Explicit non-abelian Lubin-Tate theory for GL(2)

TL;DR

This paper constructs a stable curve over an algebraic closure of a finite field that encodes the local Langlands and Jacquet-Langlands correspondences for GL(2) over a non-Archimedean local field, providing explicit geometric realizations.

Contribution

It provides an explicit geometric construction of a stable curve whose cohomology realizes the local Langlands and Jacquet-Langlands correspondences for GL(2) over a non-Archimedean local field.

Findings

Realizes the correspondences in the cohomology of a constructed stable curve.

Connects geometric objects with deep representation-theoretic correspondences.

Provides explicit geometric models for non-abelian Lubin-Tate theory.

Abstract

Let $F$ be a non-Archimedean local field with residue field $k$ of odd characteristic, and let $B/F$ be the division algebra of rank 4. We explicitly construct a stable curve $\mathfrak{X}$ over the algebraic closure of $k$ admitting an action of $GL_2(F)\times B^\times \times W_F$ which realizes the Jacquet-Langlands correspondence and the local Langlands correspondence in its cohomology.