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

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 $\mathbb{H}^d_K$.