Geodesic cycles and Eisenstein classes for SL(2,Z)

The geodesic cycles (resp. Eisenstein classes) for SL(2,Z) are special classes in the homology (resp. cohomology) of modular curve (for SL(2,Z)) defined by the closed geodesics (resp. Eisenstein series). It is known that the pairing between these geodesic cycles and Eisenstein classes gives the special values of partial zeta functions of real quadratic fields, and this has many applications. In this talk, I would like to report on some recent observations on the size of the homology subgroup generated by geodesic cycles and their applications. This is a joint work with Ryotaro Sakamoto.