de Rham theory and locally analytic vectors

Abstract

Let $K_\infty/K$ be a $p$-adic Lie extension of a $p$-adic field $K$. We study the subring of pro-analytic vectors in the de Rham period ring $\mathbf{B}_{\mathrm{dR}}^+(K_\infty)$. We show that the pro-analytic subring admits a Galois-equivariant isomorphism with a formal power series ring $\widehat{K}_{\infty}^{\mathrm{la}} [[t_{K_\infty}]]$ if and only if $K_\infty$ satisfies a certain orientability condition, which says that the $\widehat{K}_\infty$-level Sen operator admits a Galois-equivariant $\mathbf{B}_{\mathrm{dR}}^+$-lift. A key input is the vanishing of higher locally analytic vectors of $\widehat{K}_\infty$-representations. As an application, we show that the lifted Sen operator induces regular connections on pro-analytic vectors of $\mathbf{B}_{\mathrm{dR}}^+$-representations, and can be used to compute Galois cohomology.