# A quotient of the Lubin-Tate tower II

## TL;DR

This paper constructs a perfectoid quotient of the infinite-level Lubin-Tate space by a parabolic subgroup, extending previous results to arbitrary n and K/Q_p, and explores implications for the mod p Langlands program.

## Contribution

It generalizes the construction of perfectoid quotients of Lubin-Tate spaces to all dimensions and provides new vanishing results relevant to the mod p Langlands correspondence.

## Key findings

- Constructed the quotient M_1/P(K) as a perfectoid space for arbitrary n and K/Q_p.
- Proved perfectoidness results for Harris-Taylor Shimura varieties at infinite level.
- Established a vanishing theorem for mod p Jacquet-Langlands and local Langlands candidates.

## Abstract

In this article we construct the quotient M_1/P(K) of the infinite-level Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form (n-1,1) as a perfectoid space, generalizing results of one of the authors (JL) to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results for certain Harris-Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze's candidate for the mod p Jacquet-Langlands and the mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of M_1/P(K) when n = 2, and shows that M_1/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.