Invertible functions on nonarchimedean symmetric spaces

Gekeler, Ernst-Ulrich

doi:10.2140/ant.2020.14.2481

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Invertible functions on nonarchimedean symmetric spaces

Ernst-Ulrich Gekeler

Vol. 14 (2020), No. 9, 2481–2504

DOI: 10.2140/ant.2020.14.2481

Abstract

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

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Keywords

Drinfeld symmetric space, van der Put transform, Bruhat–Tits building

Mathematical Subject Classification 2010

Primary: 32P05

Secondary: 11F23, 11F85, 32C30, 32C36

Milestones

Received: 16 September 2019

Revised: 30 March 2020

Accepted: 11 May 2020

Published: 13 October 2020

Authors

Ernst-Ulrich Gekeler
Mathematik Universität des Saarlandes Saarbrücken Germany

Vol. 14, No. 9, 2020