  In this paper, we define a family of affinoids in the tubular neighborhoods
of CM points in the Lubin-Tate curve with suitable level structures, and
compute the reductions of them in the equal characteristic case. By using etale
cohomology theory of adic spaces due to Huber, we show that the cohomology of
the reductions contributes to the cohomology of the Lubin-Tate curve. As an
application, with the help of explicit descriptions of the local Langlands
correspondence and the local Jacquet-Langlands correspondence due to
Bushnell-Henniart via the theory of types, we prove the non-abelian Lubin-Tate
theory for GL(2) in the odd equal characteristic case in a purely local manner.
Conversely, if we admit the non-abelian Lubin-Tate theory, we can recover the
explicit descriptions of the two correspondences geometrically.
