# Invertible functions on non-archimedean symmetric spaces

## TL;DR

This paper explores the relationship between invertible holomorphic functions on Drinfeld spaces and harmonic cochains on associated Bruhat-Tits buildings, extending previous work to higher dimensions and linking to cohomology structures.

## Contribution

It generalizes van der Put's construction from dimension 2 to higher dimensions, relating invertible functions to harmonic cochains and establishing a natural integer structure on cohomology.

## Key findings

- Establishes an isomorphism between invertible functions and harmonic 1-cochains.
- Provides a natural -structure on the cohomology of Drinfeld spaces.
- Extends known results to higher-dimensional non-archimedean symmetric spaces.

## Abstract

Let $u$ be a nowhere vanishing holomorphic function on the Drinfeld space $\Omega^{r}$ of dimension $r-1$, where $r \geq 2$. The logarithm $\log_{q}\lvert u \rvert$ of its absolute value may be regarded as an affine function on the attached Bruhat-Tits building $\mathcal{BT}^{r}$. Generalizing a construction of van der Put in case $r=2$, we relate the group $\mathcal{O}(\Omega^{r})^{*}$ of such $u$ with the group $\mathbf{H}(\mathcal{BT}^{r}, \mathbb{Z})$ of integer-valued harmonic 1-cochains on $\mathcal{BT}^{r}$. This also gives rise to a natural $\mathbb{Z}$-structure on the first ($\ell$-adic or de Rham) cohomology of $\Omega^{r}$.