Invertible functions on non-archimedean symmetric spaces

Abstract

Let $u$ be a nowhere vanishing holomorphic function on the Drinfeld space ${\Omega }^r$ of dimension $r-1$, where $r\ge 2$. The logarithm ${\log }_q|u|$ 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 $\mathcal{O}({\Omega }^r{)}^{\ast }$ of such $u$ with the group $\mathbf{H}({\mathcal{B}\mathcal{T}}^r,\mathbb{Z})$ 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 ${\Omega }^r$.

20 pages, accepted for publication in Algebra and Number Theory

Cited by in corpus (1)

• Cohomologie analytique des arrangements d'hyperplans