Title:
CM values of higher automorphic Green functions

Abstract:
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.

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