A quotient of the Lubin-Tate tower

TL;DR

This paper demonstrates that the quotient of the Lubin-Tate space at infinite level by the Borel subgroup forms a perfectoid space and applies this to show concentration of certain étale cohomology functors in degree one for specific representations.

Contribution

It establishes the existence of a perfectoid quotient space of the Lubin-Tate tower and applies this to analyze the cohomology of projective lines with specific representations.

Findings

The quotient of the Lubin-Tate space by the Borel subgroup is a perfectoid space.

Scholze's functor H^i_et(P^1,F(pi)) is concentrated in degree one for principal series and Steinberg twists.

Provides new geometric insights into the structure of Lubin-Tate towers and their cohomology.

Abstract

In this article we show that the quotient of the Lubin-Tate space at infinite level by the Borel subgroup of upper triangular matrices in GL(2,Q_p) exists as a perfectoid space. As an application we show that Scholze's functor H^i_et(P^1,F(pi)) is concentrated in degree one whenever pi is a principal series representation or a twist of the Steinberg representation of GL(2,Q_p).