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A QUOTIENT OF THE LUBIN–TATE TOWER

Part of: Algebraic number theory: local and $p$-adic fields Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  27 July 2017

JUDITH LUDWIG
JUDITH LUDWIG*
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany; ludwig@mpim-bonn.mpg.de

Abstract

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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 $H_{\acute{\text{e}}\text{t}}^{i}(\mathbb{P}_{\mathbb{C}_{p}}^{1},{\mathcal{F}}_{\unicode[STIX]{x1D70B}})$ is concentrated in degree one whenever $\unicode[STIX]{x1D70B}$ is an irreducible principal series representation or a twist of the Steinberg representation of $\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

MSC classification

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory
Secondary: 14G22: Rigid analytic geometry
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e17
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2017

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