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}
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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}
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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 