Deformation spaces of one-dimensional formal modules and their cohomology

TL;DR

This paper investigates the geometry and cohomology of deformation spaces of one-dimensional formal modules, demonstrating how the Jacquet-Langlands correspondence is reflected in their cohomological properties without relying on global moduli spaces.

Contribution

It provides a new proof of the Jacquet-Langlands correspondence for supercuspidal representations using deformation space cohomology, avoiding global moduli spaces.

Findings

Jacquet-Langlands correspondence realized by Euler-Poincare characteristic

Under finiteness assumptions, correspondence appears in a single cohomological degree

Analysis of boundary and compactifications of deformation spaces is crucial

Abstract

We study the geometry and cohomology of the (generic fibres) of formal deformation schemes of one-dimensional formal modules of finite height. By the work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic etale cohomology of these spaces realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence for supercuspidal representations is realized by the Euler-Poincare characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the deformation spaces.