Green’s functions are crucial in arithmetic intersection theory, particularly at Archimedean places. In this talk, I will introduce a combinatorial version of Green’s functions that arises in the context of intersection pairing at some non-Archimedean places. I will first motivate the discussion with a generalization problem of the Gross-Zagier formula to CM cycles over Shimura curves, focusing on the part where Green’s functions will come into play. We will then move on to concrete definitions in harmonic analysis on local systems over graphs, as well as an explicit construction of Green's function in this context.
Green’s functions in the arithmetic intersection of cycles over Shimura curves
Boya Wen, University of Wisconsin-Madison
2023 AWM Research Symposium
Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]