A QUOTIENT OF THE LUBIN–TATE TOWER. (27th July 2017)

• Record Type: Journal Article
• Authors: LUDWIG, JUDITH
• Is Part Of: Forum of mathematics. Volume 5(2017)
• Publication Date: 2017-07-27
• Subjects: 11S37 (primary), -- 14G22 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2017.15
• Languages: English
• ISSNs: 2050-5094

Abstract

Abstract : 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 $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ .