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 