Lattices in rigid analytic representations

For a profinite group $G$ and a rigid analytic space $X$, we study when an ${\mathcal{O}}_X(X)$-linear representation $V$ of $G$ admits a lattice, i.e. an ${\mathcal{O}}_{\mathcal{X}(\mathcal{X})}$-linear model for a suitable formal model $\mathcal{X}$ of $X$ in the sense of Berthelot. We give a positive answer, under mild assumptions, when $X$ is strictly quasi-Stein. As a consequence, we are able to describe explicit open rational subdomains of $X$ over which $V$ is constant after reduction modulo a power of $p$. We give applications in two different directions. First, we prove explicit results on the reduction modulo powers of $p$ of sheaves of crystalline and semistable representations of fixed weight. Second, we deduce a result on the pseudorepresentation carried by the Coleman–Mazur eigencurve, which can be made explicit whenever equations for a rational subdomain of the eigencurve are given.