  Higher Green functions are real-valued functions of two variables on the
upper half plane which are bi-invariant under the action of a congruence
subgroup, have logarithmic singularity along the diagonal, but instead of the
usual equation $\Delta f=0$ we have equation $\Delta f = k(1-k) f$. Here $k$ is
a positive integer. Properties of these functions are related to the space of
modular forms of weight $2k$. In the case when there are no cusp forms of
weight $2k$ it was conjectured that the values of the Green function at points
of complex multiplication are algebraic multiples of logarithms of algebraic
numbers. We show that this conjecture can be proved in any particular case if
one constructs a family of elements of certain higher Chow groups on the power
of a family of elliptic curves. These families have to satisfy certain
properties. A different family of elements of Higher Chow groups is needed for
a different point of complex multiplication. We give an example of such
families, thereby proving the conjecture for the case when the group is
$PSL_2(\mathbf Z)$, $k=2$ and one of the arguments is $\sqrt{-1}$.
