Th\'eorie de Sen et vecteurs localement analytiques

TL;DR

This paper extends Sen's theory to higher-dimensional p-adic Lie extensions by utilizing locally analytic vectors, providing a broader framework for understanding Galois representations.

Contribution

It generalizes Sen's theory to p-adic Lie extensions of arbitrary dimension using locally analytic vectors, especially focusing on Lubin-Tate extensions.

Findings

Describes the field of locally analytic vectors in general cases.

Provides a detailed analysis for Lubin-Tate extensions.

Establishes a new framework for p-adic Galois representations.

Abstract

We generalize Sen theory to extensions $K_\infty/K$ whose Galois group is a $p$-adic Lie group of arbitrary dimension. To do so, we replace Sen's space of $K$-finite vectors by Schneider and Teitelbaum's space of locally analytic vectors. One then gets a vector space over the field of locally analytic vectors of $\hat{K}_\infty$. We describe this field in general and pay a special attention to the case of Lubin-Tate extensions.