Geodesic
A QUOTIENT OF THE LUBIN–TATE TOWER
Forum of Mathematics, Sigma, Tome 5 (2017)

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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 $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ . 
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     title = {A {QUOTIENT} {OF} {THE} {LUBIN{\textendash}TATE} {TOWER}},
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JUDITH LUDWIG. A QUOTIENT OF THE LUBIN–TATE TOWER. Forum of Mathematics, Sigma, Tome 5 (2017). doi: 10.1017/fms.2017.15

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