Super-Hölder vectors and the field of norms

Abstract

Let $E$ be a field of characteristic $p$. In a previous paper of ours, we defined and studied super-Hölder vectors in certain $E$-linear representations of ${\mathbb{Z}}_p$. In the present paper, we define and study super-Hölder vectors in certain $E$-linear representations of a general $p$-adic Lie group. We then consider certain $p$-adic Lie extensions $K_{\infty }∕K$ of a $p$-adic field $K$, and compute the super-Hölder vectors in the tilt of $K_{\infty }$. We show that these super-Hölder vectors are the perfection of the field of norms of $K_{\infty }∕K$. By specializing to the case of a Lubin–Tate extension, we are able to recover $E((Y))$ inside the $Y$-adic completion of its perfection, seen as a valued $E$-vector space endowed with the action of ${\mathcal{𝒪}}_K^{\times }$ given by the endomorphisms of the corresponding Lubin–Tate group.