A quotient of the Lubin-Tate tower II

Journal article, 2021

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) subset of GL(n)(K) of block form (n - 1, 1) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) 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 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.