A quotient of the Lubin-Tate tower. (English) Zbl 1434.11110
Summary: In this article we show that the quotient \(\mathcal{M}_\infty/B(\mathbb{Q}_p)\) of the Lubin-Tate space at infinite level \(\mathcal{M}_\infty\) by the Borel subgroup of upper triangular matrices \(B(\mathbb{Q}_p)\subset \operatorname{GL}_2(\mathbb{Q}_p)\) exists as a perfectoid space. As an application we show that Scholze’s functor \(H_{\text{é}\text{t}}^i(\mathbb{P}_{\mathbb{C}_{p}}^{1},\mathcal{F}_\pi)\) is concentrated in degree one whenever \(\pi\) is an irreducible principal series representation or a twist of the Steinberg representation of \(\operatorname{GL}_2(\mathbb{Q}_p)\).
MSC:
| 11F77 | Automorphic forms and their relations with perfectoid spaces |
| 14G45 | Perfectoid spaces and mixed characteristic |
| 11S37 | Langlands-Weil conjectures, nonabelian class field theory |
| 14G22 | Rigid analytic geometry |
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