*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 \[compaclas\] 