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

Let u be a nowhere vanishing holomorphic function on the Drinfeld space Ωr of dimension r 1, where r 2. The logarithm logq|u| of its absolute value may be regarded as an affine function on the attached Bruhat–Tits building 𝒯r. Generalizing a construction of van der Put in case r = 2, we relate the group 𝒪(Ωr) of such u with the group H(𝒯r, ) of integer-valued harmonic 1-cochains on 𝒯r. This also gives rise to a natural -structure on the first (-adic or de Rham) cohomology of Ωr.

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Drinfeld symmetric space, van der Put transform, Bruhat–Tits building
Mathematical Subject Classification 2010
Primary: 32P05
Secondary: 11F23, 11F85, 32C30, 32C36
Received: 16 September 2019
Revised: 30 March 2020
Accepted: 11 May 2020
Published: 13 October 2020
Ernst-Ulrich Gekeler
Mathematik Universität des Saarlandes