Équations pour le premier revêtement de l'espace symétrique de Drinfeld

Abstract

The goal of this work is to study some aspects of the geometry of the first cover ${\Sigma }^1$ in the Drinfeld tower over ${\mathbb{H}}_K^d$ the Drinfeld symmetric space over $K$ a finite extension of ${\mathbb{Q}}_p$. It is a cyclic étale cover of order prime to $p$ and even of Kummer type from the vanishing of the Picard group of ${\mathbb{H}}_K^d$ shown in a previous work of the author. It is then completely described by a certain class of invertible functions on ${\mathbb{H}}_K^d$ via the Kummer exact sequence and the main result of this article gives an explicit description of this class thus providing "equations" for ${\Sigma }^1$. This statement extends and uses crucially the local description over a vertex obtained by Wang (and originally by Teitelbaum in dimension 1). One of the main consequence of our global equation is the description of invertible functions of ${\Sigma }^1$ in terms of the invertible functions of ${\mathbb{H}}_K^d$.

35 pages, in french, new version : acknoledgement added