Traces of Green's functions on $\mathbb{H}^q$ over point-hypersurface pairs

Source abstract

The famous algebraicity conjecture of Gross and Zagier for the values at CM points of higher Green's functions was recently proven by Bruinier, Li and Yang. Motivated by this the values of the automorphic Green's function at pairs of special points in hyperbolic $3$-space $\mathbb{H}^3$ were studied by Herrero, Imamoglu, von Pippich and Schwagenscheidt, and an analogous algebraicity result for traces of Green's functions was established. In this paper, we consider higher automorphic Green's functions $G_{s}(P,Q)$ on hyperbolic $q$-space $\mathbb{H}^q$ for $q\geq 2$ when the spectral parameter $s$ is a positive even integer, and prove an explicit finite formula for their traces over geodesic hypersurfaces in one of the variables. This formula provides a geometric interpretation of the traces of Green's functions for every choice of a point in $\mathbb{H}^q$ for the second variable. As an application, we obtain rationality results when the second variable is evaluated at a special point providing further evidence that such results for traces of Green's functions do not depend on any underlying complex structure.