On the cuspidal representations of ${\rm GL}_2(F)$ of level 1 or 1/2 in the cohomology of the Lubin-Tate space $\mathcal{X}(\pi^2)$

TL;DR

This paper explicitly computes the stable reduction of the Lubin-Tate curve of level two in mixed characteristic, revealing how the local Langlands and Jacquet-Langlands correspondences are realized for certain cuspidal representations of ${ m GL}_2(F)$.

Contribution

It provides explicit calculations of the stable reduction components and actions, demonstrating the realization of local correspondences for specific cuspidal representations.

Findings

Explicit description of the stable reduction of the Lubin-Tate curve of level two.

Realization of local Langlands and Jacquet-Langlands correspondences in cohomology.

Explicit action of the division algebra and ${ m GL}_2$ on the cohomology.

Abstract

In this paper, we compute irreducible components which appear in the stable reduction of the Lubin-Tate curve of level two, in the mixed characteristic case. We also compute the action of the central division algebra of invariant 1/2, the action of ${\rm GL}_2$, and the inertia action explicitly. As a result, in a sense, we observe that, in the cohomology group of the stable reduction of the Lubin-Tate curve for ${\rm GL}_2$, the local Langlands correspondence and the local Jacquet-Langlands correspondence for ${\rm GL}_2$ are realized for the cuspidal representations of level 1 or 1/2.