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 ${\hat{K}}_{\infty }^{\mathrm{la}}[[t_{K_{\infty }}]]$ if and only if $K_{\infty }$ satisfies a certain orientability condition, which says that the ${\hat{K}}_{\infty }$-level Sen operator admits a Galois-equivariant ${\mathbf{B}}_{\mathrm{dR}}^{+}$-lift. A key input is the vanishing of higher locally analytic vectors of ${\hat{K}}_{\infty }$-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.