Lattices in rigid analytic representations

TL;DR

We develop a framework to study lattices in rigid analytic G-representations, showing that for strictly quasi-Stein spaces $X$ and absolutely irreducible residually multiplicity-free families, there exist lattices over formal models. This leads to explicit constancy results modulo powers of $p$ on clearly described residue neighborhoods, enabling precise control of reductions of crystalline and semistable two-dimensional Galois representations and of pseudorepresentations along eigenvarieties such as the Colemann–Mazur eigencurve. The approach hinges on connecting formal deformation theory, rigid-analytic geometry, and generalized matrix algebras to produce integral models that glue across $X$, and on transferring these lattice structures to local ($p$-adic) neighborhoods where reductions stabilize. The results yield explicit local radii for constancy, explicit reductions in arithmetic families, and concrete consequences for trianguline representations and for congruences between automorphic forms on eigenvarieties, including an explicit example involving a Hida-family equation. Altogether, the paper provides a robust toolkit for analyzing $p$-adic variation of G-representations in families with precise mod $p^n$ control and explicit arithmetic applications.

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. 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.

Theorems & Definitions (131)

• Theorem 1.1: tortired
• Definition 1.2
• Theorem 1.3: cf. Theorem \ref{['wideopenlatt']}
• Definition 1.4: cf. Definition \ref{['lcformdef']}
• Theorem 1.5: cf. Theorem \ref{['Unconst']}
• Theorem 1.6: cf. Theorem \ref{['bll']}
• Corollary 1.7: cf. Corollary \ref{['corhida']}
• Definition 2.1
• Remark 2.2
• Remark 2.3