\'Equations pour le premier rev\^etement de l'espace sym\'etrique de Drinfeld

TL;DR

This paper explicitly describes the equations defining the first cover in the Drinfeld tower over symmetric spaces, linking invertible functions on the cover to those on the base space, extending local descriptions to a global context.

Contribution

It provides an explicit description of the class of invertible functions defining the first Drinfeld cover using Kummer theory, extending local descriptions to a global setting.

Findings

Explicit equations for the first Drinfeld cover $\\Sigma^1$

Description of invertible functions on $\\Sigma^1$ in terms of base space functions

Extension of local descriptions to a global framework

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 $\mathbb{H}^d_K$ the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. It is a cyclic \'etale cover of order prime to $p$ and even of Kummer type from the vanishing of the Picard group of $\mathbb{H}^d_K$ shown in a previous work of the author. It is then completely described by a certain class of invertible functions on $\mathbb{H}^d_K$ 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 $\mathbb{H}^d_K$.