Kähler differentials and ${\mathbf{Z}}_p$-extensions

Abstract

Let $K$ be a $p$-adic field, and let $K_{\infty }/K$ be a Galois extension that is almost totally ramified, and whose Galois group is a $p$-adic Lie group of dimension $1$. We prove that $K_{\infty }$ is not dense in $({\mathbf{B}}_{\mathrm{dR}}^{+}/{\mathrm{Fil}}^2{\mathbf{B}}_{\mathrm{dR}}^{+}{)}^{\mathrm{Gal}(\overline{K}/K_{\infty })}$. Moreover, the restriction of $\theta$ to the closure of $K_{\infty }$ is injective, and its image via $\theta$ is the set of vectors of ${\hat{K}}_{\infty }$ that are $C^1$ with zero derivative for the action of $\mathrm{Gal}(K_{\infty }/K)$. The main ingredient for proving these results is the construction of an explicit lattice of ${\mathcal{O}}_{K_{\infty }}$ that is commensurable with ${\mathcal{O}}_{K_{\infty }}^{d=0}$, where $d:{\mathcal{O}}_{K_{\infty }}\to {\Omega }_{{\mathcal{O}}_{K_{\infty }}/{\mathcal{O}}_K}$ is the differential.