Traces of Green's functions on ℍq over point-hypersurface pairs
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 ℍ3 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 Gs(P,Q) on hyperbolic q-space ℍq for q≥ 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 ℍq 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.
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BibTeX
@misc{alfes2026traces,
title = {{Traces of Green's functions on $\mathbbH^q$ over point-hypersurface pairs}},
author = {Claudia Alfes and Jens Funke and Sebastián Herrero and Özlem Imamoglu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32825},
eprint = {2610.32825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32825},
}
RIS
TY - UNPB
TI - Traces of Green's functions on ℍ^q over point-hypersurface pairs
AU - Claudia Alfes
AU - Jens Funke
AU - Sebastián Herrero
AU - Özlem Imamoglu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32825
UR - https://arxiv.org/abs/2610.32825
AB - 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 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 ℍ^q for q≥ 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 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.
ER -