CM values of higher Green's functions and regularized Petersson products

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 a logarithmic singularity along the diagonal, and satisfy the equation f = k(1−k) f ; here is a hyperbolic Laplace operator and k is a positive integer. The significant arithmetic properties of these functions were disclosed in the paper of B. Gross and D. Zagier “Heegner points and derivatives of L series” (1986). In the particular case when k = 2 and one of the CM points is equal to √−1, the conjecture has been proved by A. Mellit in his Ph.D. thesis. In this lecture we prove that conjecture for arbitrary k, assuming that all the pairs of CM points lie in the same quadratic field. The two main parts of the proof are as follows. We first show that the regularized Petersson scalar product of a binary theta-series and a weight one weakly holomorphic cusp form is equal to the logarithm of the absolute value of an algebraic integer and then prove that the special values of weight k Green’s function, occurring in the conjecture of Gross and Zagier, can be written as the Petersson product of that type, where the form of weight one is the k − 1st Rankin-Cohen bracket of an explicitly given holomorphic modular form of weight 2 − 2k and a binary theta-series. Algebraicity of regularized Petersson products was also proved at about the same time by W. Duke and Y. Li by a different method; however, our result is stronger since we also give a formula for the factorization of the algebraic number in question.