On the existence of analytic families of G-stable lattices and their reductions

Abstract

In this article, we prove the existence of rigid analytic families of $G$-stable lattices with locally constant reductions inside families of representations of a topologically compact group $G$, extending a result of Hellman obtained in the semi-simple residual case. Implementing this generalization in the context of Galois representations, we prove a local constancy result for reductions modulo prime powers of trianguline representations of generic dimension $d$. Moreover, we present two explicit applications. First, in dimension two, we extend to a prime power setting and to the whole rigid projective line a recent result of Bergdall, Levin and Liu concerning reductions of semi-stable representations of $\text{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ with fixed Hodge-Tate weights and large $\mathcal{L}$-invariant. Second, in dimension $d$, let $V_n$ be a sequence of crystalline representations converging in a certain geometric sense to a crystalline representation $V$. We show that for any refined version $(V,\sigma )$ of $V$ (or equivalently for any chosen triangulation of its attached $(\phi ,\Gamma )$-module $D_{\text{rig}}(V)$ over the Robba ring), there exists a sequence of refinement ${\sigma }_n$ of each of the $V_n$ such that the limit as refined representations $(V_n,{\sigma }_n)$ converges to the $(V,\sigma )$. This result does not hold under the weaker assumption that $V_n$ converges only uniformly $p$-adically to $V$ (in the sense of Chenevier, Khare and Larsen).

References in corpus (4)

• Converging sequences of p-adic Galois representations and density theorems
• Reductions of certain 3-dimensional crystalline representations
• Reduction modulo $p$ of certain semi-stable representations
• A formal model of Coleman families and applications to Iwasawa invariants