_\infty[\mathfrak{p}_{y'}]$). Indeed, if it doesn’t, then the computation of the Jordan-Hölder factors of ${\mathcal{F}}_{\overline B_p}^{G_p}(\epsilon')$ ([@BreuilAnalytiqueII Cor.4.6]) together with [@BreuilAnalytiqueI Cor.3.4] show that there exits a point $x''=(y',\epsilon'')\in X_{p}({\overline{\rho}})$ such that $\epsilon''$ is locally algebraic of [*nondominant*]{} weight. In particular there is some $v\in S_p$ such that the image of $x''$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ is of the form $(r'_v, \imath_v^{-1}(\epsilon''_v))$ with $\imath_v^{-1}(\epsilon''_v)$ locally algebraic [*not*]{} strictly dominant. This contradicts [@BHS Lem.2.11].

Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant point such that for all $v\in S_p$ the eigenvalues of the geometric Frobenius on ${\rm WD}(\rho|_{{\mathcal{G}}_{F_{\tilde v}}})$ are pairwise distinct. Recall that we have associated in §\[weyl\] a Weyl group element $w_{x_v}$ to the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ via (\[eigenvartotrianguline\]). We write: $$\label{wx}
w_x:=(w_{x_v})_{v\in S_p}\in \prod_{v\in S_p}\Big(\prod_{F_{\tilde v}\hookrightarrow L}{\mathcal{S}}_n\Big)$$ for the corresponding element of the Weyl group of $({\mathrm{Res}}_{F^+/{\mathbb{Q}}}G)_L\buildrel\sim\over\rightarrow \prod_{v\in S_p} ({\mathrm{Res}}_{F_{\tilde v}/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,F_{\tilde v}})_L$. We then obtain the following corollary, which is our main result.

\[mainclassic\] Let $x=(\rho,\delta)\in Y(U^p,{\overline{\rho}})$ be a crystalline strictly dominant very regular point. Assume that the Weyl group element $w_{x}$ in (\[wx\]) is a product of pairwise distinct simple reflections. Then $x$ is classical. Moreover all eigenvectors associated to $x$ are classical, that is we have (see the proof of Proposition \[compaclas\] for $\widehat S(U^p,L)_{\mathfrak{m}^S}^{\lambda-{\rm la}}$): $$\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(S(U^p,L)^{\lambda-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\buildrel\sim\over\longrightarrow\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big).$$

Keep the notation of Theorem \[classicalitycrit\]. By Proposition \[acconZtri\] below, for each $v\in S_p$ there is a sufficiently small open neighbourhood $V_v$ of $x_v$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ such that the irreducible component $Z_{{\rm tri},V_v}(x_v)$ of $V_v$ in (ii) of Corollary \[defnZtri(x)\] is defined and satisfies the accumulation property at $x_v$ (Definition \[accu\]).

Seeing $x$ in $X_p({\overline{\rho}})$ via the closed embedding $Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})$, by (\[union\]) there exist irreducible components ${\mathfrak{X}}^p$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}$ and $Z=\prod_{v\in S_p}Z_v$ of $X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ such that: $$x\in {\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g\subseteq {\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times{\mathbb{U}}^g\buildrel (\ref{union}) \over\subseteq X_p({\overline{\rho}}).$$ Then it follows from Proposition \[acconZaut\] and Remark \[obvious\] below that $Z_v$ satisfies the accumulation property at $x_v$ for all $v\in S_p$. Let $Y_v\subseteq Z_v\cap V_v$ be a nonempty union of irreducible components of $V_v$, then $X_v:=Y_v\cup Z_{{\rm tri},V_v}(x_v)$ satisfies the accumulation property at $x_v$ since both $Y_v$ and $Z_{{\rm tri},V_v}(x_v)$ do. But $X_v$ is smooth at $x_v$ by the assumption on $w_{x_v}$ and Corollary \[Xtrismooth\] applied with $(X,x)=(X_v,x_v)$, hence there can only be one irreducible component of $X_v$ passing through $x_v$. We deduce in particular $Z_{{\rm tri},V_v}(x_v)\subseteq Y_v\subseteq Z_v$, hence $\prod_{v\in S_p}Z_{{\rm tri},V_v}(x_v)\subseteq \prod_{v\in S_p}Z_v\subseteq X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ and $x$ is classical by Theorem \[classicalitycrit\]. We also deduce that the only possible $Z=\prod_{v\in S_p}Z_v$ passing through $(x_v)_{v\in S_p}$ is smooth at $(x_v)_{v\in S_p}$, hence that $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ is smooth at $(x_v)_{v\in S_p}$.

Let us now prove the last statement. From the injection: $$\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(S(U^p,L)^{\lambda-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)\hookrightarrow\mathrm{Hom}_{T_p}\big(\delta,J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_\rho]\otimes_{k(\mathfrak{p}_\rho)}k(x))\big)$$ it is enough to prove that these two $k(x)$-vector spaces have the same (finite) dimension. Recall from [@BHS §3.2] that for any $x'=(y',\delta')\in X_p({\overline{\rho}})$ we have an isomorphism of $k(x')$-vector spaces: $$\label{iso1}
\mathrm{Hom}_{T_p}\big(\delta',J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x'))\big)\cong \mathcal{M}_\infty\otimes_{\mathcal{O}_{X_p(\overline{\rho})}} k(x').$$ If moreover $x'=(\rho',\delta')\in Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})$ we have $\widehat S(U^p,L)_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]=\Pi_{\infty}[\mathfrak{p}_{y'}]$, hence an isomorphism of $k(x')$-vector spaces: $$\label{iso2}
\mathrm{Hom}_{T_p}\big(\delta',J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)\simeq\mathrm{Hom}_{T_p}\big(\delta',J_{B_p}(\Pi_\infty^{R_\infty-{\rm an}}[\mathfrak{p}_{y'}]\otimes_{k(\mathfrak{p}_{y'})}k(x'))\big).$$ We first claim that $x$ is a smooth point of $X_p(\overline{\rho})$. Indeed, by what we proved above, it is enough to show that its component $y^p=(y_v)_{v\in S\backslash S_p}$ in $\mathfrak{X}_{\overline{\rho}^p}$ is a smooth point. As $x$ is classical, by Proposition \[compaclas\] (in particular the end of the proof) it corresponds to an automorphic representation $\pi$ of $G(\mathbb{A}_{F^+})$ with cuspidal strong base change $\Pi$ to $\mathrm{GL}_n(\mathbb{A}_F)$. It then follows from [@Caraiani Th.1.2] that $\Pi$ is tempered, in particular generic, at all finite places of $F$. Then [@BLGGT Lem.1.3.2(1)] implies that $y_v$ for $v\in S\backslash S_p$ is a smooth point of $({\mathrm{Spf}}\, R_{{\overline{\rho}}_{\tilde v}}^{\overline\square})^{{\mathrm{rig}}}$. As $\mathcal{M}_\infty$ is Cohen-Macaulay (Lemma \[CM\]) and $x$ is smooth on $X_p(\overline{\rho})$, we conclude from [@EGAIV1 Cor.17.3.5(i)] that $\mathcal{M}_\infty$ is actually locally free at $x$. Consequently there exists an open affinoid neighbourhood of $x$ in $X_p(\overline{\rho})$ on which the dimension of the fibers of $\mathcal{M}_\infty$ is constant. Intersecting this neighbourhood with $Y(U^p,\overline{\rho})$ and using (\[iso1\]) and (\[iso2\]), we obtain an open affinoid neighbourhood $V_x$ of $x$ in $Y(U^p,\overline{\rho})$ on which $\dim_{k(x')}\mathrm{Hom}_{T_p}(\delta',J_{B_p}(\widehat{S}(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x')))$ is constant for $x'=(\rho',\delta')\in V_x$.

Now let $x'\in V_x$ be a very classical point in the sense of [@BHS Def.3.16] and write $\omega(x')=\delta_{\lambda'}$ with dominant $\lambda'\in \prod_{v\in S_p}({\mathbb{Z}}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$. It follows from [*loc. cit.*]{} and [@BreuilAnalytiqueII Th.4.3] that we have: $$\begin{aligned}
\label{adjla}
{\scriptstyle {\mathrm{Hom}}_{T_p}\big(\delta',\ J_{B_p}(\widehat S(U^p,L)^{\rm an}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)}&\cong&{\scriptstyle {\mathrm{Hom}}_{G_p}\big({\rm LA}(\delta'),\ \widehat S(U^p,L)_{\mathfrak{m}^S}^{\rm an}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x')\big)}\\
&\cong& {\scriptstyle {\mathrm{Hom}}_{G_p}\big({\rm LA}(\delta'),\ \widehat S(U^p,L)_{\mathfrak{m}^S}^{\lambda'-{\rm la}}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x')\big)}\\
&\cong &{\scriptstyle {\mathrm{Hom}}_{T_p}\big(\delta',\ J_{B_p}(\widehat S(U^p,L)^{\lambda'-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)}.\end{aligned}$$ From what is proved above, it is thus enough to find a very classical point $x'$ in $V_x$ such that: $$\begin{gathered}
\label{dimequal}
\dim_{k(x')}{\mathrm{Hom}}_{T_p}\big(\delta',J_{B_p}(\widehat S(U^p,L)^{\lambda'-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho'}]\otimes_{k(\mathfrak{p}_{\rho'})}k(x'))\big)=\\
\dim_{k(x)}{\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\widehat S(U^p,L)^{\lambda-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho}]\otimes_{k(\mathfrak{p}_{\rho})}k(x))\big).\end{gathered}$$ Let $x''=(\rho'',\delta'')\in Y(U^p,{\overline{\rho}})$ be any classical crystalline strictly dominant point and let $\omega(x'')=\delta_{\lambda''}$. By Proposition \[compaclas\] it corresponds to a unique automorphic representation $\pi''$ which moreover has multiplicity $1$, hence we have (with the notation of the proof of Proposition \[compaclas\]): $$J_{B_p}\big(S(U^p,L)_{\mathfrak{m}^S}^{\lambda''-{\rm la}}[\mathfrak{p_{\rho''}}]\otimes_{k(\mathfrak{p_{\rho''}})}\overline{\mathbb{Q}}_p\big)\simeq J_{B_p}\big(L(\lambda'')\otimes_L\bigotimes_{v\in S_p}\pi''_{v}\big)\otimes_{{\overline{\mathbb{Q}}}}({\pi_f''}^{p})^{U^p}\otimes_{{\overline{\mathbb{Q}}},j_p}\overline{\mathbb{Q}}_p.$$ From the definition of $S$ together with [@EmertonJacquetI Prop.4.3.6] and property (iv) in Proposition \[compaclas\], it then easily follows that: $$\label{calcdim}
\dim_{k(x'')}{\mathrm{Hom}}_{T_p}\big(\delta'',J_{B_p}(\widehat S(U^p,L)^{\lambda''-{\rm la}}_{\mathfrak{m}^S}[\mathfrak{p}_{\rho''}]\otimes_{k(\mathfrak{p}_{\rho''})}k(x''))\big)=\dim_{{\overline{\mathbb{Q}}}}\Big(\bigotimes_{v\in S\backslash S_p} {\pi_v''}^{U_v}\Big).$$ Let $Z$ be the union of $x$ and of the very classical points in $V_x$, by [@BHS Thm.3.18] this set $Z$ accumulates at $x$. By [@Caraiani Th.1.2], we can apply [@Chenevierfern Lem.4.5(ii)] to the intersection of $Z$ with one irreducible component of $V_x$, and obtain that, for $v\nmid p$, the value $\dim_{{\overline{\mathbb{Q}}}}{\pi''_{v}}^{U_v}$ is constant on this intersection. In particular $\dim_{{\overline{\mathbb{Q}}}}(\bigotimes_{v\in S\backslash S_p} {\pi_v''}^{U_v})$ is also constant on this intersection, which finishes the proof by (\[calcdim\]) and (\[dimequal\]).

\[uniqueirredcompo\] [(i) Keeping the notation of Theorem \[classicalitycrit\], if there is a [*unique*]{} irreducible component $Z$ of $X_{\rm tri}^\square({\overline{\rho}}_p)$ passing through the image of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_p)$, or equivalently if for each $v\in S_p$ there is a unique irreducible component of $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ passing through $x_v$, then $x$ is classical. Indeed, in that case there is an irreducible component ${\mathfrak{X}}^p$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}$ such that $x\in {\mathfrak{X}}^p\times Z\times {\mathbb{U}}^g={\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times{\mathbb{U}}^g$. In particular, for a sufficiently small open neighbourhood $V_v$ of $x_v$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$, we have $\prod_{v\in S_p} V_v \subseteq Z=X_{\rm tri}^{{\mathfrak{X}}^p\rm-aut}({\overline{\rho}}_p)$ and we see that the assumption in Theorem \[classicalitycrit\] is [*a fortiori*]{} satisfied.\
(ii) Let us recall the various global hypothesis underlying the statements of Theorem \[classicalitycrit\] and Corollary \[mainclassic\]: $p>2$, $G$ is quasi-split at all finite places of $F^+$, $F/F^+$ is unramified, $U_v$ is hyperspecial if $v$ is inert in $F$, ${\overline{\rho}}(\mathcal{G}_{F(\zeta_p)})$ is adequate and $\zeta_p\notin \overline F^{\ker({\overline{\rho}}\otimes {\overline{\rho}}')}$.]{}

Accumulation properties
-----------------------

We prove some accumulation properties (as in Definition \[accu\]) that are used in the proof of Corollary \[mainclassic\] in order to apply Corollary \[Xtrismooth\].

We first go back to the purely local set up of §\[localpart1\]. We call a point $x=(r,\delta_1,\dots,\delta_n)\in X_{\rm tri}^\square(\overline r)$ *saturated* if there exists a triangulation of the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ with parameter $(\delta_1,\dots,\delta_n)$ (cf. §\[begin\]). Note that, if $x$ is crystalline strictly dominant with pairwise distinct Frobenius eigenvalues, then $x$ is saturated if and only if $x$ is noncritical (cf. §§\[variant\], \[weyl\]). Recall also from §\[begin\] that if $x$ is saturated and if $(\delta_1,\dots,\delta_n)\in {\mathcal{T}}_{\rm reg}^n$ then $x\in U_{\rm tri}^\square(\overline r)$.

\[numnoncrit\] Let $x=(r,\delta_1,...,\delta_n)\in X_{\rm tri}^\square(\overline r)$ with $\omega(x)=\delta_{\bf k}$ for some ${\bf k}\!=\!(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\!\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. Assume that: $$\label{eqnnumnoncrit}
k_{\tau,i}-k_{\tau,i+1}> [K:K_0] {\mathrm{val}}\big(\delta_1(\varpi_K)\cdots \delta_i(\varpi_K)\big)$$ for $i\in \{1,\dots,n-1\}$, $\tau\in {\mathrm{Hom}}(K,L)$. Then $x$ is saturated and $r$ is semi-stable. If moreover $(\delta_1,\dots,\delta_n)\in {\mathcal{T}}^n_{\rm reg}$, then $r$ is crystalline strictly dominant noncritical.

By [@KPX Th.6.3.13] and [@BHS Prop.2.9] the representation $r$ is trianguline with parameter $(\delta'_1,\dots, \delta'_n)$ where $\delta'_i=\delta_i z^{{\bf k}_{w^{-1}(i)}-{\bf k}_i}$ for some $w=(w_\tau)_{\tau:\, K\hookrightarrow L}\in W=\prod_{\tau:K\hookrightarrow L} {\mathcal{S}}_n$. As $D_{{\mathrm{rig}}}(r)$, and hence $\wedge_{\mathcal{R}_{k(x),K}}^i D_{{\mathrm{rig}}}(r)$, are $\varphi$-modules over $\mathcal{R}_{k(x),K}$ which are pure of slope zero (being étale $(\varphi,\Gamma_K)$-modules), it follows that for all $i$: $$1\leq \big|\delta'_1(\varpi_K)\cdots\delta'_i(\varpi_K)\big|_K.$$ Since $\delta'_1(\varpi_K)\cdots\delta'_i(\varpi_K)=\delta_1(\varpi_K)\cdots\delta_i(\varpi_K)\cdot \prod_{j=1}^i\prod_\tau (\tau(\varpi_K)^{k_{\tau,w_{\tau}^{-1}(j)}-k_{\tau,j}})$ we obtain: $$\label{slope}
{\mathrm{val}}\big(\delta_1(\varpi_K)\cdots \delta_i(\varpi_K)\big) \geq \tfrac{1}{[K:K_0]}\sum_{j=1}^i\sum_{\tau}(k_{\tau,j}-k_{\tau,w_\tau^{-1}(j)}).$$ We now prove by induction on $i$ that $w_\tau^{-1}(i)=i$ for all $\tau$. The inequality (\[slope\]) for $i=1$ gives ${\mathrm{val}}(\delta_1(\varpi_K))\geq \tfrac{1}{[K:K_0]}\sum_\tau (k_{\tau,1}-k_{\tau,w_\tau^{-1}(1)})$. But assumption $(\ref{eqnnumnoncrit})$ with $i=1$ implies ${\mathrm{val}}(\delta_1(\varpi_K))< \tfrac{1}{[K:K_0]}\sum_\tau (k_{\tau,1}-k_{\tau,j})$ for $j\in \{2,\dots,n\}$ which forces $w_\tau^{-1}(1)=1$ for all $\tau$. Assume by induction that $w_\tau^{-1}(j)=j$ for all $j\leq i-1$ and all $\tau$. Then (\[slope\]) gives: $${\mathrm{val}}(\delta_1(\varpi_K)\cdots\delta_i(\varpi_K))\geq \tfrac{1}{[K:K_0]}\sum_\tau (k_{\tau,i}-k_{\tau,w_\tau^{-1}(i)})$$ and again $(\ref{eqnnumnoncrit})$ implies ${\mathrm{val}}(\delta_1(\varpi_K)\cdots\delta_i(\varpi_K))< \tfrac{1}{[K:K_0]}\sum_\tau (k_{i,\tau}-k_{j,\tau})$ for $j\in \{i,\dots,n\}$ which forces $w_\tau^{-1}(i)=i$ for all $\tau$. We thus have $(\delta_1,\dots, \delta_n)=(\delta'_1,\dots, \delta'_n)$ which implies that the point $x=(r,\delta_1,\dots, \delta_n)$ is saturated. Since $\delta$ is strictly dominant, we obtain that $r$ is semi-stable by the argument in the proof of [@Chenevier Th.3.14] (see also the proof of [@HellmSchrDensity Cor.2.7(i)]). By a slight generalisation of the proof of Lemma \[paramofcrystpt\] (that we leave to the reader), we have $\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i)$ where the $\varphi_i$ are the eigenvalues of the linearized Frobenius $\varphi^{[K_0:{\mathbb{Q}}_p]}$ on the $K_0\otimes_{{\mathbb{Q}}_p}k(x)$-module $D_{\rm st}(r):=(B_{\rm st}\otimes_{{\mathbb{Q}}_p}r)^{\mathcal{G}_K}$. If in addition $(\delta_1,\dots,\delta_n)\in {\mathcal{T}}^n_{\rm reg}$, then it follows from Remark \[varphi\] that $\varphi_i\varphi_j^{-1}\ne p^{-[K_0:{\mathbb{Q}}_p]}$ for $1\leq i\leq j\leq n$ and the argument of [@Chenevier Th.3.14], [@HellmSchrDensity Cor.2.7(i)] then shows that the monodromy operator $N$ on $D_{\rm st}(r)$ must be zero, i.e. that $r$ is crystalline. This finishes the proof.

\[acconZtri\] Let $x=(r,\delta)\in X_{\rm tri}^\square(\overline r)$ be a crystalline strictly dominant point such that the eigenvalues of the geometric Frobenius on ${\rm WD}(r)$ are pairwise distinct. Then there exists a sufficiently small open neighbourhood $U$ of $x$ in $X_{\rm tri}^\square(\overline r)$ such that the irreducible component $Z_{{\rm tri},U}(x)$ of $U$ in (ii) of Corollary \[defnZtri(x)\] is defined and satisfies the accumulation property at $x$.

Recall that we have to prove that, for any positive real number $C$, the set of points $x'=(r',\delta')\in Z_{{\rm tri},U}(x)$ such that $r'$ is crystalline with pairwise distinct geometric Frobenius eigenvalues on ${\rm WD}(r')$ and $x'$ is noncritical with $\omega(x')=\delta_{\bf k'}$ strictly dominant satisfying: $$\label{accumulation}
k'_{\tau,i}-k'_{\tau,i+1}>C$$ for all $i=1,\dots,n-1$, $\tau:K\hookrightarrow L$ accumulates at $x$.

Let $U$ be an open subset of $x$ in $X_{\rm tri}^\square(\overline r)$ as in (iii) of Corollary \[defnZtri(x)\], i.e. such that for any open $U'\subseteq U$ containing $x$ we have $Z_{{\rm tri},U}(x)\cap U'=Z_{{\rm tri},U'}(x)$. Let $\widetilde Z_{\rm cris}(x)$ as in (i) of Corollary \[defnZtri(x)\], by Lemma $\ref{genericopensinXcris}$, the space $V:=\iota_{\bf k}(\widetilde Z_{\rm cris}(x)\cap \widetilde V_{\overline r}^{\square,{\bf k}{\rm -cr}})$ is Zariski-open and Zariski-dense in $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$, hence accumulates in $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$ at any point of $\iota_{\bf k}(\widetilde Z_{\rm cris}(x))$, in particular at $x$. We claim that it is enough to prove that the points $x'\in U$ as above accumulate in $U$ at every point of $V\cap U$. Indeed, if $U'\subseteq U$ is an open neighbourhood containing $x$, then $U'$ also contains a point $v\in V$. By the accumulation statement at $v\in V\cap U$, the Zariski closure in $U'$ of the points $x'$ contains a small neighbourhood around $v$, hence contains an irreducible component of $U'$ containing $v$. But since $v$ is a smooth point of $U'$ (over $L$) as $v\in U_{\rm tri}^\square(\overline r)$ by the last statement of Lemma \[genericopensinXcris\], there is only one such irreducible component, and since $v\in \iota_{\bf k}(\widetilde Z_{\rm cris}(x))\cap U'\subseteq Z_{{\rm tri},U'}(x)$, we see that this irreducible component must be $Z_{{\rm tri},U'}(x)$. Thus the Zariski closure in $U'$ of the points $x'$ always contains $Z_{{\rm tri},U'}(x)$. This easily implies the proposition since $Z_{{\rm tri},U'}(x)=Z_{{\rm tri},U}(x)\cap U'$.

Since $U_{\rm tri}^\square(\overline r)$ is open in $X_{\rm tri}^\square(\overline r)$, it is enough to prove that the crystalline points $x'$ in $U\cap U_{\rm tri}^\square(\overline r)$ satisfying the conditions in the first paragraph of this proof accumulate at any crystalline strictly dominant point $x$ of $U\cap U_{\rm tri}^\square(\overline r)$. The condition on their Frobenius eigenvalues is then in fact automatic by Remark \[varphi\]. Shrinking $U$ further if necessary, we can take $U$ to be contained in some quasi-compact open neighbourhood of $x$ in $X_{\rm tri}^\square(\overline r)$, and thus we may assume that for $i\in \{1,\dots,n\}$ the functions $y=(r_y,(\delta_{y,1},\dots,\delta_{y,n}))\mapsto \delta_{y,i}(\varpi_K)$ are uniformly bounded on $U$. Hence by Lemma $\ref{numnoncrit}$ we may assume that $C$ is sufficiently large so that the points $x'\in U\cap U_{\rm tri}^\square(\overline r)$ with $\omega(x)=\delta_{\bf k'}$ algebraic satisfying $(\ref{accumulation})$ are in fact also automatically crystalline noncritical. Changing notation, we see that it is finally enough to prove that the points $x'\in U_{\rm tri}^\square(\overline r)$ satisfying (\[accumulation\]) for $C$ big enough accumulate at any crystalline strictly dominant point $x$ of $U_{\rm tri}^\square(\overline r)$.

We now consider the rigid analytic spaces ${\mathcal{S}}_n$, ${\mathcal{S}}^\square(\overline r)$ appearing in the proof of [@BHS Th.2.6] (to which we refer the reader for more details; do not confuse here ${\mathcal{S}}_n$ with the permutation group!). In [*loc. cit.*]{} there is a diagram of rigid spaces over ${\mathcal{T}}_L^n$: $$\begin{xy}\xymatrix{&{\mathcal{S}}^\square(\overline r)\ar[dl]_{\pi_{\overline r}}\ar[dr]^g&\\ U_{\rm tri}^\square(\overline r) & & {\mathcal{S}}_n}
\end{xy}$$ where $\pi_{\overline r}$ is a ${\mathbb{G}}_m^n$-torsor and $g$ is a composition ${\mathcal{S}}^\square(\overline r)\hookrightarrow {\mathcal{S}}_n^{\square,{\rm adm}}\rightarrow {\mathcal{S}}_n^{{\rm adm}}\hookrightarrow {\mathcal{S}}_n$ where the first and last maps are open embeddings and the middle one is a ${\mathrm{GL}}_n$-torsor.

Let us choose a point $\tilde x\in \pi_{\overline r}^{-1}(x)$. As $\pi_{\overline r}$ is a ${\mathbb{G}}_m^n$-torsor, it is enough to prove that the points in ${\mathcal{S}}^\square(\overline r)$ satisfying $(\ref{accumulation})$ accumulate at $\tilde x$. The same argument shows that it is enough to prove that the points of ${\mathcal{S}}_n$ satisfying $(\ref{accumulation})$ accumulate at $g(\tilde x)$. But the morphism ${\mathcal{S}}_n\rightarrow {\mathcal{T}}^n_L$ is a composition of open embeddings and structure morphisms of geometric vector bundles (compare the proof of [@HellmSchrDensity Th.2.4]). It follows that $g(\tilde x)$ has a basis of neighbourhoods $(U_i)_{i\in I}$ in ${\mathcal{S}}_n$ such that $V_i:=\omega(U_i)$ is a basis of neighbourhoods of $\omega( x)$ in ${\mathcal{W}}_L^n$ and such that the rigid space $U_i$ is isomorphic to a product $V_i\times{\mathbb{B}}$ of rigid spaces over $L$ where ${\mathbb{B}}$ is some closed polydisc (compare [@Chenevier Cor.3.5] and [@HellmSchrDensity Lem.2.18]). Write $\omega( x)=\delta_{\bf k}$, it is thus enough to prove that the algebraic weights $\delta_{\bf k'}\in {\mathcal{W}}_L^n$ satisfying $(\ref{accumulation})$ accumulate at $\delta_{\bf k}$ in ${\mathcal{W}}_L^n$, which is obvious.

We are now back to the global setting of §\[firstclassical\]. Similarly to Definition \[accu\], we say that a union $X$ of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ satisfies the accumulation property at a point $x\in X$ if, for any positive real number $C>0$, $X$ contains crystalline strictly dominant points $x'=(x'_v)_{v\in S_p}$ with pairwise distinct Frobenius eigenvalues, which are noncritical, such that $\omega(x'_v)=\delta_{{\bf k}'_v}$ with $k'_{v,\tau,i}-k'_{v,\tau,i+1}>C$ for $v\in S_p$, $i\in \{1,\dots,n-1\}$, $\tau\in {\mathrm{Hom}}(F_{\tilde v},L)$ and that accumulate at $x$ in $X$.

\[acconZaut\] Let ${\mathfrak{X}}^p\subset {\mathfrak{X}}_{{\overline{\rho}}^p}$ be an irreducible component and $x\in X_{\rm tri}^{{\mathfrak{X}}^p\rm- aut}({\overline{\rho}}_p)$ be a crystalline strictly dominant point. Then $X_{\rm tri}^{{\mathfrak{X}}^p\rm-aut}({\overline{\rho}}_p)$ satisfies the accumulation property at $x$.

It is enough to show that, for $C$ large enough, the points of ${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g$ such that their projection to $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ is a point $x'=(x'_v)_{v\in S_p}$ of the same form as above accumulate at any point of ${\mathfrak{X}}^p\times X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)\times {\mathbb{U}}^g$ mapping to $x$ in $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$. Using (\[union\]) this claim is contained in the proof of [@BHS Th.3.18] (which itself is a consequence of [@BHS Prop.3.10]).

\[obvious\] [It is obvious from the definition that if a union $X$ of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_p)=\prod_{v\in S_p}X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ satisfies the accumulation property at some point $x\in X$, then for each $v\in S_p$ the image of $X$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ (which is a union of irreducible components of $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$) satisfies the accumulation property at the image of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$.]{}

On the local geometry of the trianguline variety {#phiGammacohomology}
================================================

This section is entirely local and devoted to the proof of Theorem \[upperbound\] above giving an upper bound on some local tangent spaces. We use the notation of §\[localpart1\].

Tangent spaces {#start}
--------------

We start with easy preliminary lemmas on some tangent spaces.

If $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$, we denote by ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ the usual $k(x)$-vector space of $\mathcal{G}_K$-extensions $0\rightarrow r\rightarrow *\rightarrow r\rightarrow  0$.

\[ext1\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be any point, then there is an exact sequence of $k(x)$-vector spaces $0\rightarrow K(r)\rightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\rightarrow {\rm Ext}^1_{\mathcal{G}_K}(r,r)\rightarrow 0$ where $K(r)$ is a $k(x)$-subvector space of $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$ of dimension $\dim_{k(x)}{\mathrm{End}}_{k(x)}(r)-\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)=n^2-\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)$.

It easily follows from [@KisinModularity Lem.2.3.3 & Prop.2.3.5] that there is a topological isomorphism $\widehat{\mathcal{O}}_{\mathfrak{X}^\square_{{\overline{r}}},r}\cong R^\square_{r}$ where the former is the completed local ring at $r$ to the rigid analytic variety $\mathfrak{X}^\square_{{\overline{r}}}$ and the latter is the framed local deformation ring of $r$ in equal characteristic $0$. In particular from (\[tangent\]) we have $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\cong {\mathrm{Hom}}_{k(x)}\big(R^\square_{r},k(x)[\varepsilon]/(\varepsilon^2)\big)$. Then the result follows by the same argument as in [@KisinModularity §2.3.4], seeing an element of ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ as a deformation of $r$ with values in $k(x)[\varepsilon]/(\varepsilon^2)$.

\[dim\] Let $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ be a point such that $H^2(\mathcal{G}_K,r\otimes r')=0$ ($r'$ being the dual of $r$), then $\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)=\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)+n^2[K:{\mathbb{Q}}_p]$.

This follows by the usual argument computing $\dim_{k(x)}H^1(\mathcal{G}_K,r\otimes r')$ from the Euler characteristic formula of Galois cohomology using $\dim_{k(x)}H^0(\mathcal{G}_K,r\otimes r')=\dim_{k(x)}{\mathrm{End}}_{\mathcal{G}_K}(r)$ and $\dim_{k(x)}H^2(\mathcal{G}_K,r\otimes r')=0$.

\[cascr\] [Lemma \[dim\] in particular holds if $x$ is crystalline and the Frobenius eigenvalues $(\varphi_i)_{1\leq i\leq n}$ (see Lemma \[paramofcrystpt\]) satisfy $\varphi_i\varphi_j^{-1}\ne q$ for $1\leq i,j\leq n$. In particular it holds if $x$ is crystalline strictly dominant very regular (cf. Definition \[veryreg\]).]{}

We now fix a point $x=(r,\delta)\in X_{\rm tri}^\square({\overline{r}})$ which is crystalline strictly dominant very regular and a union $X$ of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$ such that $X$ satisfies the accumulation property at $x$ (Definition \[accu\]). It obviously doesn’t change the tangent space $T_{X,x}$ of $X$ at $x$ if we replace $X$ by the union of its irreducible components that contain $x$, hence we may (and do) assume that $x$ belongs to each irreducible component of $X$.

\[inj\] There is an injection of $k(x)$-vector spaces $T_{X,x}\hookrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$.

The embedding $X\hookrightarrow X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ induces an injection on tangent spaces (with obvious notation): $$T_{X,x}\hookrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\oplus T_{\mathcal{T}^n_L,\delta}.$$ We thus have to show that the composition with the projection $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\oplus T_{\mathcal{T}^n_L,\delta}\twoheadrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$ remains injective. Let $\vec{v}\in T_{X,x}$ which maps to $0\in T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$, and thus [*a fortiori*]{} to $0$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ via the surjection in Lemma \[ext1\]. We have to show that the image of $\vec{v}$ in $T_{\mathcal{T}^n_L,\delta}$ is also $0$. We know that the image of $\vec{v}$ in $T_{\mathcal{W}^n_L,\omega(x)}$ is zero since the Hodge-Tate weights don’t vary (that is, the $d_{\tau,i,\vec{v}}$ below are all zero, see the beginning of §\[wedgesection\]). To conclude that the image in $T_{\mathcal{T}^n_L,\delta}$ is also $0$, we can for instance use Bergdall’s Theorem \[bergsplit\] below (which uses the accumulation property of $X$ at $x$) together with an obvious induction on $i$.

\[boundtx\] Assume that the $k(x)$-vector space image of $T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ has dimension smaller or equal than: $$\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big).$$ Then Theorem \[upperbound\] is true.

From Lemma \[inj\] and Lemma \[ext1\] we obtain a short exact sequence: $$\begin{aligned}
\label{exactseq}
0\longrightarrow K(r)\cap T_{X,x}\longrightarrow T_{X,x}\longrightarrow {\rm Ext}^1_{\mathcal{G}_K}(r,r).\end{aligned}$$ Hence the assumption implies: $$\dim_{k(x)}T_{X,x}\leq \dim_{k(x)}K(r)+\lg(w_x)-d_x+\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-[K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}.$$ But from Lemma \[ext1\], Lemma \[dim\] and Remark \[cascr\] we have: $$\begin{gathered}
\dim_{k(x}K(r)+\lg(w_x)-d_x+\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-[K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}=\\
\lg(w_x)-d_x+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}\end{gathered}$$ which gives Theorem \[upperbound\].

We will see below that $d_x$ correspond to the “weight conditions” and $[K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)$ correspond to the “splitting conditions”.

Tangent spaces and local triangulations {#wedgesection}
---------------------------------------

We recall some of the results of [@Bergdall] that we use to prove a technical statement on the image of $T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ (Corollary \[inclv\]).

We keep the notation of §\[start\], in particular $x=(r,\delta)=(r,\delta_1,\dots,\delta_n)\in X_{\rm tri}^\square({\overline{r}})$ is a crystalline strictly dominant very regular point and $X$ is a union of irreducible components of an open subset of $X_{\rm tri}^\square({\overline{r}})$, each component in $X$ satisfying the accumulation property at $x$. Taking a look at [@Bergdall §§5.1,6.1], it is easy to see from the properties of $X_{\rm tri}^\square({\overline{r}})$ and from Definition \[accu\] (together with the discussion that follows) that one can apply all the results of [@Bergdall §7] at $X$ and the point $x$ (called the “center” and denoted by $x_0$ in [*loc. cit.*]{}). We let $w_x=(w_{x,\tau})_{\tau:\, K\hookrightarrow L}\in \prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$ be the Weyl group element associated to $x$ (§\[weyl\]).

Recall that $D_{{\mathrm{rig}}}(r)$ is the étale $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{k(x),K}=k(x)\otimes_{{\mathbb{Q}}_p}\mathcal{R}_{K}$ associated to $r$. Note that ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\cong {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ where the right hand side denotes the extension in the category of $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{k(x),K}$ (see [@BelChe Prop.5.2.6] for $K={\mathbb{Q}}_p$, the proof for any $K$ is analogous). We write $\omega(x)=\delta_{\bf k}$ for ${\bf k}=(k_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in(\mathbb{Z}^n)^{{\mathrm{Hom}}(K,L)}$. Let $\vec{v}\in T_{X,x}$, seeing the image of $\vec{v}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ as a $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$, we can write its Sen weights as $(k_{\tau,i}+\varepsilon d_{\tau,i,\vec{v}})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ for some $d_{\tau,i,\vec{v}}\in k(x)$. The tangent space $T_{{\mathcal W}_L^n,\omega(x)}$ to ${\mathcal W}_L^n$ at $\omega(x)$ is isomorphic to $k(x)^{[K:{\mathbb{Q}}_p]n}$ and the $k(x)$-linear map of tangent spaces $d\omega : T_{X,x}\longrightarrow T_{{\mathcal W}_L,\omega(x)}$ induced by the weight map $\omega\vert_X$ sends $\vec{v}$ to the tuple $(d_{\tau,i,\vec{v}})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$. The following theorem is a direct application of [@Bergdall Th.7.1].

\[bergweight\] For any $\vec{v}\in T_{X,x}$, we have $d_{\tau,i,\vec{v}}=d_{\tau,w_{x,\tau}^{-1}(i),\vec{v}}$ for $1\leq i\leq n$ and $\tau:\, K\hookrightarrow L$.

Let $\vec{v}\in T_{X,x}$, we can see $\vec{v}$ as a $k(x)[\varepsilon]/(\varepsilon^2)$-valued point of $X$, and the composition: $${\mathrm{Sp}}\,k(x)[\varepsilon]/(\varepsilon^2)\buildrel \vec{v}\over \longrightarrow X\hookrightarrow X_{\rm tri}^\square({\overline{r}})\longrightarrow \mathcal{T}^n_L$$ gives rise to continuous characters $\delta_{i,\vec{v}}:K^\times \rightarrow (k(x)[\varepsilon]/(\varepsilon^2))^\times$ for $1\leq i\leq n$. The following theorem again follows from an examination of the proof of [@Bergdall Th.7.1].

\[bergsplit\] For any $\vec{v}\in T_{X,x}$ and $1\leq i\leq n$ we have an injection of $(\varphi,\Gamma_K)$-modules over $\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}=k(x)[\varepsilon]/(\varepsilon^2)\otimes_{{\mathbb{Q}}_p}\mathcal{R}_{K}$: $$\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}\big(\delta_{1,\vec{v}}\delta_{2,\vec{v}}\cdots \delta_{i,\vec{v}}\big)\hookrightarrow D_{{\mathrm{rig}}}(\wedge_{k(x)[\varepsilon]/(\varepsilon^2)}^ir_{\vec{v}})\cong \wedge_{\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}}^iD_{{\mathrm{rig}}}(r_{\vec{v}})$$ where the left hand side is the rank one $(\varphi,\Gamma_K)$-module defined by the character $\delta_{1,\vec{v}}\delta_{2,\vec{v}}\cdots \delta_{i,\vec{v}}$ ([@KPX Cons.6.2.4]) and where $r_{\vec{v}}$ is the $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$ associated to $\vec{v}$.

From [@BelChe Prop.2.4.1] (which readily extends to $K\ne {\mathbb{Q}}_p$) or arguing as in §\[variant\], the $(\varphi,\Gamma_K)$-module $D_{{\mathrm{rig}}}(r)$ has a triangulation ${\mathrm{Fil}}_{\bullet}$ for $\bullet\in \{1,\cdots,n\}$, the graded pieces being: $$\begin{aligned}
\label{triang}
\mathcal{R}_{k(x),K}\big(z^{{\bf k}_{w_x^{-1}(1)}}{\rm unr}(\varphi_1)\big),\dots,\mathcal{R}_{k(x),K}\big(z^{{\bf k}_{w_x^{-1}(n)}}{\rm unr}(\varphi_n)\big)\end{aligned}$$ where ${\bf k}_{w_x^{-1}(i)}:=(k_{\tau,w_{x,\tau}^{-1}(i)})_{\tau:\, K\hookrightarrow L}$ (see (\[charc\]) for $z^{{\bf k}_{j}}$). Note that we have: $$\label{dexat}
\delta_{i}(z)=z^{{\bf k}_{i}-{\bf k}_{w_x^{-1}(i)}}(z^{{\bf k}_{w_x^{-1}(i)}}{\rm unr}(\varphi_i)).$$ For $1\leq i\leq n$ we let $D_{{\mathrm{rig}}}(r)^{\leq i}:={\mathrm{Fil}}_{i}\subseteq D_{{\mathrm{rig}}}(r)$, and we set $D_{{\mathrm{rig}}}(r)^{\leq 0}:=0$. We thus have for $1\leq i\leq n$: $${\rm gr}_iD_{{\mathrm{rig}}}(r):=D_{{\mathrm{rig}}}(r)^{\leq i}/D_{{\mathrm{rig}}}(r)^{\leq i-1}=\mathcal{R}_{k(x),K}\big(z^{{\bf k}_{w_x^{-1}(i)}}{\rm unr}(\varphi_i)\big).$$ For $\tau:\, K\hookrightarrow L$ we fix a Lubin-Tate element $t_\tau\in \mathcal{R}_{L,K}$ as in [@KPX Not.6.2.7] (recall that the ideal $t_\tau\mathcal{R}_{L,K}$ is uniquely determined). If ${\bf k}:=(k_\tau)_{\tau:\, K\hookrightarrow L}\in{\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$, we let $t^{\bf k}:=\prod_{\tau:\, K\hookrightarrow L}t_\tau^{k_{\tau}}$. We set for $1\leq i\leq n$: $$\Sigma_i({\bf k},w_x):=\sum_{j=1}^i({\bf k}_{j}-{\bf k}_{w_x^{-1}(j)})\in {\mathbb{Z}}_{\geq 0}^{{\mathrm{Hom}}(K,L)}$$ (where nonnegativity comes from $k_{\tau,i}\geq k_{\tau,i+1}$ for every $i,\tau$) and we can thus define $t^{\Sigma_i({\bf k},w_x)}\in \mathcal{R}_{L,K}$. In particular we deduce from (\[dexat\]) (and the properties of the $t_\tau$): $$\begin{aligned}
\label{wedgedelta}
\mathcal{R}_{k(x),K}(\delta_{1}\cdots \delta_{i})\cong t^{\Sigma_i({\bf k},w_x)}\wedge^i_{\mathcal{R}_{k(x),K}}D_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow \wedge^i_{\mathcal{R}_{k(x),K}}D_{{\mathrm{rig}}}(r).\end{aligned}$$

We consider for $1\leq i\leq n$ the cartesian square (which defines $V_i$): $$\xymatrix{{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\ar[r]&{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\big)\\ 
V_i \ar[r]\ar@{^{(}->}[u]& 
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)^{\leq i}\big)\ar@{^{(}->}[u]}$$ where the first horizontal map is the restriction map and where the injection on the right easily follows from the very regularity assumption (Definition \[veryreg\]). Equivalently we have: $$\begin{gathered}
\label{viker}
V_i\cong \ker\!\Big({\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\longrightarrow
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\Big)\end{gathered}$$ where the map is defined by pushforward along $D_{{\mathrm{rig}}}(r)\twoheadrightarrow D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}$ and pullback along $t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow D_{{\mathrm{rig}}}(r)$.

\[inclv\] The image of any $\vec{v}\in T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\cong {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ is in $V_1\cap \cdots \cap V_{n-1}$ (where the intersection is within ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$).

Note that $V_1\cap V_2\cap \cdots \cap V_{n}=V_1\cap V_2\cap \cdots \cap V_{n-1}$. Let $\vec{v}\in T_{X,x}$, $r_{\vec{v}}$ the associated $k(x)[\varepsilon]/(\varepsilon^2)$-deformation and see $D_{{\mathrm{rig}}}(r_{\vec{v}})$ as an element of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$. We have to prove that the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$  in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})$ is zero for any $1\leq i\leq n$ (see (\[viker\])). The proof is by induction on $i\geq 1$. The case $i=1$ follows immediately from Theorem \[bergsplit\] and (\[wedgedelta\]) (together with Definition \[veryreg\]). We prove that the statement for $i-1$ implies the statement for $i$.

So, assume $i\geq 2$ and that the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i-1}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1}\big)$$ is zero. Then by Corollary \[zero2\] the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ is also zero. From the exact sequence: $$\begin{gathered}
0\rightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{gathered}$$ (where the injectivity on the left follows from Definition \[veryreg\]), we see that the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ in ${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$ comes from a unique extension: $$D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}\in {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big).$$ We thus have to prove that $D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}=0$.

The twist by the rank one $(\varphi,\Gamma_K)$-module $\wedge_{\mathcal{R}_{k(x),K}}^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1}$ is easily seen (by elementary linear algebra) to induce an isomorphism: $$\begin{gathered}
\label{wedgetwist}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\buildrel\sim\over\longrightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)\end{gathered}$$ where we write $\wedge^\cdot D_{{\mathrm{rig}}}(r)$ for $\wedge_{\mathcal{R}_{k(x),K}}^\cdot D_{{\mathrm{rig}}}(r)$ and where $(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})$ stands for the quotient: $$\begin{gathered}
\big(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)/\big(D_{{\mathrm{rig}}}(r)^{\leq i}\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)\cong\\
\big(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}.\end{gathered}$$ (here, $D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})$ and $D_{{\mathrm{rig}}}(r)^{\leq i}\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})$ are seen inside $\wedge^{i}D_{{\mathrm{rig}}}(r)$). Moreover the injective map $\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1}\hookrightarrow \wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i}$ still induces an injection (using Definition \[veryreg\]): $$\begin{gathered}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i-1})\big)\hookrightarrow\\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big).\end{gathered}$$ Denote by: $$\label{rtilde}
\widetilde D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}\ \in \ {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)$$ the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}$ (using the isomorphism (\[wedgetwist\])). It is thus equivalent to prove that $\widetilde D_{{\mathrm{rig}}}(r_{\vec{v}})^{(i)}=0$. Note that: $$\begin{gathered}
\label{wedgequot}
(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\cong \big(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}.\end{gathered}$$

For $1\leq i\leq n$, we have a $k(x)$-linear map ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\rightarrow {\rm Ext}^1_{\mathcal{G}_K}(\wedge_{k(x)}^ir,\wedge_{k(x)}^ir)$ defined by mapping a $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$ to its $i$-th exterior power over $k(x)[\varepsilon]/(\varepsilon^2)$. This induces an $\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}$-linear map: $${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\longrightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\wedge^i\!D_{{\mathrm{rig}}}(r),\wedge^iD_{{\mathrm{rig}}}(r)\big).$$ Let $D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})\in  {\rm Ext}^1_{(\varphi,\Gamma_K)}(\wedge^i\!D_{{\mathrm{rig}}}(r),\wedge^iD_{{\mathrm{rig}}}(r))$ be the image of $D_{{\mathrm{rig}}}(r_{\vec{v}})$. The pull-back along $\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow \wedge^iD_{{\mathrm{rig}}}(r)$ sends $D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})$ to an element in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^{i}D_{{\mathrm{rig}}}(r))$. Elementary linear algebra (recall $\varepsilon^2=0$!) shows this element in fact belongs to: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)$$ (which embeds into ${\rm Ext}^1_{(\varphi,\Gamma_K)}(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^{i}D_{{\mathrm{rig}}}(r))$ again by Definition \[veryreg\]). The pushforward along: $$D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\twoheadrightarrow (D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i}))/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}$$ now gives by (\[wedgequot\]) an element in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big)$$ and further pull-back along $t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}\hookrightarrow \wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}$ finally gives an element: $$\label{wedgetilde}
\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})\ \in \ {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i})\big).$$ Now, again manipulations of elementary linear algebra show we recover the element $\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}$ of (\[rtilde\]), that is, we have $\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})=\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}$.

But we know from Theorem \[bergsplit\] (using (\[wedgedelta\]) and Definition \[veryreg\]) that the image of $D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})$ (by pullback) in ${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^iD_{{\mathrm{rig}}}(r)\big)$ actually sits in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ (in fact even in the image of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i})$). In particular its image $\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}\wedge^iD_{{\mathrm{rig}}}(r)^{\leq i},(D_{{\mathrm{rig}}}(r)\wedge (\wedge^{i-1}D_{{\mathrm{rig}}}(r)^{\leq i}))/\wedge^{i}D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ must be zero. Since $\widetilde D_{{\mathrm{rig}}}(\wedge^ir_{\vec{v}})=\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}$, we obtain $\widetilde D_{\rm rig}(r_{\vec{v}})^{(i)}=0$.

Proof of the main local theorem {#endofproof}
-------------------------------

We compute various dimensions and finish the proof of Theorem \[upperbound\].

Seeing an element of ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ as a $k(x)[\varepsilon]/(\varepsilon^2)$-valued representation of $\mathcal{G}_K$, we can write its Sen weights as $(k_{\tau,i}+\varepsilon d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ for some $d_{\tau,i}\in k(x)$. We let $V$ be the $k(x)$-subvector space of ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ (or of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$) of extensions such that $d_{\tau,i}=d_{\tau,w_{x,\tau}^{-1}(i)}$ for $1\leq i\leq n$ and $\tau:\, K\hookrightarrow L$.

\[condweight\] We have $\dim_{k(x)}V =\dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x$.

The Sen map ${\rm Ext}^1_{\mathcal{G}_K}(r,r)\cong {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\longrightarrow k(x)^{[K:{\mathbb{Q}}_p]n}$ sending an extension to $(d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ is easily checked to be surjective (by a dévissage argument using Définition \[veryreg\], we are reduced to the rank one case where it is obvious). The  $k(x)$-subvector  space  of  $k(x)^{[K:{\mathbb{Q}}_p]n}$  of  tuples $(d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}$ such that $d_{\tau,i}=d_{\tau,w_{x,\tau}^{-1}(i)}$ for $1\leq i\leq n$ and $\tau:\, K\hookrightarrow L$ has dimension $[K:{\mathbb{Q}}_p]n-d_x$ (argue as in the beginning of the proof of Lemma \[coxeter\]). The result follows.

Recall that a $(\varphi,\Gamma_K)$-module $D$ over $\mathcal{R}_{k(x),K}$ is called [*crystalline*]{} if $D[1/\prod_{\tau:\, K\hookrightarrow L}t_\tau]^{\Gamma_K}$ is free over $K_0\otimes_{{\mathbb{Q}}_p}k(x)$ of the same rank as $D$. If $D,D'$ are two crystalline $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{k(x),K}$, one can define the $k(x)$-subvector space of [*crystalline extensions*]{} ${\rm Ext}^1_{\rm cris}(D,D')\subseteq {\rm Ext}^1_{(\varphi,\Gamma_K)}(D,D')$. Note that ${\rm Ext}^1_{\rm cris}(\cdot,\cdot)$ respects surjectivities on the right entry (resp. sends injectivities to surjectivities on the left entry) as there is no ${\rm Ext}^2_{\rm cris}$, see [@Benois Cor.1.4.6].

\[dim1\] For $1\leq i\leq \ell\leq n$ we have: $$\begin{gathered}
\dim_{k(x)}{\rm Ext}^1_{(\varphi,\Gamma_K)}\big({\rm gr}_iD_{{\mathrm{rig}}}(r)/(t^{\Sigma_{\ell}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)=\\
\sum_{\tau:\, K\hookrightarrow L}\big\vert\{i_1\in \{\ell+1,\dots,n\}, w_{x,\tau}^{-1}(i_1)<w_{x,\tau}^{-1}(i)\}\big\vert.\end{gathered}$$

It follows from Proposition \[cris\] below (applied with $(i,\ell)=(i,i)$ and $(i,\ell)=(i-1,i)$) together with the two exact sequences: $$\begin{gathered}
0\rightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big({\rm gr}_iD_{{\mathrm{rig}}}(r)/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}^{\leq i}(r)/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big),\end{gathered}$$ $$\begin{gathered}
0\rightarrow {\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\rightarrow {\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)
\rightarrow \\
{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{gathered}$$ (injectivity on the left following again from Definition \[veryreg\]), that we have: $$\begin{gathered}
\label{dimcrisi}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big({\rm gr}_iD_{{\mathrm{rig}}}(r)/(t^{\Sigma_{\ell}({\bf k},w_x)}),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\cong \\
{\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big).\end{gathered}$$ By dévissage on $D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}$ using that ${\rm Ext}^1_{\rm cris}$ respects here short exact sequences (by Definition \[veryreg\] and the discussion above), we have: $$\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)=
\sum_{i_1=\ell+1}^{n}\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big({\rm gr}_iD_{{\mathrm{rig}}}(r),{\rm gr}_{i_1}D_{{\mathrm{rig}}}(r)\big).$$ The result follows from (\[dimcris\]) below.

\[condsplit\] We have: $$\dim_{k(x)}\big( V_1\cap V_2\cap \cdots \cap V_{n-1}\big) = \dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big).$$

To lighten notation in this proof, we write $D_{{\mathrm{rig}}}$ instead of $D_{{\mathrm{rig}}}(r)$ and drop the subscript $(\varphi,\Gamma_K)$. We first prove that, for $1\leq i\leq n$, we have an isomorphism of $k(x)$-vector spaces: $$\begin{gathered}
\label{isoi}
V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i} \buildrel\sim\over\longrightarrow\\
{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\end{gathered}$$ where $V_1\cap\cdots \cap V_{i-1}:={\rm Ext}^1(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}})$ if $i=1$. We first define the map. We have the following commutative diagram: $$\begin{matrix}
&&\scriptstyle 0 &&\scriptstyle 0  &&\scriptstyle 0  &&\\
&&\downarrow &&\downarrow &&\downarrow &&\\
\!\!\!\scriptstyle{0}&\!\!\!\scriptstyle{\rightarrow} &\!\!\!\scriptstyle{{\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})}&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i-1}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle 0\\
&&\downarrow &&\downarrow &&\downarrow &&\\
\!\!\!\scriptstyle 0&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle 0\\
&&\downarrow &&\downarrow &&\downarrow &&\\
\!\!\!\scriptstyle 0&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}{\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle{\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})&\!\!\!\scriptstyle\rightarrow &\!\!\!\scriptstyle 0
\end{matrix}$$ where the injections on top and left and the surjections on the two bottom lines all follow from Definition \[veryreg\], and where the surjection on the top right corner follows from Corollary \[zero3\] below. Denote by $E_i$ the inverse image of ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i-1}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$, then we have an isomorphism: $$\begin{gathered}
\label{coim}
{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\buildrel\sim\over\rightarrow \\
E_i\big/{\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big).\end{gathered}$$ We consider the composition: $$\begin{gathered}
\label{mapi}
V_1\cap\cdots \cap V_{i-1}\hookrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)\twoheadrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\twoheadrightarrow \\
{\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\end{gathered}$$ and note that the image of $V_1\cap\cdots \cap V_{i-1}$ falls in $E_i\big/{\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ by Corollary \[zero2\] below. If $v\in V_1\cap\cdots \cap V_{i-1}$ is also in $V_i$, then its image in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ maps to $0$ in ${\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$, hence belongs to ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$. By (\[coim\]), we thus have a canonical induced map: $$\begin{gathered}
\label{imap}
V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i} \rightarrow \\
{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\big/{\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big).\end{gathered}$$ Let us prove that (\[imap\]) is surjective. One easily checks that ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\subseteq V_1\cap\cdots \cap V_{i-1}$ and that the natural map ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\rightarrow {\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)$ is surjective (again by Definition \[veryreg\]). This implies that [*a fortiori*]{} (\[imap\]) must also be surjective. Let us prove that (\[imap\]) is injective. If $v\in V_1\cap\cdots \cap V_{i-1}$ maps to zero, then the image of $v$ in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ belongs to ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i}/(t^{\Sigma_{i}({\bf k},w_x)}),D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ by (\[coim\]), i.e. maps to zero in ${\rm Ext}^1(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$, i.e. $v\in V_i$ by (\[viker\]), hence $v\in V_1\cap\cdots \cap V_{i}$.

We now prove the statement of the proposition. From (\[isoi\]) and Lemma \[dim1\], we obtain for $1\leq i\leq n$: $$\begin{gathered}
\dim_{k(x)}\big(V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i}\big)=\\
[K:{\mathbb{Q}}_p](n-i)-\sum_{\tau:\, K\hookrightarrow L}\big\vert\{j\in \{i+1,\dots,n\}, w_{x,\tau}^{-1}(j)<w_{x,\tau}^{-1}(i)\}\big\vert=\\
\sum_{\tau:\, K\hookrightarrow L}\big\vert\{j\in \{i+1,\dots,n\}, w_{x,\tau}^{-1}(i)<w_{x,\tau}^{-1}(j)\}\big\vert.\end{gathered}$$ Summing up $\dim_{k(x)}(V_1\cap\cdots \cap V_{i-1}/V_1\cap\cdots \cap V_{i})$ for $i=1$ to $n-1$ thus yields: $$\begin{gathered}
\dim_{k(x)}{\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)-\dim_{k(x)}(V_1\cap\cdots \cap V_{n-1})= \\
\sum_{\tau:\, K\hookrightarrow L}\big\vert\{1\leq i_1<i_2\leq n,\ w_{x,\tau}^{-1}(i_1)<w_{x,\tau}^{-1}(i_2)\}\big\vert.\end{gathered}$$ But $\vert\{1\leq i_1<i_2\leq n,\ w_{x,\tau}^{-1}(i_1)<w_{x,\tau}^{-1}(i_2)\}\vert= \frac{n(n-1)}{2}-\lg(w_{x,\tau})$ (see e.g. [@HumBGG §0.3]), and thus we get: $$\begin{aligned}
\dim_{k(x)}(V_1\cap\cdots \cap V_{n-1})&= &\dim_{k(x)}{\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)-\sum_{\tau:\, K\hookrightarrow L}\big(\frac{n(n-1)}{2}-\lg(w_{x,\tau})\big)\\
&= &\dim_{k(x)}{\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big)\end{aligned}$$ which finishes the proof.

\[inter\] We have: $$\dim_{k(x)}\!\big(V \cap (V_1\cap \cdots \cap V_{n-1})\big)\!= \dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big).$$

Consider the following cartesian diagram which defines $W_i$: $$\xymatrix{{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\ar[r]&{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\big)\\ 
W_i \ar[r]\ar@{^{(}->}[u]& 
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)^{\leq i}\big)\ar@{^{(}->}[u]},$$ then $W_i\subseteq V_i$, hence $W_1\cap \cdots \cap W_{n-1}\subseteq V_1\cap \cdots \cap V_{n-1}$. In fact, $W_1\cap \cdots \cap W_{n-1}$ is the $k(x)$-subvector space of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ of extensions which respect the triangulation $(D_{{\mathrm{rig}}}(r)^{\leq i})_{1\leq i\leq n}$ on $D_{{\mathrm{rig}}}(r)$. A dévissage argument (using Definition \[veryreg\]) that we leave to the reader then shows that the composition: $$W_1\cap \cdots \cap W_{n-1}\hookrightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\longrightarrow k(x)^{[K:{\mathbb{Q}}_p]n}$$ (where the second map is the Sen map in the proof of Proposition \[condweight\]) remains surjective. [*A fortiori*]{}, $V_1\cap \cdots \cap V_{n-1}\hookrightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\rightarrow k(x)^{[K:{\mathbb{Q}}_p]n}$ is also surjective. By the same proof as that of Proposition \[condweight\] we get: $$\dim_{k(x)}\!\big(V \cap (V_1\cap \cdots \cap V_{n-1})\big)\!= \dim_{k(x)}(V_1\cap \cdots \cap V_{n-1})-d_x$$ and the result follows from Proposition \[condsplit\].

\[ouf\] The image of $T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ has dimension $\leq \dim_{k(x)}{\rm Ext}^1_{\mathcal{G}_K}(r,r)-d_x-\big([K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}-\lg(w_x)\big)$.

It follows from Theorem \[bergweight\] that the image of any $\vec{v}\in T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ is in $V$. It follows from Corollary \[inclv\] that the image of any $\vec{v}\in T_{X,x}$ in ${\rm Ext}^1_{\mathcal{G}_K}(r,r)$ is also in $V_1\cap \cdots \cap V_{n-1}$. One concludes with Proposition \[inter\].

By Lemma \[boundtx\] this finishes the proof of Theorem \[upperbound\].

[The collection of $(\varphi,\Gamma_K)$-submodules $(t^{\Sigma_i({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i})_{1\leq i\leq n}$ of $D_{{\mathrm{rig}}}(r)$ plays an important role in the proof of Corollary \[ouf\]. One can wonder if they “globalize” in a neighbourhood of the point $x$ in $X$? Note that those for which $\Sigma_i({\bf k},w_x)=0$ do by [@Bergdall Th.A].]{}

Calculation of some Ext groups
------------------------------

We prove several technical but crucial results of Galois cohomology that were used above.

For a continuous character $\delta:K^\times\rightarrow L^\times$ and $\tau:\, K\hookrightarrow L$, we define its (Sen) weight ${\rm wt}_\tau(\delta)\in L$ in the direction $\tau$ by taking the [*opposite*]{} of the weight defined in [@Bergdall §2.3]. For instance ${\rm wt}_\tau(\tau(z)^{k_\tau})=k_\tau$ ($k_\tau\in {\mathbb{Z}}$).

\[null\] Let $\tau:\, K\hookrightarrow L$ and $k_\tau\in {\mathbb{Z}}_{>0}$.\
(i) For $j\in \{0,1\}$ we have ${\rm Ext}^j_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau})\big)\ne 0$ if and only if ${\rm wt}_\tau(\delta)\in \{-(k_\tau-1),\dots,0\}$ and we have ${\rm Ext}^2_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau})\big)=0$ for all $\delta$.\
(ii) For $j\in \{1,2\}$ we have ${\rm Ext}^j_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}\big)\ne 0$ if and only if ${\rm wt}_\tau(\delta)\in \{-k_\tau,\dots,-1\}$ and we have ${\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}\big)=0$ for all $\delta$.\
(iii) When either of these spaces is nonzero, it has dimension $1$ over $L$.

The first part of (i) is in [@Bergdall Prop.2.7] (and initially in [@Co Prop.2.18] for $K={\mathbb{Q}}_p$) and the second part in [@Liuduality Th.3.7(2)]. The second part of (ii) is obvious, let us prove the first. We have an exact sequence: $$\label{exactR}
0\longrightarrow \mathcal{R}_{L,K}(\delta^{-1})\longrightarrow \mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})\longrightarrow \mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\longrightarrow 0.$$ The cup product with (\[exactR\]) yields canonical morphisms of $L$-vector spaces: $$\begin{aligned}
\label{mapL}
{\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\rightarrow & {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big)\\
\nonumber {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\rightarrow & {\rm Ext}^2_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big).\end{aligned}$$ There is an obvious isomorphism of $L$-vector spaces: $$\begin{aligned}
{\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\cong \!\!& {\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)\end{aligned}$$ and an analysis of the cokernel of the multiplication by $t_\tau^{k_\tau}$ map on a short exact sequence $0\rightarrow  \mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\rightarrow {\mathcal E}\rightarrow \mathcal{R}_{L,K}\rightarrow 0$ of $(\varphi,\Gamma_K)$-module over $\mathcal{R}_{L,K}$ yields a canonical morphism of $L$-vector spaces: $$\begin{aligned}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\!\rightarrow \!\!\!& {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big).\end{aligned}$$ Thus we have canonical morphisms of $L$-vector spaces: $$\begin{aligned}
\label{mapL2}
{\rm Ext}^0_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\!\rightarrow \!\!\!& {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big)\\
\nonumber {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K},\mathcal{R}_{L,K}(\tau(z)^{-k_\tau}\delta^{-1})/(t_\tau^{k_\tau})\big)&\!\!\!\rightarrow \!\!\!& {\rm Ext}^2_{(\varphi,\Gamma_K)}\big(\mathcal{R}_{L,K}/(t_\tau^{k_\tau}),\mathcal{R}_{L,K}(\delta^{-1})\big).\end{aligned}$$ It is then a simple exercise of linear algebra to check that the morphisms in (\[mapL\]) fit into a natural morphism of complexes of $L$-vector spaces from the long exact sequence of ${\rm Ext}^j_{(\varphi,\Gamma_K)}(\mathcal{R}_{L,K},\cdot)$ applied to the short exact sequence (\[exactR\]) to the long exact sequence of ${\rm Ext}^j_{(\varphi,\Gamma_K)}(\cdot,\mathcal{R}_{L,K}(\delta^{-1}))$ applied to the short exact sequence $0\rightarrow t_\tau^{k_\tau}\mathcal{R}_{L,K}\rightarrow \mathcal{R}_{L,K}\rightarrow \mathcal{R}_{L,K}/(t_\tau^{k_\tau})\rightarrow 0$ (note that there is a shift in this map of complexes). Since all the morphisms are obviously isomorphisms except possibly the morphisms (\[mapL2\]), we deduce that the latter are also isomorphisms. Twisting by $\mathcal{R}_{L,K}(\delta)$ on the right hand side of (\[mapL2\]) and using (i) applied to the left hand side, the first part of (ii) easily follows. Finally (iii) follows from [@Bergdall Prop.2.7] and from the previous isomorphisms (\[mapL2\]).

Recall that for $i,\ell\in \{1,\dots,n\}$ we have an exact sequence: $$\begin{gathered}
\label{ses}
0\rightarrow {\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\rightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\end{gathered}$$ where the injection on the left follows as usual from Definition \[veryreg\].

\[cris\] For  $1\leq i\leq \ell\leq n$,  we  have  an  isomorphism  of  subspaces  of ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell})$: $$\begin{gathered}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big)\cong \\
{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq \ell}\big).\end{gathered}$$

To lighten notation, we write $D_{{\mathrm{rig}}}$ instead of $D_{{\mathrm{rig}}}(r)$ and drop the subscript $(\varphi,\Gamma_K)$. By the exact sequence (\[ses\]) and a dévissage on $D_{{\mathrm{rig}}}^{\leq i}$ and $D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}$ (recall from Definition \[veryreg\] and the discussion preceding Lemma \[dim1\] that ${\rm Ext}^1_{\rm cris}$ respects short exact sequences here), it is enough to prove (i) that the composition: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big)\subseteq {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big)\longrightarrow  {\rm Ext}^1\big(t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big)$$ is zero and (ii) that: $${\rm Ext}^1\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)\cong {\rm Ext}^1_{\rm cris}\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)$$ (inside ${\rm Ext}^1({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}})$) for all $\ell',\ell''$ such that $\ell'\leq \ell$ and $\ell''\geq \ell+1$.

We prove (i). The map clearly factors through: $${\rm Ext}^1_{\rm cris}\big(t^{\Sigma_{\ell}({\bf k},w_x)}D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq \ell}\big),$$ let us prove that the latter vector space is zero. By dévissage again, it is enough to prove that: $${\rm Ext}^1_{\rm cris}\big(t^{\Sigma_{\ell}({\bf k},w_x)}{\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)=0$$ for $\ell',\ell''$ such that $\ell'\leq \ell$ and $\ell''\geq \ell+1$. It is enough to prove that, for all $\tau:\, K\hookrightarrow L$, we have ${\rm wt}_\tau(t^{\Sigma_{\ell}({\bf k},w_x)}{\rm gr}_{\ell'}D_{{\mathrm{rig}}})\geq {\rm wt}_\tau({\rm gr}_{\ell''}D_{{\mathrm{rig}}})$ (using Definition \[veryreg\] when these two weights are equal). This is equivalent to: $$\begin{aligned}
\label{bound}
\sum_{j=1}^\ell(k_{\tau,j} - k_{\tau,w_{x,\tau}^{-1}(j)})+k_{\tau,w_{x,\tau}^{-1}(\ell')}\geq k_{\tau,w_{x,\tau}^{-1}(\ell'')}\end{aligned}$$ which indeed holds for $\ell'$, $\ell''$ as above because $k_{\tau,1}>k_{\tau,2}>\cdots>k_{\tau,n}$.

We prove (ii). From (i) we have in particular an inclusion: $$\begin{aligned}
\label{incl}
{\rm Ext}^1_{\rm cris}\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)\subseteq {\rm Ext}^1\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big).\end{aligned}$$ It is an easy (and well-known) exercise that we leave to the reader to check that: $$\begin{aligned}
\label{dimcris}
\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)=\big\vert\{\tau:\, K\hookrightarrow L, w_{x,\tau}^{-1}(\ell'')<w_{x,\tau}^{-1}(\ell')\}\big\vert.\end{aligned}$$ On the other hand, from (ii) and (iii) of Lemma \[null\], using (\[bound\]) and $\mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau}t_\sigma^{k_\sigma})\cong \mathcal{R}_{L,K}(\delta)/(t_\tau^{k_\tau})\times \mathcal{R}_{L,K}(\delta)/(t_\sigma^{k_\sigma})$ if $\tau\ne\sigma$, we deduce: $$\begin{gathered}
\label{dimrig}
\dim_{k(x)}{\rm Ext}^1\big({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}}\big)=\big\vert\{\tau:\, K\hookrightarrow L, w_{x,\tau}^{-1}(\ell'')<w_{x,\tau}^{-1}(\ell')\}\big\vert.\end{gathered}$$ (\[incl\]), (\[dimcris\]) and (\[dimrig\]) imply ${\rm Ext}^1_{\rm cris}({\rm gr}_{\ell'}D_{{\mathrm{rig}}},{\rm gr}_{\ell''}D_{{\mathrm{rig}}})\cong {\rm Ext}^1({\rm gr}_{\ell'}D_{{\mathrm{rig}}}/(t^{\Sigma_{\ell}({\bf k},w_x)}),{\rm gr}_{\ell''}D_{{\mathrm{rig}}})$ which finishes the proof.

\[zero2\] Let $i\in \{1,\dots,n\}$, ${\mathcal E}\in {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ and assume that the image of $\mathcal E$ (by pullback and pushforward) in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i-1}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1}\big)$$ is zero. Then the image of $\mathcal E$ in: $${\rm Ext}^1_{(\varphi,\Gamma_K)}\big(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ is also zero.

By  Proposition  \[cris\]  applied  with  $(i,\ell)=(i-1,i-1)$,  the  image  of  $\mathcal E$  in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1})$ sits in: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i-1}\big).$$ Hence its image in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})$ sits in: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big).$$ It  follows  from  Proposition  \[cris\]  again  applied  with  $(i,\ell)=(i-1,i)$  that  it  maps  to zero in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i})$.

\[zero3\] For  $2\leq i\leq n$ we have a surjection: $$\begin{gathered}
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\twoheadrightarrow \\
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{gathered}$$ where the map is the pullback along: $$D_{{\mathrm{rig}}}(r)^{\leq i-1}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i-1}\hookrightarrow D_{{\mathrm{rig}}}(r)^{\leq i}/t^{\Sigma_{i}({\bf k},w_x)}D_{{\mathrm{rig}}}(r)^{\leq i}.$$

This follows from Proposition \[cris\] (applied with $(i,\ell)=(i,i)$ and $(i,\ell)=(i-1,i)$) and the fact that the map: $${\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\longrightarrow
{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r)^{\leq i-1},D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)$$ is surjective.

Modularity and local geometry of the trianguline variety {#modularity}
========================================================

We prove that the main conjecture of [@BHS] (see [@BHS Conj.3.22]), and thus the classical modularity conjectures by [@BHS Prop.3.26], imply Conjecture \[mainconj\] when ${\overline{r}}$ “globalizes” and $x$ is very regular.

A closed embedding {#closed}
------------------

Assuming the main conjecture of [@BHS] and using Theorem \[intercompanion\] below we construct a certain closed embedding in the trianguline variety (Proposition \[companion\]).

We fix a continuous representation ${\overline{r}}:\mathcal{G}_K\rightarrow {\mathrm{GL}}_n(k_L)$ as in §\[begin\] and keep the local notation of §\[localpart1\] and §\[phiGammacohomology\]. We also assume that there exist number fields $F/F^+$, a unitary group $G/F^+$, a tame level $U^p$, a set of finite places $S$ and an irreducible representation ${\overline{\rho}}$ as in §\[classic\] such that all the assumptions in §\[classic\] and §\[firstclassical\] are satisfied, and such that for each place $v\in S_p$ there is a place $\tilde v$ of $F$ dividing $v$ satisfying $F_{\tilde v}\cong K$ and ${\overline{\rho}}_{\tilde v}\cong {\overline{r}}$. Note that this implies in particular $(2n,p)=1$ (as $p>2$ and as $(n,p)=1$ by the proof of [@GHTT Th.9]). Assuming $(2n,p)=1$, it follows from [@CEGGPS Lem.2.2] and [@CEGGPS §2.3] that such $(F/F^+,G,U^p,S,{\overline{\rho}})$ always exist if $n=2$ or if ${\overline{r}}$ is (absolutely) semi-simple (increasing $L$ if necessary).

We recall the statement of [@BHS Conj.3.22] (see §\[weyl\] for $\widetilde X_{\rm tri}^\square({\overline{r}})$).

\[BHS\] The rigid subvariety $X_{\rm tri}^{{\mathfrak{X}}^p \rm-aut}({\overline{\rho}}_p)$ of $X_{\rm tri}^\square({\overline{\rho}}_p)$ doesn’t depend on ${\mathfrak{X}}^p$ and is isomorphic to $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p):= \prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$.

\[conjvariant\] [(i) By (\[union\]), Conjecture \[BHS\] is thus equivalent to $X_p({\overline{\rho}})\buildrel{\sim}\over \rightarrow  {\mathfrak{X}}_{{\overline{\rho}}^p}\times \widetilde X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g$.\
(ii) The authors do not know if $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)$ is really strictly smaller than $X_{\rm tri}^\square({\overline{\rho}}_p)$.\
(iii) Finally, recall that Conjecture \[BHS\] is [*implied*]{} by the classical modularity lifting conjectures for ${\overline{\rho}}$ (in all weights with trivial inertial type), see [@BHS Prop.3.26].]{}

Let ${\bf k}:=({\bf k}_i)_{1\leq i\leq n}$ where ${\bf k}_i:=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ is such that $k_{\tau,i}> k_{\tau,i+1}$ for all $i$ and $\tau$. For $w=(w_{\tau})_{\tau:\, K\hookrightarrow L}\in W=\prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$, denote by $\mathcal{W}^n_{w,{\bf k},L}\subset \mathcal{W}^n_L$ the Zariski-closed (reduced) subset of characters $(\eta_1,\dots,\eta_n)$ defined by the equations: $$\label{equations}
{\rm wt}_\tau(\eta_{w_\tau(i)}\eta_i^{-1})=k_{\tau,i}-k_{\tau,w_\tau^{-1}(i)},\ \ 1\leq i\leq n,\ \ \tau:\, K\hookrightarrow L.$$ For instance one always has: $$\label{point}
(z^{{\bf k}_{w^{-1}(1)}}\chi_1,\dots,z^{{\bf k}_{w^{-1}(n)}}\chi_n)\in \mathcal{W}^n_{w,{\bf k},L}$$ where $\chi_i\in \mathcal{W}_L$ are finite order characters. Note that $\mathcal{W}^n_{1,{\bf k},L}=\mathcal{W}^n_L$. We define an automorphism $\jmath_{w,{\bf k}}:\mathcal{T}^n_L\buildrel\sim\over\rightarrow \mathcal{T}^n_L$, $\eta=(\eta_1,\dots,\eta_n)\mapsto \jmath_{w,{\bf k}}(\eta)=\jmath_{w,{\bf k}}(\eta_1,\dots,\eta_n)$ by: $$\jmath_{w,{\bf k}}(\eta_1,\dots,\eta_n):= (z^{{\bf k}_1-{\bf k}_{w^{-1}(1)}}\eta_1,\dots,z^{{\bf k}_n-{\bf k}_{w^{-1}(n)}}\eta_n)$$ which we extend to an automorphism $\jmath_{w,{\bf k}}:\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\buildrel\sim\over\rightarrow\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$, $(r,\eta)\mapsto (r,\jmath_{w,{\bf k}}(\eta))$. We will be particularly interested in applying $\jmath_{w,{\bf k}}$ to points whose image in $\mathcal{W}^n_L$ lies in $\mathcal{W}^n_{w,{\bf k},L}$.

\[example\] [Consider the case $[K:{\mathbb{Q}}_p]=2$ (so ${\mathrm{Hom}}(K,L)=\{\tau,\tau'\}$), $n=3$ and $w=(w_\tau,w_{\tau'})$ with $w_{\tau}=s_1s_2s_1$, $w_{\tau'}=s_2s_1$ ($s_1,s_2$ being the simple reflections in ${\mathcal S}_3$). Then $\mathcal{W}^3_{w,{\bf k},L}$ is the set of characters of the form: $$\eta=(\eta_1,\eta_2,\eta_3) =  \big(\tau(z)^{k_{\tau,3}}\tau'(z)^{k_{\tau',2}}\chi_1, \ \tau(z)^{k_{\tau,2}}\tau'(z)^{k_{\tau',3}}\chi_2, \ \tau(z)^{k_{\tau,1}}\tau'(z)^{k_{\tau',1}}\chi_3\big)$$ where ${\rm wt}_\tau(\chi_1)={\rm wt}_\tau(\chi_3)$ and ${\rm wt}_{\tau'}(\chi_1)={\rm wt}_{\tau'}(\chi_2)={\rm wt}_{\tau'}(\chi_3)$. Note that there is no condition on ${\rm wt}_\tau(\chi_2)$ (so one could as well rewrite the middle character as just $\tau'(z)^{k_{\tau',3}}\chi_2 $). One has (when the $\eta_i$, or equivalently the $\chi_i$, come from characters in $\mathcal{T}_L$): $$\jmath_{w,{\bf k}}(\eta)= \big(\tau(z)^{k_{\tau,1}}\tau'(z)^{k_{\tau',1}}\chi_1,\ \tau(z)^{k_{\tau,2}}\tau'(z)^{k_{\tau',2}}\chi_2,\ \tau(z)^{k_{\tau,3}}\tau'(z)^{k_{\tau',3}}\chi_3\big).$$]{}

Let $\widetilde U_{\rm tri}^\square({\overline{r}}):=U_{\rm tri}^\square({\overline{r}})\cap \widetilde X_{\rm tri}^\square({\overline{r}})$ (a union of connected components of $U_{\rm tri}^\square({\overline{r}})$), then $\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}$ is reduced (since smooth over $\mathcal{W}^n_{w,{\bf k},L}$) and Zariski-open (but not necessarily Zariski-dense) in $(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})^{\mathrm{red}}$ where $(-)^{{\mathrm{red}}}$ means the associated reduced closed analytic subvariety. We denote by $\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}$ its Zariski-closure, so that we have a chain of Zariski-closed embeddings: $$\begin{gathered}
\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\subseteq (\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})^{\mathrm{red}}\subseteq \widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}\subseteq \widetilde X_{\rm tri}^\square({\overline{r}})\\
\subseteq X_{\rm tri}^\square({\overline{r}})\subseteq \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L.\end{gathered}$$

\[companion\] Assume Conjecture \[BHS\], then for $w\in W$ the automorphism $\jmath_{w,{\bf k}}:\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\buildrel\sim\over\rightarrow\mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L$ induces a closed embedding of reduced rigid analytic spaces over $L$: $$\jmath_{w,{\bf k}}: \overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\hookrightarrow \widetilde X_{\rm tri}^\square({\overline{r}})\subseteq X_{\rm tri}^\square({\overline{r}}).$$

Since $\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}$ is Zariski-dense in $\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}$, it is enough to prove $\jmath_{w,{\bf k}}(\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})\subseteq \widetilde X_{\rm tri}^\square({\overline{r}})$, i.e. that any point $x'=(r',\delta')$ in $\widetilde U_{\rm tri}^\square({\overline{r}})$ with $\omega(x')\in \mathcal{W}^n_{w,{\bf k},L}$ is such that $\jmath_{w,{\bf k}}(x')$ is still in $\widetilde X_{\rm tri}^\square({\overline{r}})$.

Recall that by assumption: $$\label{rappelconj}
X_p({\overline{\rho}}) \buildrel{\substack{(\ref{union})\\ \sim}}\over \longrightarrow  {\mathfrak{X}}_{{\overline{\rho}}^p}\times \widetilde X_{\rm tri}^\square({\overline{\rho}}_p)\times {\mathbb{U}}^g\subseteq {\mathfrak{X}}_{{\overline{\rho}}^p}\times ({\mathfrak{X}}_{{\overline{\rho}}_p}\times \widehat T_{p,L})\times {\mathbb{U}}^g.$$ Let $y'\in X_p({\overline{\rho}})$ be any point such that its image in $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)$ by (\[rappelconj\]) is $(x')_{v\in S_p}$. Write again $w$ for the element $(w)_{v\in S_p}\in \prod_{v\in S_p}(\prod_{F_{\tilde v}\hookrightarrow L}{\mathcal{S}}_n)$ (that is, for each $v$ we have the same element $w=(w_{\tau})_{\tau:\, K\hookrightarrow L}\in \prod_{\tau: K\hookrightarrow L}{\mathcal{S}}_n$), ${\bf k}$ for $({\bf k})_{v\in S_p}\in \prod_{v\in S_p}{\mathbb{Z}}^{{\mathrm{Hom}}(K,L)}$ (ibid.), $\jmath_{w,{\bf k}}$ for the automorphism $(\jmath_{w,{\bf k}})_{v\in S_p}$ of $\widehat T_{p,L}\cong \prod_{v\in S_p}\widehat T_{v,L}\cong \prod_{v\in S_p}\mathcal{T}^n_L$ and (again) $\jmath_{w,{\bf k}}$ for the automorphism ${\mathrm{id}}\times ({\mathrm{id}}\times \jmath_{w,{\bf k}})\times {\mathrm{id}}$ of ${\mathfrak{X}}_{{\overline{\rho}}^p}\times ({\mathfrak{X}}_{{\overline{\rho}}_p}\times \widehat T_{p,L})\times {\mathbb{U}}^g$. Then it is enough to prove that $\jmath_{w,{\bf k}}(y')\in X_p({\overline{\rho}})$ (via (\[rappelconj\])). Writing $y'=(\mathfrak{m}',\epsilon')\in X_p({\overline{\rho}})\subseteq {\mathfrak{X}}_\infty\times \widehat T_{p,L}$ where $\mathfrak{m}'\subset R_\infty[1/p]$ is the maximal ideal corresponding to the projection of $y'$ in ${\mathfrak{X}}_\infty$ and $\epsilon'=(\imath_v(\delta'))_{v\in S_p}$, we have ${\mathrm{Hom}}_{T_p}(\epsilon',J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}']\otimes_{k(\mathfrak{m}')}k(y')))\ne 0$ (see (\[pointpatching\])) and we have to prove (note that $\jmath_{w,{\bf k}}\circ\imath_v^{-1}=\imath_v^{-1}\circ\jmath_{w,{\bf k}}$ on $\widehat T_{v,L}$ and that $k(y')=k(\jmath_{w,{\bf k}}(y'))$): $$\label{nonzero}
{\mathrm{Hom}}_{T_p}\big(\jmath_{w,{\bf k}}(\epsilon'),J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}']\otimes_{k(\mathfrak{m}')}k(y'))\big)\ne 0.$$

From Theorem \[intercompanion\] below, it is enough to prove $\epsilon' \uparrow \jmath_{w,{\bf k}}(\epsilon')$ in the sense of Definition \[stronglink\] below. Since $\epsilon'\jmath_{w,{\bf k}}(\epsilon')^{-1}$ is clearly an algebraic character of $T_p$ by definition of $\jmath_{w,{\bf k}}$, it is enough to prove $\imath_v(\delta') \uparrow_{{\mathfrak t}_v} \jmath_{w,{\bf k}}(\imath_v(\delta'))$ (see §\[jw\] for the notation) for one, or equivalently all here, $v\in S_p$. From (\[equations\]), we see that we can write: $$\delta'=(z^{{\bf k}_{w^{-1}(1)}}\chi_1,\dots,z^{{\bf k}_{w^{-1}(n)}}\chi_n)\ \ {\rm and}\ \ \jmath_{w,{\bf k}}(\imath_v(\delta'))=(z^{{\bf k}_1}\chi_1,\dots,z^{{\bf k}_n}\chi_n)$$ where ${\rm wt}_\tau(\chi_i)={\rm wt}_\tau(\chi_{w_\tau(i)})$ for $1\leq i\leq n$ and $\tau:\, K=F_{\tilde v}\hookrightarrow L$ (compare Example \[example\]). As we only care about the ${\mathfrak t}_{v,L}$-action, setting $s_{\tau,i}:={\rm wt}_\tau(\chi_i)\in L$ and using usual additive notation, we can write $\imath_v(\delta')\vert_{{\mathfrak t}_{v,L}}=(\imath_v(\delta')_\tau)_{\tau:\, F_{\tilde v}\hookrightarrow L}$ and $\jmath_{w,{\bf k}}(\imath_v(\delta'))\vert_{{\mathfrak t}_{v,L}}=(\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau)_{\tau:\, F_{\tilde v}\hookrightarrow L}$ with: $$\begin{aligned}
\imath_v(\delta')_\tau&=&(k_{\tau,w_{\tau}^{-1}(1)}+s_{\tau,1},k_{\tau,w_{\tau}^{-1}(2)}+s_{\tau,2}+1,\dots,k_{\tau,w_{\tau}^{-1}(n)}+s_{\tau,n}+n-1)\\
\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau&=&(k_{\tau,1}+s_{\tau,1},k_{\tau,2}+s_{\tau,2}+1,\dots,k_{\tau,n}+s_{\tau,n}+n-1)\end{aligned}$$ (see the beginning of §\[jw\]). Since $s_{\tau,i}=s_{\tau,w_\tau^{-1}(i)}$ for all $i,\tau$, we can rewrite: $$\imath_v(\delta')_\tau=(k_{\tau,w_{\tau}^{-1}(1)}+s_{\tau,w_{\tau}^{-1}(1)},k_{\tau,w_{\tau}^{-1}(2)}+s_{\tau,w_{\tau}^{-1}(2)}+1,\dots,k_{\tau,w_{\tau}^{-1}(n)}+s_{\tau,w_{\tau}^{-1}(n)}+n-1)$$ hence we have $\imath_v(\delta')_\tau=w_\tau\cdot \jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau$ for the “dot action” $\cdot$ with respect to the upper triangular matrices in ${\mathrm{GL}}_{n,F_{\tilde v}}\times_{F_{\tilde v},\tau}L$ (see [@HumBGG §1.8]). Let us write the permutation $w_\tau$ on $\{1,\dots,n\}$ as a product of commuting cycles $c_1\circ\cdots \circ c_m$ with pairwise disjoint support ${\rm supp}(c_i)\subseteq \{1,\dots,n\}$. Let us denote by ${\mathcal S}_{n,i}\subseteq {\mathcal S}_{n}$ the subgroup of permutations which fixes the elements in $\{1,\dots,n\}$ [*not*]{} in ${\rm supp}(c_i)$ and set ${\mathcal S}_{n,w_\tau}:= \prod_{i=1}^m{\mathcal S}_{n,i}\subseteq {\mathcal S}_n$. Then, arguing in each ${\rm supp}(c_i)$, it is not difficult to see that one can write $w_\tau$ as a product: $$w_\tau = s_{\alpha_d}s_{\alpha_{d-1}}\cdots s_{\alpha_1}$$ where the $\alpha_i$ are (not necessarily simple) roots of the upper triangular matrices in ${\mathrm{GL}}_{n,F_{\tilde v}}\times_{F_{\tilde v},\tau}L$, the associated reflections $s_{\alpha_i}$ are in ${\mathcal S}_{n,w_\tau}$ and where $s_{\alpha_i+1}s_{\alpha_{i}}\cdots s_{\alpha_1}> s_{\alpha_i}\cdots s_{\alpha_1}$ for the Bruhat order in ${\mathcal S}_n$ ($1\leq i\leq n-1$). By an argument analogous [*mutatis mutandis*]{} to the one in [@HumBGG §5.2], it then follows from the above assumptions (in particular $s_{\tau,i}=s_{\tau,w_\tau^{-1}(i)}$ for all $i$) that we have for $1\leq i\leq n-1$ with obvious notation: $$(s_{\alpha_i+1}\cdots s_{\alpha_1})\cdot \jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau \leq (s_{\alpha_i}\cdots s_{\alpha_1})\cdot \jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau.$$ By definition this implies that $w_\tau\cdot\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau$ is [*strongly linked*]{} to $\jmath_{w,{\bf k}}(\imath_v(\delta'))_\tau$ ([@HumBGG §5.1]). As this holds for all $\tau$, we have $\imath_v(\delta') \uparrow_{{\mathfrak t}_v} \jmath_{w,{\bf k}}(\imath_v(\delta'))$.

[It would be very interesting to find a purely local proof of the local statement of Proposition \[companion\] without assuming Conjecture \[BHS\].]{}

Companion points on the patched eigenvariety {#jw}
--------------------------------------------

We prove that the existence of certain points on the patched eigenvariety $X_p({\overline{\rho}})$ implies the existence of others (Theorem \[intercompanion\]). This result is crucially used in the proof of Proposition \[companion\] above.

We use the notation of §\[globalpart\]. We denote by ${\mathfrak g}$ (resp. ${\mathfrak b}$, resp. ${\mathfrak t}$) the ${\mathbb{Q}}_p$-Lie algebra of $G_p$ (resp. $B_p$, resp. $T_p$). We also denote by ${\mathfrak n}$ (resp. $\overline{\mathfrak n}$) the ${\mathbb{Q}}_p$-Lie algebra of the inverse image $N_p$ in $B_p$ (resp. $\overline N_p$ in $\overline B_p$) of the subgroup of upper (resp. lower) unipotent matrices of $\prod_{v\in S_p}{\mathrm{GL}}_n(F_{\tilde v})$. We add an index $L$ for the $L$-Lie algebras obtained by scalar extension $\cdot \otimes_{{\mathbb{Q}}_p}L$ (e.g. ${\mathfrak g}_L$, etc.) and we denote by $U(\cdot)$ the corresponding enveloping algebras.

For $v\in S_p$ we denote by ${\mathfrak t}_v$ the ${\mathbb{Q}}_p$-Lie algebra of the torus $T_v$, so that ${\mathfrak t}=\prod_{v\in S_p}{\mathfrak t}_v$. Recall that ${\mathfrak t}_v$ is an $F_{\tilde v}$-vector space, and thus ${\mathfrak t}_{v,L}={\mathfrak t}_v\otimes_{{\mathbb{Q}}_p}L\cong \prod_{\tau:\, F_{\tilde v}\hookrightarrow L}{\mathfrak t}_v\otimes_{F_{\tilde v},\tau}L$. We can see any $\eta=(\eta_v)_{v\in S_p}=(\eta_{v,1},\dots,\eta_{v,n})_{v\in S_p}\in \widehat T_{p,L}$ as an $L$-valued additive character of ${\mathfrak t}$, and thus of ${\mathfrak t}_{L}$ by $L$-linearity, via the usual derivative action $({\mathfrak z}_{v,1},\dots,{\mathfrak z}_{v,n})_{v\in S_p}\mapsto \sum_{v\in S_p}\sum_{i=1}^n\frac{d}{dt}\eta_{v,i}(\exp(t{\mathfrak z}_{v,i}))\vert_{t=0}$. Recall that the character ${\mathfrak z}_{v,i}\in F_{\tilde v}\mapsto \frac{d}{dt}\eta_{v,i}(\exp(t{\mathfrak z}_{v,i}))\vert_{t=0}$ is nothing else than $\sum_{\tau:\, F_{\tilde v}\hookrightarrow L}\tau({\mathfrak z}_{v,i}){\rm wt}_\tau(\eta_{v,i})\in L$.

In what follows we use notation and definitions from [@STdist] concerning $L$-Banach representations of $p$-adic Lie groups and their locally ${\mathbb{Q}}_p$-analytic vectors. If $\Pi$ is an admissible continuous representation of $G_p$ on a $L$-Banach space we denote by $\Pi^{\rm an}\subseteq \Pi$ its invariant subspace of locally ${\mathbb{Q}}_p$-analytic vectors.

\[injective\] Let $\Pi$ be an admissible continuous representation of $G_p$ on a $L$-Banach space and assume that the continuous dual $\Pi'$ is a finite projective $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$-module. Let $\lambda$, $\mu$ be $L$-valued characters of ${\mathfrak t}_L$ that we see as $L$-valued characters of ${\mathfrak b}_L$ by sending ${\mathfrak n}_L$ to $0$. If $U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\mu\hookrightarrow U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda$ is an injection of $U(\mathfrak{g}_{L})$-modules, then the map: $${\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi^{\rm an}\big)\longrightarrow{\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\mu,\Pi^{\rm an}\big)$$ induced by functoriality is surjective.

We have as in [@STdist Prop.6.5] a $K_p$-equivariant isomorphism: $$\label{limr}
\Pi^{\rm an}\cong {\lim_{\stackrel{r\rightarrow 1}{r<1}}}\ \Pi_r$$ where each $\Pi_r\subseteq \Pi^{\rm an}$ is a Banach space over $L$ endowed with an admissible locally ${\mathbb{Q}}_p$-analytic action of $K_p$. In particular each $\Pi_r$ is stable under $U(\mathfrak{g}_L)$ in $\Pi^{\rm an}$. If $f:U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda\rightarrow\Pi^{\rm an}$ is a $U(\mathfrak{g}_L)$-equivariant morphism, the source, being of finite type over $U(\mathfrak{g}_{L})$, factors through some $\Pi_r$ by (\[limr\]). Moreover the action of $U(\mathfrak{g}_{L})$ on $\Pi_r$ extends to an action of the $L$-Banach algebra $U_r(\mathfrak{g}_{L})$ which is the topological closure of $U(\mathfrak{g}_{L})$ in the completed distribution algebra $D_r(K_p,L)$ (see [@STdist §5]). Consequently $f$ extends to a $U_r(\mathfrak{g}_{L})$-equivariant morphism: $$f_r:\, U_r(\mathfrak{g}_{L})\otimes_{U(\mathfrak{g}_{L})}(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda)\cong
U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)} \lambda\longrightarrow\Pi_r$$ where $U_r(\mathfrak{b}_L)$ is the closure of $U(\mathfrak{b}_L)$ in $D_r(K_p,L)$ and the first isomorphism follows from $\lambda\buildrel\sim\over\rightarrow U_r(\mathfrak{b}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$ (as $\lambda$ is both finite dimensional with dense image). We deduce from [@OrlikStrauch Prop.3.4.8] (applied with $w=1$) that the injection $U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda\hookrightarrow U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\mu$ extends to an injection of $U_r(\mathfrak{g}_{L})$-modules $U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\lambda\hookrightarrow U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\mu$. Moreover, as $U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\lambda$ and $U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\mu$ are $U_r(\mathfrak{g}_{L})$-modules of finite type, they have a unique topology of Banach module over $U_r(\mathfrak{g}_{L})$ and every $U_r(\mathfrak{g}_{L})$-linear map of one of them into $\Pi_r$ is automatically continuous (see [@STdist Prop.2.1]). We deduce from all this isomorphisms: $$\begin{aligned}
{\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi^{\rm an}\big)&\buildrel\sim\over\longrightarrow &\varinjlim_r{\mathrm{Hom}}_{U(\mathfrak{g}_{L})}\big(U(\mathfrak{g}_{L})\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi_r\big)\\
&\buildrel\sim\over\longrightarrow &\varinjlim_r{\mathrm{Hom}}_{U_r(\mathfrak{g}_{L})-{\rm cont}}\big(U_r(\mathfrak{g}_{L})\otimes_{U_r(\mathfrak{b}_L)}\lambda,\Pi_r\big)\end{aligned}$$ where ${\mathrm{Hom}}_{U_r(\mathfrak{g}_{L})-{\rm cont}}$ means continuous homomorphisms of $U_r(\mathfrak{g}_{L})$-Banach modules, and likewise with $\mu$ instead of $\lambda$. By exactitude of $\varinjlim_r$, we see that it is enough to prove that $\Pi_r$ is an injective object (with respect to injections which have closed image) in the category of $U_r(\mathfrak{g}_{L})$-Banach modules with continuous maps.

By assumption the dual $\Pi'$ is a projective module of finite type over $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$, hence a direct summand of $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]^{\oplus s}$ for some $s>0$. From the proof of [@STdist Prop.6.5] together with [@STdist Th.7.1(iii)], we also know that $\Pi_r$ is the continuous dual of the $D_r(K_p,L)$-Banach module: $$\Pi_r':=D_r(K_p,L)\otimes_{\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]}\Pi'.$$ We get that the $D_r(K_p,L)$-module $\Pi_r'$ is a direct summand of $D_r(K_p,L)^{\oplus s}$. Now it easily follows from the results in [@Ko §1.4] that $D_r(K_p,L)$ is itself a free $U_r(\mathfrak{g}_{L})$-module of finite rank. Dualizing, we finally obtain that there is a finite dimensional $L$-vector space $W$ such that the left $U_r(\mathfrak{g}_{L})$-Banach module $\Pi_r$ is a direct factor of the left $U_r(\mathfrak{g}_{L})$-Banach module ${\rm Hom}_{\rm cont}(U_r(\mathfrak{g}_{L})\otimes_L W,L)$ (which is seen as a left $U_r(\mathfrak{g}_{L})$-module via the automorphism on $U_r(\mathfrak{g}_{L})$ extending the multiplication by $-1$ on $\mathfrak{g}_{L}$). Since direct summands and finite sums of injective modules are still injective, it is enough to prove the injectivity of ${\rm Hom}_{\rm cont}(U_r(\mathfrak{g}_{L}),L)$ in the category of $U_r(\mathfrak{g}_{L})$-Banach modules with continuous maps.

If $V$ is any $U_r(\mathfrak{g}_L)$-Banach module, it is not difficult to see that there is a canonical isomorphism of Banach spaces over $L$: $$\label{dualdual}
{{\mathrm{Hom}}}_{U_r(\mathfrak{g}_{L})-{\rm cont}}\big(V,{{\mathrm{Hom}}}_{\rm cont}(U_r(\mathfrak{g}_L),L)\big)\buildrel\sim\over\longrightarrow {{\mathrm{Hom}}}_{\rm cont}(V,L)$$ so that the required injectivity property is a consequence of the Hahn-Banach Theorem (see for example [@Schneidernfa Prop.9.2]).

We go on with two technical lemmas which require more notation. Fix a compact open uniform normal pro-$p$ subgroup $H_p$ of $K_p$ such that $H_p=(\overline N_p\cap H_p)(T_p\cap H_p)(N_p\cap H_p)$. For example $H_p$ can be chosen of the form $\prod_{v\in S_p}H_{v}$ where $H_v$ is the inverse image in $K_v$ of matrices of ${\mathrm{GL}}_n(\mathcal{O}_{F_{\tilde{v}}})$ congruent to $1$ mod $p^m$ for $m$ big enough. Let $N_0:=N_p\cap H_p$, $T_{p,0}:=T_p\cap H_p$, $\overline{N}_0:=\overline{N}_p\cap H_p$ (which are still uniform pro-p-groups) and $T_p^+:=\{ t\in T_p\ {\rm such\ that}\ tN_0t^{-1}\subseteq N_0\}$ (which is a multiplicative monoid in $T_p$). We also fix $z\in T_p^+$ such that $zN_0z^{-1}\subseteq N_0^p$ and we assume moreover $z^{-1}H_pz\subseteq K_p$ so that the elements of $z^{-1}H_pz$ normalize $H_p$ (as $H_p$ is normal in $K_p$). Note that such a $z$ always exists, for instance take $z$ such that $zN_0z^{-1}\subseteq N_0^p$, choose $r$ such that $H_p^{p^r}\subseteq zK_pz^{-1}$ and replace $H_p$ by $H_p^{p^r}$: with this new choice we still have $zN_0z^{-1}\subseteq N_0^p$.

For any uniform pro-$p$-group $H$ we denote by $\mathcal{C}(H,L)$ the Banach space of continuous $L$-valued functions on $H$ and, if $h\geq 1$, by $\mathcal{C}^{(h)}(H,L)$ the Banach space of $h$-analytic $L$-valued functions on $H$ defined in [@CD §0.3]. We have $\mathcal{C}(H_p,L)\cong \mathcal{C}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L)$ and likewise with $\mathcal{C}^{(h)}(\cdot,L)$.

\[restriction\] Let $f\in\mathcal{C}(H_p,L)$ such that for each left coset $(zH_pz^{-1}\cap H_p)n\subset H_p$, there exists $f_n\in\mathcal{C}^{(h)}(H_p,L)$ such that $f(gn)=f_n(z^{-1}gz)$ for $g\in zH_pz^{-1}\cap H_p$. Then we have: $$f\in\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L).$$

Representatives of the quotient $(zH_pz^{-1}\cap H_p)\backslash H_p$ can be chosen in $N_0$, whence the above notation $n$ (do not confuse with the $n$ of ${\mathrm{GL}}_n$!). Restricting $f$ to the left coset $(zH_pz^{-1}\cap H_p)n$ for some $n\in N_0$ and translating on the right by $n$ we can assume that the support of $f$ is contained in $zH_pz^{-1}\cap H_p$. Then if $g\in zH_pz^{-1}\cap H_p$, we have by assumption $f(g)=F(z^{-1}gz)$ for some $F\in\mathcal{C}^{(h)}(H_p,L)$. Consequently $f|_{zH_pz^{-1}\cap H_p}$ can be extended to an $h$-analytic function on $zH_pz^{-1}$ and $f$ can be extended (by $0$) on $zH_pz^{-1}N_0=z\overline N_0z^{-1} T_{p,0} N_0$ as an element of: $$\mathcal{C}^{(h)}(z\overline{N}_0z^{-1},L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L).$$ We deduce that $f$ is in the image of the restriction map (note that $zN_0z^{-1}\subseteq N_0$ implies $\overline{N}_0\subseteq z\overline{N_0}z^{-1}$): $$\mathcal{C}^{(h)}(z\overline{N}_0z^{-1},L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L)\longrightarrow\mathcal{C}^{(h)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L).$$ Now the stronger condition $zN_0z^{-1}\subseteq N_0^p$ implies $\overline{N}_0\subseteq z\overline{N_0}^pz^{-1}=(z\overline{N}_0z^{-1})^p$. But by [@CD Rem.IV.12] the restriction to $(z\overline{N}_0z^{-1})^p$ (and [*a fortiori*]{} to $\overline N_0$) of an $h$-analytic function on $z\overline{N}_0z^{-1}$ is $(h-1)$-analytic and we can conclude.

If $\Pi$ is an admissible continuous representation of $G_p$ on a $L$-Banach space and if $h\geq1$, we denote by $\Pi_{H_p}^{(h)}$ the $H_p$-invariant Banach subspace of $\Pi^{\rm an}$ defined in [@CD §0.3]. If $V$ is any (left) $U(\mathfrak{t}_L)$-module over $L$ and $\lambda:\mathfrak{t}_L\rightarrow L$ is a character, we let $V_\lambda$ be the $L$-subvector space of $V$ on which $\mathfrak{t}_L$ acts via the multiplication by $\lambda$. Recall that if $V$ is any $L[G_p]$-module, the monoid $T_p^+$ acts on $V^{N_0}$ via $v\longmapsto t\cdot v:=\delta_{B_p}(t)\sum_{n_0\in N_0/tN_0t^{-1}}n_0tv$ ($v\in V^{N_0}$, $t\in T_p^+$, see §\[classic\] for $\delta_{B_p}$). This $T_p^+$-action respects the subspace $(\Pi_\lambda^{\rm an})^{N_0}$ of $(\Pi^{\rm an})^{N_0}$ (use that $tN_0t^{-1}=N_0$ for $t\in T_p^0$).

We don’t claim any originality on the following lemma which is a variant of classical results (see e.g. [@EmertonJacquetI]), however we couldn’t find its exact statement in the literature.

\[compact\] Let $\Pi$ be an admissible continuous representation of $G_p$ on a $L$-Banach space, $\lambda$ an $L$-valued character of $\mathfrak{t}_L$ and $h\geq1$. Then the action of $z$ on $(\Pi^{\rm an})^{N_0}$ preserves the subspace $(\Pi^{(h)}_{H_p})^{N_0}_\lambda=(\Pi^{(h)}_{H_p})^{N_0}\cap \Pi^{\rm an}_\lambda$ and is a [compact]{} operator on this subspace.

Let $l_1,\dots,l_s$ be a system of generators of the continous dual $\Pi'$ as a module over the algebra $\mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}H_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$. Define a closed embedding of $\Pi$ into $\mathcal{C}(H_p,L)^{\oplus s}$ via the map $v\mapsto (g\mapsto l_i(gv))_{1\leq i\leq s}$. This embedding is $H_p$-equivariant for the left action of $H_p$ on $\mathcal{C}(H_p,L)$ by right translation on functions. By [@CD Prop.IV.5], we have $\Pi_{H_p}^{(h)}=\Pi\cap \mathcal{C}^{(h)}(H_p,L)^{\oplus s}$. If $v\in\Pi_{H_p}^{(h)}$, $n\in N_0$ and $g\in H_p\cap z H_pz^{-1}$, we have $l_i(gnzv)=l_i(z(z^{-1}gz)(z^{-1}nz)v)$. Let $v\in\Pi_{H_p}^{(h)}$ and $n\in N_0$. As $z^{-1}N_0z$ normalizes $H_p$ (by the choice of $z$) we have $w:=(z^{-1}nz)v\in\Pi_{H_p}^{(h)}$ (see [@CD Prop.IV.16]). As $l_i(z\cdot)$ is a continuous linear form on $\Pi$, using [@CD Thm.IV.6(i)] the function $f_n:H_p\rightarrow L$, $g\mapsto f_n(g):=l_i(zgw)$ is in $\mathcal{C}^{(h)}(H_p,L)$ and $l_i(gnzv)=f_n(z^{-1}gz)$ for $g\in zH_pz^{-1}\cap H_p$. We deduce from Lemma \[restriction\] applied to the functions $f:H_p\rightarrow L$, $g\mapsto l_i(gzv)$ for $1\leq i\leq s$ that: $$\label{incl(h)}
z\Pi_{H_p}^{(h)}\subset \big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\widehat{\otimes}_L\mathcal{C}(N_0,L)\big)^{\oplus s}.$$ Let $v\in(\Pi_{H_p}^{(h)})^{N_0}$, the space on the right hand side of (\[incl(h)\]) being stable under $N_0$ (acting by right translation on functions), it still contains $z\cdot v=\sum_{n_0\in N_0/tN_0t^{-1}}n_0zv$. Since $z\cdot v$ is fixed under $N_0$, we deduce: $$z\cdot v\ \in\ \big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\big)^{\oplus s}\subseteq \ \mathcal{C}^{(h)}(H_p,L)^{\oplus s}.$$ In particular $z\cdot(\Pi_{H_p}^{(h)})^{N_0}_\lambda \subseteq (\Pi^{\rm an}_\lambda)^{N_0}\cap \mathcal{C}^{(h)}(H_p,L)^{\oplus s}=(\Pi_{H_p}^{(h)})^{N_0}_\lambda$ which shows the first statement. We also deduce: $$z\cdot(\Pi_{H_p}^{(h)})^{N_0}_\lambda\subseteq\big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\mathcal{C}^{(h)}(T_{p,0},L)\big)_{\!\lambda}^{\oplus s}\cong \mathcal{C}^{(h-1)}(\overline{N}_0,L)\widehat{\otimes}_L\big(\mathcal{C}^{(h)}(T_{p,0},L)_\lambda^{\oplus s}\big).$$ But by [@CD Prop. IV.13.(i)], we have $\mathcal{C}^{(h)}(T_{p,0},L)'\simeq D_r(T_{p,0},L)$ for $r=p^{-1/p^h}$ where $D_r(T_{p,0},L)$ is as in [@STdist §4]. Let $U_r(\mathfrak{t}_L)$ be the closure of $U(\mathfrak{t}_L)$ in $D_r(T_{p,0},L)$, then (as in the proof of Lemma \[injective\]) $D_r(T_{p,0},L)$ is a finite free $U_r(\mathfrak{t}_L)$-module ([@Ko §1.4]). Using $\lambda\buildrel\sim\over\rightarrow U_r(\mathfrak{t}_L)\otimes_{U(\mathfrak{t}_L)}\lambda$, it follows that $(\mathcal{C}^{(h)}(T_{p,0},L)_\lambda)'$, and hence $\mathcal{C}^{(h)}(T_{p,0},L)_\lambda^{\oplus s}$, are finite dimensional $L$-vector spaces. We denote the latter by $W_\lambda$.

We thus have $z\cdot(\Pi_{H_p}^{(h)})^{N_0}_\lambda\subseteq \mathcal{C}^{(h-1)}(\overline{N}_0,L)\otimes_L W_\lambda$: the endomorphism induced by $z$ on $(\Pi_{H_p}^{(h)})^{N_0}_\lambda$ factors through the subspace $(\Pi_{H_p}^{(h)})^{N_0}_\lambda\cap\big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\otimes_L W_\lambda\big)$. As the inclusion of $\mathcal{C}^{(h-1)}(\overline{N}_0,L)$ into $\mathcal{C}^{(h)}(\overline{N}_0,L)$ is compact and $W_\lambda$ is finite dimensional over $L$, the inclusion of $(\Pi_{H_p}^{(h)})^{N_0}_\lambda\cap\big(\mathcal{C}^{(h-1)}(\overline{N}_0,L)\otimes_L W\big)$ into $(\Pi_{H_p}^{(h)})^{N_0}_\lambda$ is compact, which proves the result.

If $\delta_v,\epsilon_v\in \widehat T_{v,L}$, we write $\epsilon_v \uparrow_{{\mathfrak t}_v} \delta_v$ if, seeing $\delta_v,\epsilon_v$ as $U({\mathfrak t}_{v,L})$-modules, we have $\epsilon_v \uparrow \delta_v$ in the sense of [@HumBGG §5.1] with respect to the roots of the upper triangular matrices in $({\mathrm{Res}}_{F_{\tilde v}/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,F_{\tilde v}})_L$. Likewise if $\delta,\epsilon\in \widehat T_{p,L}$, we write $\epsilon \uparrow_{{\mathfrak t}} \delta$ if, seeing $\delta,\epsilon$ as $U({\mathfrak t}_{L})$-modules, we have $\epsilon \uparrow \delta$ in the sense of [@HumBGG §5.1] with respect to the roots of the upper triangular matrices in $\prod_{v\in S_p}({\mathrm{Res}}_{F_{\tilde v}/{\mathbb{Q}}_p}{\mathrm{GL}}_{n,F_{\tilde v}})_L$. Thus writing $\delta=(\delta_v)_{v\in S_p},\epsilon=(\epsilon_v)_{v\in S_p}$, we have $\epsilon \uparrow_{{\mathfrak t}} \delta$ if and only if $\epsilon_v \uparrow_{{\mathfrak t}_v} \delta_v$ for all $v\in S_p$.

\[stronglink\] Let $\delta,\epsilon\in \widehat T_{p,L}$, we write $\epsilon \uparrow \delta$ if $\epsilon \uparrow_{{\mathfrak t}} \delta$ and if $\epsilon\delta^{-1}$ is an algebraic character of $T_p$, i.e. $\epsilon\delta^{-1}=\delta_{\lambda}$ for some ${\lambda}=({\lambda}_v)_{v\in S_p}\in\prod_{v\in S_p}(\mathbb{Z}^n)^{{\mathrm{Hom}}(F_{\tilde v},L)}$.

We can now prove the main theorem of this section.

\[intercompanion\] Let ${\mathfrak m}\subseteq R_\infty[1/p]$ be a maximal ideal, $\delta,\epsilon\in \widehat T_{p,L}$ such that $\epsilon \uparrow \delta$ and $L'$ a finite extension of $L$ containing the residue fields $k(\delta)=k(\epsilon)$ and $k({\mathfrak m})$. Then we have: $${\mathrm{Hom}}_{T_p}\big(\epsilon,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]\otimes_{k({\mathfrak m})}L')\big)\ne 0\Longrightarrow {\mathrm{Hom}}_{T_p}\big(\delta,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]\otimes_{k({\mathfrak m})}L')\big)\ne 0.$$

We assume first $k(\delta)=k(\epsilon)=L$ and $L'=k({\mathfrak m})$, so that we can forget about $L'$. Let $\Pi$ be a locally ${\mathbb{Q}}_p$-analytic representation of $B_p$ over $L$. The subspace $\Pi^{\mathfrak{n}_L}$ of vectors killed by $\mathfrak{n}_L$ is a smooth representation of the group $N_0$ and we denote by $\pi_{N_0}:\Pi^{\mathfrak{n}_L}\twoheadrightarrow \Pi^{N_0}\subseteq \Pi^{\mathfrak{n}_L}$ the unique $N_0$-equivariant projection on its subspace $\Pi^{N_0}$. It is preserved by the action of $T_p$ inside $\Pi$, hence also by the action of $\mathfrak{t}_L$ and one easily checks that: $$\label{commutt}
\pi_{N_0}\circ \mathfrak{x}=\mathfrak{x}\circ \pi_{N_0}\ \ \ \ (\mathfrak{x}\in \mathfrak{t}_L)$$ (use $tN_0t^{-1}=N_0$ for $t\in T_p^0$). The subspace $\Pi_{\lambda}^{\mathfrak{n}_L}:=\Pi_\lambda\cap \Pi^{\mathfrak{n}_L}\subseteq \Pi^{\mathfrak{n}_L}$ is still preserved by $T_p$ and by (\[commutt\]) the projection $\pi_{N_0}$ sends $\Pi_{\lambda}^{\mathfrak{n}_L}$ onto $\Pi_{\lambda}^{N_0}:=\Pi^{N_0}\cap \Pi_{\lambda}^{\mathfrak{n}_L}\subseteq \Pi_{\lambda}^{\mathfrak{n}_L}$. We have $t\cdot v=\pi_{N_0}(tv)$ for $t\in T_p^+$, $v\in \Pi^{N_0}_\lambda$ and in the rest of the proof we view $\Pi^{N_0}_\lambda$ as an $L[T_p^+]$-module via this monoid action.

The locally ${\mathbb{Q}}_p$-analytic character $\delta:T_p\rightarrow L^\times$ determines a surjection of $L$-algebras $L[T_p]\twoheadrightarrow L$ and we denote its kernel by $\mathfrak{m}_\delta$ (a maximal ideal of the $L$-algebra $L[T_p]$). We still write $\mathfrak{m}_\delta$ for its intersection with $L[T_p^+]$, which is then a maximal ideal of $L[T_p^+]$. Let $\lambda:\mathfrak{t}_L\rightarrow L$ be the derivative of $\delta$, arguing as in [@EmertonJacquetI Prop.3.2.12] we get for $s\geq 1$: $$\label{jm}
J_{B_p}(\Pi)[\mathfrak{m}_\delta^s]\cong \Pi^{N_0}[\mathfrak{m}_{\delta}^s]\cong \Pi_\lambda^{N_0}[\mathfrak{m}_{\delta}^s],$$ (in particular ${\mathrm{Hom}}_{T_p}(\delta,J_{B_p}(\Pi))\cong \Pi^{N_0}[\mathfrak{m}_{\delta}]\cong \Pi_\lambda^{N_0}[\mathfrak{m}_{\delta}]$). Likewise we have $J_{B_p}(\Pi)[\mathfrak{m}_{\epsilon}^s]\cong \Pi^{N_0}[\mathfrak{m}_{\epsilon}^s]\cong \Pi_\mu^{N_0}[\mathfrak{m}_{\epsilon}^s]$ if $\mu:\mathfrak{t}_L\rightarrow L$ is the derivative of $\epsilon$.

Let $\mathfrak{I}\subset S_\infty[1/p]$ be an ideal such that $\dim_L(S_\infty[1/p]/\mathfrak{I})<\infty$ and define $\Pi_{\mathfrak{I}}:=\Pi_\infty[\mathfrak{I}]$. As the continuous dual $\Pi_\infty'$ is a finite projective $S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$-module (property (ii) in §\[firstclassical\]), the continuous dual $\Pi_\infty[\mathfrak{I}]'$ of the $G_p$-representation $\Pi_\infty[\mathfrak{I}]$, which is isomorphic to $\Pi_\infty'/\mathfrak{I}$ by the discussion at the end of [@BHS §3.1], is a finite projective $S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]/\mathfrak{I}S_\infty{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]\cong \mathcal{O}_L{{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\rm [\hspace{-0.15em}[}}
                              {\mbox{\scriptsize\rm [\hspace{-0.15em}[}}
                              {\mbox{\tiny\rm [\hspace{-0.15em}[}}}}K_p{{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\rm ]\hspace{-0.15em}]}}
                              {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}}
                              {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}}[1/p]$-module (in particular it is an admissible continuous representation of $G_p$ over $L$). Moreover it is immediate to check that $\Pi_\infty^{R_\infty-\rm an}[\mathfrak{I}]$ is isomorphic to the subspace $\Pi_{\mathfrak{I}}^{\rm an}$ of locally ${\mathbb{Q}}_p$-analytic vectors of $\Pi_{\mathfrak{I}}$.

Taking the image of a vector in (the underlying $L$-vector space of) $\lambda$ or $\mu$ gives natural isomorphisms: $${\mathrm{Hom}}_{U(\mathfrak{g}_L)}\big(U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda,\Pi_{\mathfrak{I}}^{\rm an}\big)\buildrel\sim\over\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{\mathfrak{n}_L}\ {\rm and}\ {\mathrm{Hom}}_{U(\mathfrak{g}_L)}\big(U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\mu,\Pi_{\mathfrak{I}}^{\rm an}\big)\buildrel\sim\over\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{\mathfrak{n}_L}.$$ As $\mu$ is strongly linked to $\lambda$ by assumption, [@HumBGG Th.5.1] implies the existence of a unique (up to $L$-homothety) $U(\mathfrak{g}_L)$-equivariant injection: $$\label{mulambdau}
\iota_{\mu, \lambda}:\,U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\mu\hookrightarrow U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$$ which induces an $L$-linear map: $$\label{imulambda}
\iota^*_{\mu, \lambda}:\,(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{\mathfrak{n}_L}\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{\mathfrak{n}_L}.$$

We claim that (\[imulambda\]) maps the subspace $(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}$ to the subspace $(\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}$. It is enough to prove: $$\label{mulambda}
\iota^*_{\mu, \lambda}\circ\pi_{N_0}=\pi_{N_0}\circ \iota^*_{\mu, \lambda}.$$ Let $\mathfrak{v}$ be the image by (\[mulambdau\]) of a nonzero vector $v$ in the underlying $L$-vector space of $\mu$. Writing $U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda\cong U(\mathfrak{n}_L^-)$ we see that $\mathfrak{v}\in U(\mathfrak{n}_L^-)_{\mu-\lambda}$ (with obvious notation), that $\mathfrak{v}$ is killed by $\mathfrak{n}_L$ in $U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$ and that the morphism $\iota^*_{\mu, \lambda}$ is given by the action (on the left) by $\mathfrak{v}$. To get (\[mulambda\]) it is enough to prove $\pi_{N_0}\circ\mathfrak{v}=\mathfrak{v}\circ\pi_{N_0}$ on $(\Pi_\mathfrak{I}^{\rm an})^{\mathfrak{n}_L}_{\lambda}$, which itself follows from $n\circ \mathfrak{v}=\mathfrak{v}\circ n$ for $n\in N_0$ ($n$ acting via the underlying $G_p$-action on $\Pi_{\mathfrak{I}}^{\rm an}$), or equivalently $\mathrm{Ad}(n)(\mathfrak{v})=\mathfrak{v}$ on $(\Pi_\mathfrak{I}^{\rm an})^{\mathfrak{n}_L}_{\lambda}$. Writing $n=\exp(\mathfrak{m})$ with $\mathfrak{m}\in\mathfrak{n}_L$ (do not confuse here with the maximal ideal $\mathfrak{m}$!), recall we have $\mathrm{Ad}(n)(\mathfrak{v})=\exp(\mathrm{ad}(\mathfrak{m}))(\mathfrak{v})$ (using standard notation). Since $\mathfrak{v}$ is killed by left multiplication by $\mathfrak{n}_L$ in $U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda$, we have: $$\exp(\mathrm{ad}(\mathfrak{m}))(\mathfrak{v})\in\mathfrak{v}+U(\mathfrak{g}_L)(\mathfrak{n}_L+\ker(\lambda))$$ where $\ker(\lambda):=\ker(U(\mathfrak{g}_L)\rightarrow U(\mathfrak{g}_L)\otimes_{U(\mathfrak{b}_L)}\lambda, \ \mathfrak{x}\mapsto \mathfrak{x}\otimes v)$. The action of $\mathrm{Ad}(n)(\mathfrak{v})$ on $(\Pi_\mathfrak{I}^{\rm an})^{\mathfrak{n}_L}_{\lambda}$ is thus the same as that of $\mathfrak{v}$.

We still write: $$\label{mulambda0}
\iota^*_{\mu, \lambda}:\,(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}\longrightarrow (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}$$ for the map induced by (\[imulambda\]). Using $\mathfrak{v}\in U(\mathfrak{n}_L^-)_{\mu-\lambda}$ together with (\[mulambda\]), it is easy to check that $\iota^*_{\mu, \lambda}\circ t=(\delta\epsilon^{-1})(t)(t\circ \iota^*_{\mu, \lambda})$ for $t\in T_p^+$ (for the previous $L[T_p^+]$-module structure). Moreover, it follows from Lemma \[injective\] that (\[imulambda\]) is surjective, hence the top horizontal map and the two vertical maps are surjective in the commutative diagram: $$\begin{xy}\xymatrix{(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{\mathfrak{n}_L}\ar[d]_{\pi_{N_0}}\ar[r]^{(\ref{imulambda})} & (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{\mathfrak{n}_L}\ar[d]^{\pi_{N_0}}\\(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}\ar[r]^{(\ref{mulambda0})} & (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}}
\end{xy}$$ which implies that (\[mulambda0\]) is also surjective. Note also that both (\[imulambda\]) and (\[mulambda0\]) trivially commute with the action of $R_\infty$ (which factors through $R_\infty/\mathfrak{I}R_\infty$).

From [@CD §0.3] we have $\Pi_{\mathfrak{I}}^{\rm an}\cong \lim_{h\rightarrow +\infty}\Pi_{\mathfrak{I},H_p}^{(h)}$ and thus: $$(\Pi_{\mathfrak{I}}^{\rm an})_\lambda^{N_0}\cong \lim_{h\rightarrow +\infty}(\Pi_{\mathfrak{I},H_p}^{(h)})_\lambda^{N_0}\ \ {\rm and}\ \ (\Pi_{\mathfrak{I}}^{\rm an})_\mu^{N_0}\cong \lim_{h\rightarrow +\infty}(\Pi_{\mathfrak{I},H_p}^{(h)})_\mu^{N_0}.$$ By Lemma \[compact\] there is $z\in T_p^+$ which acts compactly on $(\Pi_{\mathfrak{I},H_p}^{(h)})_\lambda^{N_0}$ and $(\Pi_{\mathfrak{I},H_p}^{(h)})_\mu^{N_0}$. We deduce from this fact together with [@SerreCC Prop.9] and [@SerreCC Prop.12] that the map $\iota^*_{\mu,\lambda}$ in (\[mulambda0\]) remains surjective at the level of [*generalized eigenspaces*]{} for the action of $T_p^+$ (twisting this action by the character $\delta\epsilon^{-1}$ on the right hand side). Consequently $\iota^*_{\mu,\lambda}$ induces a surjective map: $$\bigcup_{s\geq 1}(\Pi_{\mathfrak{I}}^{\rm an})_{\lambda}^{N_0}[\mathfrak{m}_{\delta}^s]\twoheadrightarrow \bigcup_{s\geq 1}(\Pi_{\mathfrak{I}}^{\rm an})^{N_0}_\mu[\mathfrak{m}_{\epsilon}^s].$$ As both the source and target of this map are unions of finite dimensional $L$-vector spaces (as follows from the admissibility of $\Pi_{\mathfrak{I}}^{\rm an}$, [@EmertonJacquetI Th.4.3.2] and (\[jm\])) which are stable under $R_\infty$ and as $\iota^*_{\mu,\lambda}$ is $R_\infty$-equivariant, the following map induced by $\iota^*_{\mu,\lambda}$ remains surjective: $$\label{cupcup}
\bigcup_{s,t\geq 1}(\Pi_\mathfrak{I}^{\rm an})_{\lambda}^{N_0}[\mathfrak{m}_{\delta}^s, \mathfrak{m}^t]\twoheadrightarrow \bigcup_{s,t\geq 1}(\Pi_\mathfrak{I}^{\rm an})^{N_0}_\mu[\mathfrak{m}_{\epsilon}^s,\mathfrak{m}^t].$$ Since $\mathfrak{m}^t$ is an ideal of cofinite dimension in $R_\infty[1/p]$, the inverse image $\mathfrak{I}$ of $\mathfrak{m}^t$ in $S_\infty[1/p]$ is [*a fortiori*]{} of cofinite dimension in $S_\infty[1/p]$ and we can apply (\[cupcup\]) with such an $\mathfrak I$. But we have for this $\mathfrak{I}$: $$(\Pi_{\mathfrak{I}}^{\rm an})^{N_0}_\lambda[\mathfrak{m}_\delta^s,\mathfrak{m}^t]=(\Pi_\infty^{R_\infty-\rm an})^{N_0}_\lambda[\mathfrak{m}_\delta^s,\mathfrak{m}^t,\mathfrak{I}]=(\Pi_\infty^{R_\infty-\rm an})^{N_0}_\lambda[\mathfrak{m}_{\delta}^s,\mathfrak{m}^t]$$ and likewise with $\mathfrak{m}_{\epsilon}$, so that (\[cupcup\]) is a surjection: $$\bigcup_{s,t\geq 1}(\Pi_\infty^{R_\infty-\rm an})_{\lambda}^{N_0}[\mathfrak{m}_{\delta}^s, \mathfrak{m}^t]\twoheadrightarrow \bigcup_{s,t\geq 1}(\Pi_\infty^{R_\infty-\rm an})_{\mu}^{N_0}[\mathfrak{m}_{\epsilon}^s, \mathfrak{m}^t].$$ Looking at the eigenspaces on both sides, we obtain ${\mathrm{Hom}}_{T_p}(\delta,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]))\ne 0$ if ${\mathrm{Hom}}_{T_p}(\epsilon,J_{B_p}(\Pi_\infty^{R_\infty-\rm an}[\mathfrak{m}]))\ne 0$.\
Finally, when $k(\delta)=k(\epsilon)$ is larger than $L$, we replace $\Pi_\infty$ by $\Pi_\infty':=\Pi_\infty\otimes_LL'$, $S_\infty[1/p]$ by $S_\infty[1/p]\otimes_LL'$, ${\mathfrak m}$ by ${\mathfrak m}':=\ker(R_\infty[1/p]\otimes_LL'\twoheadrightarrow k({\mathfrak m})\otimes_LL'\twoheadrightarrow L')$ (the last surjection coming from the inclusion $k({\mathfrak m})\subseteq L'$) and the reader can check that all the arguments of the previous proof go through [*mutatis mutandis*]{}.

Tangent spaces on the trianguline variety {#geometry}
-----------------------------------------

We prove that Conjecture \[BHS\] implies Conjecture \[mainconj\] (when ${\overline{r}}$ “globalizes” and $x$ is very regular) and give one (conjectural) application.

We keep the notation and assumptions of §\[closed\]. We fix $x=(r,\delta)\in \widetilde X_{\rm tri}^\square({\overline{r}})\subseteq X_{\rm tri}^\square({\overline{r}})$ which is crystalline strictly dominant very regular. Recall from Lemma \[paramofcrystpt\] that $\delta=(\delta_1\dots,\delta_n)$ where $\delta_i=z^{{\bf k}_i}{\rm unr}(\varphi_i)$ with ${\bf k}_i=(k_{\tau,i})_{\tau:\, K\hookrightarrow L}\in\mathbb{Z}^{{\mathrm{Hom}}(K,L)}$ and $\varphi_i\in k(x)^\times$. The following result immediately follows from Proposition \[acconZaut\] and Theorem \[upperbound\] applied to $X=\widetilde X_{\rm tri}^\square(\overline r)$.

\[easybound\] Assume Conjecture \[BHS\], then we have: $$\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square(\overline r),x}\leq \lg(w_x)-d_x+\dim X_{\rm tri}^\square({\overline{r}})=\lg(w_x)-d_x+n^2+[K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}.$$

The rest of this section is devoted to the proof of the converse inequality (still assuming Conjecture \[BHS\]).

As in the proof of Proposition \[inter\], we consider for $1\leq i\leq n$ the cartesian diagram which defines $W_i$ (with the notation of §\[wedgesection\]): $$\xymatrix{{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\ar[r]&{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)\big)\\ 
W_i \ar[r]\ar@{^{(}->}[u]& 
{\rm Ext}^1_{(\varphi,\Gamma_K)}\big(D_{{\mathrm{rig}}}(r)^{\leq i},D_{{\mathrm{rig}}}(r)^{\leq i}\big).\ar@{^{(}->}[u]}$$ We define $W_{{\rm cris},i}\subseteq {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ as $W_i$ but replacing everywhere ${\rm Ext}^1_{(\varphi,\Gamma_K)}$ by its subspace ${\rm Ext}^1_{\rm cris}$. Note that $W_{{\rm cris},i}\subseteq W_i$ for $1\leq i\leq n$.

\[gradcris\] For $1\leq i\leq n$, we have isomorphisms of $k(x)$-vector spaces: $$\begin{aligned}
W_1\cap\cdots \cap W_{i-1}/W_1\cap\cdots \cap W_{i} &\buildrel\sim\over\longrightarrow &
{\rm Ext}_{(\varphi,\Gamma_K)}^1\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\\
\nonumber W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},i-1}/W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},i} &\buildrel\sim\over\longrightarrow &
{\rm Ext}_{\rm cris}^1\big({\rm gr}_iD_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)/D_{{\mathrm{rig}}}(r)^{\leq i}\big)\end{aligned}$$ where  $W_1\cap\cdots \cap W_{i-1}:={\rm Ext}_{(\varphi,\Gamma_K)}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$  (resp.  $W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},i-1}:={\rm Ext}_{\rm cris}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$) if $i=1$.

We write $D_{{\mathrm{rig}}}$ instead of $D_{{\mathrm{rig}}}(r)$ and drop the subscript $(\varphi,\Gamma_K)$ in this proof. We start with the first isomorphism, the proof of which is analogous to (though simpler than) the proof of (\[isoi\]) in §\[endofproof\]. We have the exact sequence (using Definition \[veryreg\]): $$\label{exactwcris}
\!\!0\rightarrow {\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\rightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\rightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)\rightarrow  0.$$ The composition: $$W_1\cap\cdots \cap W_{i-1}\hookrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}},D_{{\mathrm{rig}}}\big)\twoheadrightarrow {\rm Ext}^1\big(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big)$$ lands in ${\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ by (\[exactwcris\]). If $v\in W_1\cap\cdots \cap W_{i-1}$ is also in $W_i$, then its image in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ is $0$. We thus deduce a canonical induced map: $$\label{wimap}
W_1\cap\cdots \cap W_{i-1}/W_1\cap\cdots \cap W_{i} \rightarrow {\rm Ext}^1\big({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i}\big).$$ Let us prove that (\[wimap\]) is surjective. One easily checks that ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\subseteq W_1\cap\cdots \cap W_{i-1}$ and that the natural map ${\rm Ext}^1(D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i-1},D_{{\mathrm{rig}}})\rightarrow {\rm Ext}^1({\rm gr}_iD_{{\mathrm{rig}}},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ is surjective (again by Definition \[veryreg\]). This implies that [*a fortiori*]{} (\[wimap\]) must also be surjective. Let us prove that (\[wimap\]) is injective. If $w\in W_1\cap\cdots \cap W_{i-1}$ maps to zero, then the image of $w$ in ${\rm Ext}^1(D_{{\mathrm{rig}}}^{\leq i},D_{{\mathrm{rig}}}/D_{{\mathrm{rig}}}^{\leq i})$ is also zero, i.e. $w\in W_i$ hence $w\in W_1\cap\cdots \cap W_{i}$.

The proof for the second isomorphism is exactly the same replacing everywhere $W_j$ by $W_{{\rm cris},j}$ and ${\rm Ext}^1_{(\varphi,\Gamma_K)}$ by ${\rm Ext}^1_{\rm cris}$.

\[condcris\] We have: $$\begin{aligned}
\dim_{k(x)}\big(W_{1}\cap\cdots \cap W_{n-1}\big) &=& \dim_{k(x)}{\rm Ext}_{(\varphi,\Gamma_K)}^1\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big) - [K:{\mathbb{Q}}_p]\frac{n(n-1)}{2}\\
\dim_{k(x)}\big(W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}\big) &=& \dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)-\lg(w_x).\end{aligned}$$

This follows from Proposition \[gradcris\] together with (\[dimcrisi\]) and Lemma \[dim1\] (both for $\ell=i$) by the same argument as at the end of the proof of Proposition \[condsplit\].

\[intercris\] [Note that $W_1\cap\cdots \cap W_{n-1} \cap {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))=W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}$.]{}

Now consider $x':=(r,\delta')=(r,\delta'_1,\dots,\delta'_n)$ with $\delta'_i:=z^{{\bf k}_{w_x^{-1}(i)}}{\rm unr}(\varphi_i)$, then $x'\in \widetilde U_{\rm tri}^\square({\overline{r}})$ by (\[triang\]). We also have $\omega(x')\in \mathcal{W}^n_{w_x,{\bf k},L}$ by (\[point\]), thus $x'\in \widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L}\subseteq \widetilde U_{\rm tri}^\square({\overline{r}})$ and $\jmath_{w,{\bf k}}(x')=x$. Recall from §\[wedgesection\] and the smoothness of $U_{\rm tri}^\square({\overline{r}})$ over $\mathcal{W}^n_{L}$ that the weight map $\omega$ induces a $k(x)$-linear surjection on tangent spaces (note that $k(x')=k(x)$): $$\label{sen}
d\omega : T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\cong T_{X_{\rm tri}^\square({\overline{r}}),x'}\twoheadrightarrow T_{{\mathcal W}^n_L,\omega(x')}\cong k(x)^{[K:{\mathbb{Q}}_p]n}, \ \ \vec{v} \longmapsto (d_{\tau,i,\vec{v}})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}.$$

\[tgtx’\] We have an isomorphism of $k(x)$-subvector spaces of $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}$: $$\begin{gathered}
T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}\buildrel\sim\over\longrightarrow \\
\left\{\vec{v}\in T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\ {\rm such\ that}\ d_{\tau,i,\vec{v}}=d_{\tau,w_{x,\tau}^{-1}(i),\vec{v}},\ 1\leq i\leq n,\ \tau:\, K\hookrightarrow L\right\}.\end{gathered}$$ In particular $\dim_{k(x)}T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}=\dim X_{\rm tri}^\square({\overline{r}})-d_x$.

We write ${\mathrm{Hom}}$ instead of ${\mathrm{Hom}}_{k(x)-{\rm alg}}$ in this proof. Let $\widetilde U_{{\rm tri},w_x,{\bf k}}^\square({\overline{r}}):=\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L}$, we have: $$\label{tensorlocal}
{\mathcal O}_{\widetilde U_{{\rm tri},w_x,{\bf k}}^\square({\overline{r}}),x'}\cong {\mathcal O}_{\widetilde U_{\rm tri}^\square({\overline{r}}),x'}\otimes_{{\mathcal O}_{\mathcal{W}^n_L,\omega(x')}}{\mathcal O}_{\mathcal{W}^n_{w_x,{\bf k},L},\omega(x')}$$ and note that $T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}=T_{\widetilde U_{{\rm tri},w,{\bf k}}^\square({\overline{r}}),x'}$. Recall that, if $A,B,C,D$ are commutative $k(x)$-algebras with $B,C$ being $A$-algebras, we have: $$\label{abcd}
{\mathrm{Hom}}(B\otimes_AC,D)\buildrel\sim\over\longrightarrow {\mathrm{Hom}}(B,D)\times_{{\mathrm{Hom}}(A,D)}{\mathrm{Hom}}(C,D).$$ From (\[tensorlocal\]) and (\[abcd\]) we deduce: $$\begin{gathered}
T_{\widetilde U_{{\rm tri},w,{\bf k}}^\square({\overline{r}}),x'}={\mathrm{Hom}}\big({\mathcal O}_{\widetilde U_{{\rm tri},w_x,{\bf k}}^\square({\overline{r}}),x'},k(x)[\varepsilon]/(\varepsilon^2)\big)\cong T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\times_{T_{{\mathcal W}^n_L,\omega(x')}}T_{{\mathcal W}^n_{w_x,{\bf k},L},\omega(x')}.\end{gathered}$$ But from (\[equations\]) we have: $$T_{{\mathcal W}^n_{w_x,{\bf k},L},\omega(x')}=\left\{(d_{\tau,i})_{1\leq i\leq n,\tau:\, K\hookrightarrow L}\in T_{{\mathcal W}^n_{L},\omega(x')}\ {\rm such\ that}\ d_{\tau,i}=d_{\tau,w_{x,\tau}^{-1}(i)},\ \forall\ i,\ \forall\ \tau\right\}$$ whence the first statement. The last statement comes from $\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}=\dim \widetilde X_{\rm tri}^\square({\overline{r}})=\dim X_{\rm tri}^\square({\overline{r}})$ (since $x'$ is smooth on $X_{\rm tri}^\square({\overline{r}})$ as $x'\in U_{\rm tri}^\square({\overline{r}})$), the surjectivity of $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow T_{{\mathcal W}^n_{L},\omega(x')}$ (since the morphism $\widetilde U_{\rm tri}^\square({\overline{r}})\rightarrow {\mathcal W}^n_{L}$ is smooth by [@BHS Th.2.6(iii)]) and the same argument as in the proof of Proposition \[condweight\].

Recall from the discussion just before Conjecture \[mainconj\] that we have a closed embedding $\iota_{\bf k}:\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}\hookrightarrow \widetilde X_{\rm tri}^\square(\overline r)$ with $x\in \iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr})$. We deduce an injection of $k(x)$-vector spaces: $$T_{\iota_{\bf k}(\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\hookrightarrow T_{\widetilde X_{\rm tri}^\square(\overline r),x}.$$ Likewise we deduce from Proposition \[companion\] (assuming Conjecture \[BHS\]) another injection of $k(x)$-vector spaces: $$T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}\hookrightarrow T_{\widetilde X_{\rm tri}^\square(\overline r),x}.$$ Taking the sum in $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$ of these two subspaces of $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$, we have an injection of $k(x)$-vector spaces: $$\label{injtang}
T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\hookrightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}.$$

\[dimsum\] Assume Conjecture \[BHS\], then we have: $$\dim_{k(x)}\big(T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\big)=\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}}).$$

The composition $\widetilde X_{\rm tri}^\square({\overline{r}})\hookrightarrow \mathfrak{X}_{{\overline{r}}}^\square\times \mathcal{T}^n_L\twoheadrightarrow \mathfrak{X}_{{\overline{r}}}^\square$ induces a $k(x)$-linear morphism $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\!\rightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$. Since $x'\in \widetilde U_{\rm tri}^\square({\overline{r}})\subseteq U_{\rm tri}^\square({\overline{r}})$, it follows from [@KPX Th.Cor.6.3.10] (arguing e.g. as in the proof of [@BHS Lem.2.11]) that the triangulation $(D_{{\mathrm{rig}}}^{\leq i})_{1\leq i\leq n}$ “globalizes” in a small neighbourhood of $x'$ in $U_{\rm tri}^\square({\overline{r}})$, or equivalenty in $\widetilde U_{\rm tri}^\square({\overline{r}})$. In particular, for any $\vec{v}\in T_{\widetilde U_{\rm tri}^\square({\overline{r}}),x'}\cong T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}$ we have a triangulation of $D_{{\mathrm{rig}}}(r_{\vec{v}})$ by free $(\varphi,\Gamma_K)$-submodules over $\mathcal{R}_{k(x)[\varepsilon]/(\varepsilon^2),K}$ such that the associated parameter is $(\delta_{1,\vec{v}},\dots,\delta_{n,\vec{v}})$ (see §\[wedgesection\] for the notation). This has two consequences: (1) the proof of Lemma \[inj\] goes through and the above map $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$ is an injection of $k(x)$-vector spaces and (2) the image of the composition $T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\hookrightarrow T_{\mathfrak{X}_{{\overline{r}}}^\square,r}\twoheadrightarrow {\rm Ext}_{(\varphi,\Gamma_K)}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ (see Lemma \[ext1\]) lies in $W_{1}\cap\cdots \cap W_{n-1}\!\subseteq \!{\rm Ext}_{(\varphi,\Gamma_K)}^1(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$. From Lemma \[ext1\] we thus obtain an exact sequence: $$0\rightarrow K(r)\cap T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow W_{1}\cap\cdots \cap W_{n-1}.$$ But $\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}=\dim X_{\rm tri}^\square({\overline{r}})$ since $\widetilde X_{\rm tri}^\square({\overline{r}})$ is smooth at $x'$, and from Lemma \[ext1\], Lemma \[dim\] and Corollary \[condcris\], we have: $$\dim_{k(x)}K(r) + \dim_{k(x)}\big(W_{1}\cap\cdots \cap W_{n-1}\big) = n^2 + [K:{\mathbb{Q}}_p]\frac{n(n+1)}{2}= \dim X_{\rm tri}^\square({\overline{r}})$$ which forces a short exact sequence $0\rightarrow K(r)\rightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x'}\rightarrow W_{1}\cap\cdots \cap W_{n-1}\rightarrow 0$. It then follows from Proposition \[tgtx’\] that we have a short exact sequence of $k(x)$-vector spaces: $$\label{se1}
0\rightarrow K(r)\rightarrow T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w_x,{\bf k},L})^{\mathrm{red}},x'}\rightarrow W_{1}\cap\cdots \cap W_{n-1}\cap V\rightarrow 0$$ where $V\subseteq {\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$ is as in §\[endofproof\] (the intersection on the right hand side being in ${\rm Ext}^1_{(\varphi,\Gamma_K)}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$).

Arguing as in [@KisinModularity §2.3.5] we also have a short exact sequence (see [@Kisindef (3.3.5)]): $$\label{se2}
0\rightarrow K(r)\rightarrow T_{{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr},r}\rightarrow {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\rightarrow 0.$$ Using $T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\cong T_{{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr},r}$ (which easily follows from the fact that the Frobenius eigenvalues $(\varphi_1,\dots,\varphi_n)$ are pairwise distinct) and: $$T_{(\widetilde X_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L})^{\mathrm{red}},x'}\cong T_{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L},x'}\cong T_{\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}},x'}\buildrel\sim\over\rightarrow T_{\jmath_{w_x,{\bf k}}(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}),x},$$ we deduce from (\[se1\]) and (\[se2\]) a short exact sequence of $k(x)$-vector spaces: $$\begin{gathered}
0\rightarrow K(r)\rightarrow T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}\cap T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\rightarrow \\
W_{1}\cap\cdots \cap W_{n-1}\cap V\cap {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\rightarrow 0,\end{gathered}$$ the intersection in the middle being in $T_{\mathfrak{X}_{{\overline{r}}}^\square,r}$. But we have: $$\begin{gathered}
W_{1}\cap\cdots \cap W_{n-1}\cap V\cap {\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\buildrel\sim\over\rightarrow W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}\cap V\\
\buildrel\sim\over\rightarrow W_{{\rm cris},1}\cap\cdots \cap W_{{\rm cris},n-1}\end{gathered}$$ where   the   first   isomorphism   is   Remark   \[intercris\]   and   the   second   follows   from  ${\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))\subseteq V$ (since the Hodge-Tate weights don’t vary at all in ${\rm Ext}^1_{\rm cris}(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r))$). From Corollary \[condcris\] we thus get: $$\begin{gathered}
\label{dimker}
\dim_{k(x)}\Big(T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}\cap T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\big)=\\
\dim_{k(x)}K(r)+\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)-\lg(w_x).\end{gathered}$$ We now compute using Proposition \[tgtx’\], (\[se2\]) and (\[dimker\]): $$\begin{gathered}
\dim_{k(x)}\big(T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+ T_{\iota_{\bf k}(\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\big)=\big(\dim X_{\rm tri}^\square({\overline{r}})-d_x\big)+\\
\big(\dim_{k(x)}K(r)+\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)\big)-\\
\big(\dim_{k(x)}K(r)+\dim_{k(x)}{\rm Ext}^1_{\rm cris}\big(D_{{\mathrm{rig}}}(r),D_{{\mathrm{rig}}}(r)\big)-\lg(w_x)\big)=\\
\dim X_{\rm tri}^\square({\overline{r}})-d_x+\lg(w_x).\end{gathered}$$

\[bonnedim\] Conjecture \[BHS\] implies Conjecture \[mainconj\] for ${\overline{r}}={\overline{\rho}}_{\tilde v}$ ($v\in S_p$), i.e.: $$\dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}=\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}}).$$ In particular $x$ is smooth on $\widetilde X_{\rm tri}^\square({\overline{r}})$ if and only if $w_x$ is a product of distinct simple reflections.

It follows from (\[injtang\]) and Proposition \[dimsum\] that we have $\lg(w_x)-d_{x}+\dim X_{\rm tri}^\square({\overline{r}})\leq \dim_{k(x)}T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$. The equality follows from Corollary \[easybound\] which gives the converse inequality. Note that we also deduce $T_{\jmath_{w_x,{\bf k}}\big(\overline{\widetilde U_{\rm tri}^\square({\overline{r}})\times_{\mathcal{W}^n_L}\mathcal{W}^n_{w,{\bf k},L}}\big),x}+T_{\iota_{\bf k}(\widetilde {\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}),x}\buildrel\sim\over\longrightarrow T_{\widetilde X_{\rm tri}^\square({\overline{r}}),x}$. Finally, as we have already seen, the last statement follows from Lemma \[coxeter\].

We end up with an application of Corollary \[bonnedim\] (thus assuming Conjecture \[BHS\]) to the classical eigenvariety $Y(U^p,{\overline{\rho}})$ of §\[classic\]. We keep the notation and assumptions of §\[classic\] and §\[firstclassical\] and we consider a point $x\in Y(U^p,{\overline{\rho}})$ which is crystalline strictly dominant very regular. In a recent on-going work ([@Bergdraft]), Bergdall, inspired by the upper bound in Theorem \[upperbound\], proved an analogous upper bound for $\dim_{k(x)}T_{Y(U^p,{\overline{\rho}}),x}$, and obtained in particular that $Y(U^p,{\overline{\rho}})$ is smooth at $x$ when the Weyl group element $w_x$ in (\[wx\]) is a product of distinct simple reflections and when some Selmer group (which is always conjectured to be zero) vanishes. As a consequence of Corollary \[bonnedim\] we prove that this should [*not*]{} remain so when $w_x$ is [*not*]{} a product of distinct simple reflections.

\[singhecke\] Assume Conjecture \[BHS\] and assume that $w_x$ is [not]{} a product of distinct simple reflections. Then the eigenvariety $Y(U^p,{\overline{\rho}})$ is singular at $x$.

For $v\in S_p$ denote by $x_v$ the image of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ via (\[eigenvartotrianguline\]). Since $Y(U^p,{\overline{\rho}})\hookrightarrow X_p({\overline{\rho}})$, we have $x_v\in \widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$. It follows from Corollary \[bonnedim\] that it is enough to prove the following: if $Y(U^p,{\overline{\rho}})$ is smooth at $x$ then $\widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ is smooth at $x_v$ for all $v\in S_p$, or equivalently $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)\cong \prod_{v\in S_p}\widetilde X_{\rm tri}^\square({\overline{\rho}}_{\tilde v})$ is smooth at $(x_v)_{v\in S_p}$. Recall from §\[firstclassical\] that we have: $$\label{basechange}
Y(U^p,{\overline{\rho}})\cong X_p({\overline{\rho}})\times_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}}}{\mathrm{Sp}}\, L$$ where the map $S_\infty\twoheadrightarrow L$ is $S_\infty\twoheadrightarrow (S_\infty/{\mathfrak a})[1/p]$ and where $X_p({\overline{\rho}})\rightarrow {\mathfrak X}_\infty\rightarrow ({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}}$ is induced by the morphism $S_\infty\rightarrow R_\infty$. Let $\omega_{\infty}(x)$ be the image of $x$ in $({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}}$, by an argument similar to the one in the proof of Proposition \[tgtx’\] we deduce from (\[basechange\]): $$T_{Y(U^p,{\overline{\rho}}),x}\cong \left\{\vec{v}\in T_{X_p({\overline{\rho}}),x}\ {\rm mapping\ to}\ 0\ {\rm in}\ T_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}},\omega_\infty(x)}\otimes_{k(\omega_\infty(x))}k(x)\right\}.$$ This obviously implies: $$\label{tangentxp}
\dim_{k(x)}T_{Y(U^p,{\overline{\rho}}),x}\geq \dim_{k(x)}T_{X_p({\overline{\rho}}),x} - \dim_{k(\omega_\infty(x))}T_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}},\omega_\infty(x)}.$$ But $\dim_{k(\omega_\infty(x))}T_{({\mathrm{Spf}}\, S_\infty)^{{\mathrm{rig}}},\omega_\infty(x)}=g+[F^+:{\mathbb{Q}}]\tfrac{n(n-1)}{2}+|S|n^2$ (see beginning of §\[firstclassical\]) and $\dim_{k(x)}T_{Y(U^p,{\overline{\rho}}),x}=\dim Y(U^p,{\overline{\rho}})=n[F^+:{\mathbb{Q}}]$ since $x$ is assumed to be smooth on $Y(U^p,{\overline{\rho}})$, hence we deduce from (\[tangentxp\]): $$\dim_{k(x)}T_{X_p({\overline{\rho}}),x}\leq g+[F^+:{\mathbb{Q}}]\tfrac{n(n+1)}{2}+|S|n^2=\dim X_p({\overline{\rho}})$$ where the last equality follows from [@BHS Cor.3.11]. We thus have $\dim_{k(x)}T_{X_p({\overline{\rho}}),x}=\dim X_p({\overline{\rho}})$ which implies that $x$ is smooth on $X_p({\overline{\rho}})$, and thus by (i) of Remark \[conjvariant\] that $(x_v)_{v\in S_p}$ is smooth on $\widetilde X_{\rm tri}^\square({\overline{\rho}}_p)$.

[Singular crystalline strictly dominant points on eigenvarieties are already known to exist by [@Bel §6]. However, the singular points of [*loc. cit.*]{} are different from the points $x$ of Corollary \[singhecke\] since they have reducible associated global Galois representations.]{}

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