Affinoids in the Lubin-Tate perfectoid space and special cases of the local Langlands correspondence

TL;DR

This paper constructs specific affinoids in the Lubin-Tate perfectoid space whose cohomology realizes parts of the local Langlands and Jacquet-Langlands correspondences for certain totally ramified representations.

Contribution

It introduces a new family of affinoids and formal models that explicitly realize the local Langlands and Jacquet-Langlands correspondences for minimal admissible pairs with totally ramified extensions.

Findings

Cohomology of constructed affinoids matches local Langlands correspondence

Realizes local Jacquet-Langlands correspondence for specific representations

Provides explicit geometric models for certain tame representations

Abstract

Following Weinstein, Boyarchenko-Weinstein and Imai-Tsushima, we construct a family of affinoids in the Lubin-Tate perfectoid space and formal models such that the cohomology of the reduction of each formal model realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for certain representations. In the terminology of the essentially tame local Langlands correspondence, the representations treated here are characterized as being parametrized by minimal admissible pairs in which the field extensions are totally ramified.