Good reduction of affinoids on the Lubin-Tate tower

TL;DR

This paper studies the geometry of Lubin-Tate deformation spaces, identifying affinoids with good reduction properties and linking their L-functions to the local Langlands correspondence and Bushnell-Kutzko types.

Contribution

It explicitly constructs affinoids with good reduction in the Lubin-Tate tower and connects their L-functions to the local Langlands conjecture and representation theory.

Findings

Identified affinoids with good reduction as nonsingular hypersurfaces.

Established a link between L-functions of hypersurfaces and the local Langlands correspondence.

Provided explicit equations for the affinoids in the Lubin-Tate tower.

Abstract

We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a one-dimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Granting a conjecture on the L-functions of this hypersurface, we find a link between the conjecture of Deligne-Carayol and the theory of Bushnell-Kutzko types, at least for certain class of wildly ramified supercuspidal representations of small conductor.