Lattices in rigid analytic representations

2024/03/29 by Andrea Conti, Conti, Andrea, Emiliano Torti +1
Mathematics · #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2403.20232

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.

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