The automorphic Green function for a modular curve $X$ is a function on $X\times X$ with a logarithmic singularity along the diagonal which is a resolvent kernel of the hyperbolic Laplacian. It plays an important role in the analytic theory of automorphic forms and in the Arakelov geometry of modular curves. Gross and Zagier conjectured that for positive integral spectral parameter $s$ the values at CM points of certain linear combinations of Hecke translates of this Green function are given by logarithms of algebraic numbers in suitable class fields. In certain cases this conjecture was proved by Mellit and Viazovska. We report on joint work with S. Ehlen and T. Yang in which we establish new cases of the conjecture. We also discuss generalizations to orthogonal groups of signature $(n,2)$ and possible applications.