Lattices in rigid analytic representations

Abstract

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 and regular. 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.

arXiv comment43 pages. Replaced the old Theorem 4.14 with the simpler Proposition 5.3, requiring extra assumptions on the regularity of the space and the dimension of the pseudorepresentation