Modularity and Heights of CM cycles on Kuga-Sato varieties

We study CM cycles on Kuga-Sato varieties over X(N) via theta lifting and relative trace formula. Our first result is the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple whose irreducible components are associated to higher weight holomorphic cuspidal automorphic representations of GL2(Q). This is proved via theta lifting. Our second result is a higher weight analog of the general Gross-Zagier formula of Yuan, S. Zhang and W. Zhang. This is proved via relative trace formula, provided the modularity of CM cycles.