CM Values of Higher Green's Functions

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, and satisfy the equation
Delta f =k(1 - k)f, where Delta is a hyperbolic Laplace operator and k is a
positive integer. Such functions were introduced in the paper of
Gross and Zagier  "Heegner points and derivatives of $L-series"(1986).
Also it was conjectured in this paper that higher Green's  functions have
``algebraic'' values at CM points.  In many particular cases this conjecture
was proven by A. Mellit in his Ph. D. thesis. In this talk we will present a 
proof of the conjecture for any pair of CM points lying in the same quadratic
imaginary field.