Affinoids in the Lubin-Tate perfectoid space and simple supercuspidal representations II: wild case

TL;DR

This paper constructs specific affinoids in the Lubin-Tate perfectoid space whose cohomology realizes key local correspondences and Galois representations, also proving the Tate conjecture for related Artin-Schreier varieties.

Contribution

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

Findings

Cohomology of reductions realizes local Langlands correspondence.

Formal models are isomorphic to perfections of Artin-Schreier varieties.

Proves Tate conjecture for Artin-Schreier varieties associated to quadratic forms.

Abstract

We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of their reductions realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for the simple supercuspidal representations. The reductions of the formal models are isomorphic to the perfections of some Artin-Schreier varieties, whose cohomology realizes primitive Galois representations. We show also the Tate conjecture for Artin-Schreier varieties associated to quadratic forms.