Invertible functions on nonarchimedean symmetric spaces

Abstract

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