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In this paper, we compute irreducible components which appear in the stablereduction of the Lubin-Tate curve of level two, in the mixed characteristiccase.  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, ina sense, we observe that, in the cohomology group of the stable reduction ofthe Lubin-Tate curve for ${\rm GL}_2$, the local Langlands correspondence andthe local Jacquet-Langlands correspondence for ${\rm GL}_2$ are realized forthe cuspidal representations of level 1 or 1/2.

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