A quotient of the Lubin\u2013Tate tower II

Abstract

In this article we construct the quotient mathcal {M}_mathbf {1}/P(K) of the infinite-level Lubin–Tate space mathcal {M}_mathbf {1} by the parabolic subgroup P(K) subset mathrm {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/{mathbb {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 mathcal {M}_mathbf {1}/P(K) when n=2, and shows that mathcal {M}_mathbf {1}/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.