Un autre calcul des fonctions inversibles sur l'espace sym\'etrique de Drinfeld

TL;DR

This paper explicitly describes invertible functions on the Drinfeld symmetric space over a p-adic field, linking them to distribution spaces and establishing a new isomorphism that provides a Z-structure on related cohomology groups.

Contribution

It introduces a novel explicit description of invertible functions on the Drinfeld symmetric space via distribution spaces, extending previous methods and establishing an isomorphism that yields a Z-structure on cohomology.

Findings

Invertible functions correspond to certain distribution spaces.

Constructed an isomorphism compatible with existing cohomology computations.

Established a Z-structure on the étale and de Rham cohomology groups.

Abstract

In this article, we give an explicit description of the invertible functions on the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. We identify them with some distribution spaces over the profinite set of $K$-rationnal points of the projective space. The strategy consists of constructing a map from these distributions to the invertible functions following the methods of Schneider-Stuhler, Iovita-Spiess, de Shalit. We show that it is compatible with the isomorphisms they constructed to compute \'etale and de Rham cohomology in degree $1$ and that this property forces our desired map to be an isomorphism. In particular, we get a $\mathbb{Z}$-structure on these cohomology groups.