#### Vol. 14, No. 9, 2020

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Invertible functions on nonarchimedean symmetric spaces

### Ernst-Ulrich Gekeler

Vol. 14 (2020), No. 9, 2481–2504
##### Abstract

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

##### Keywords
Drinfeld symmetric space, van der Put transform, Bruhat–Tits building
##### Mathematical Subject Classification 2010
Primary: 32P05
Secondary: 11F23, 11F85, 32C30, 32C36