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 and a generalization of the conjecture to orthogonal groups of

signature $(n,2)$.

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