Harmonic Analysis on Local Systems on Graphs -- Towards a Gross-Zagier Formula on CM-cycles over Shimura Curves

Abstract

In Part I of the thesis, we study harmonic analysis on local systems over graphs, and construct Green's functions explicitly. In Part II, we apply the results in Part I towards a Gross-Zagier formula for CM cycles over Shimura curves, which connects the global height pairing of special cycles in Kuga varieties over Shimura curves with the derivatives of the $L$-functions associated to weight-$2k$ modular forms. More precisely, we present a suitable choice of integral model of the CM cycles, and compute local intersections at primes where the quaternion algebra associated with the Shimura curve ramifies.