In 1986, Gross and Zagier formulated an algebraicity conjecture concerning special values of higher Green's functions on modular curves, drawing a deep analogy with the behavior of the modular j-invariant in the theory of complex multiplication. An averaged version of this conjecture was later proved by Gross, Kohnen, and Zagier, and the full conjecture has recently been resolved by Bruinier, Li, and Yang.
In this talk, I will present new results on an analogous problem in a three-dimensional setting: the special values of Green’s functions on hyperbolic 3-space. We show that certain averages of these values can be expressed in terms of logarithms of primes and logarithms of units in real quadratic fields, and discuss twisted averages that yield algebraic numbers. This is joint work with Özlem Imamoglu, Sebastian Herrero, and Markus Schwagenscheidt.