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 