Action of Hecke operators on cohomology of modular curves of level two

We calculate the action of the p-th Hecke operator and the inertia group on the a""-adic cohomology of modular curve of level I"(0)(p (2)) under the assumption p a parts per thousand yen 13, using only a local geometrical method. We also calculate the action of the p-th Hecke operator and the inertia group on the a""-adic cohomology of the Lubin-Tate space of the same level over the maximal unramified extension of .