Modularity and Heights of CM cycles on Kuga-Sato varieties

We prove a higher weight general Gross--Zagier formula for CM cycles on Kuga--Sato varieties over modular curves of arbitrary levels. To formulate and prove this result, we prove several results on the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple modules whose irreducible components are associated to higher weight holomorphic cuspidal automorphic representations. These two types of results provide evidence toward two conjectures of Beilinson--Bloch. The higher weight general Gross--Zagier formula is proved using arithmetic relative trace formulas. The proof of the modularity of CM cycles is inspired by arithmetic theta lifting.