A QUOTIENT OF THE LUBIN–TATE TOWER

Judith Ludwig

doi:10.1017/fms.2017.15

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

Highlights

In this article we show that the quotient M∞/B(Qp) of the Lubin–Tate space at infinite level M∞ by the Borel subgroup of upper triangular matrices B(Qp) ⊂ GL2(Qp) exists as a perfectoid space
As an application we show that Scholze’s functor Heit(P1Cp, Fπ ) is concentrated in degree one whenever π is an irreducible principal series representation or a twist of the Steinberg representation of GL2(Qp). 2010 Mathematics Subject Classification: 11S37; 14G22
In [13], Scholze constructs a candidate for the mod p local Langlands correspondence for the group GLn(F), where n 1 and F/Qp is a finite extension

Summary

Introduction

In [13], Scholze constructs a candidate for the mod p local Langlands correspondence for the group GLn(F), where n 1 and F/Qp is a finite extension. If there is an open cover of X by affinoid subsets Spa(A, A+) ⊂ X such that the map lim Am → A −→ Assume X = Spa(R, R+) is an affinoid perfectoid space over Spa(K , OK ), with compatible maps pm : X → Xm and such that be a rational subset.

Results

Conclusion