Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations I: tame case

TL;DR

This paper constructs specific affinoids in the Lubin-Tate perfectoid space whose cohomology realizes key local correspondences for simple supercuspidal representations in the tame case, linking geometry and representation theory.

Contribution

It introduces a new geometric construction of affinoids in the Lubin-Tate space that realize local Langlands and Jacquet-Langlands correspondences for simple supercuspidal representations.

Findings

Cohomology of constructed affinoids matches local correspondences.

Reductions are isomorphic to Artin-Schreier varieties.

Provides geometric realization in the tame case.

Abstract

We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of the reductions of the formal models realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for simple supercuspidal representations in the case where the dimension of Galois representations is prime to the residue characteristic. The reductions of the formal models are isomorphic to the perfections of Artin-Schreier varieties associated to quadratic forms.