  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
$\mathbb{H}^d_K$.
