Modularity and Heights of CM cycles on Kuga-Sato varieties

Congling Qiu

Published 2021-05-26Version 1

We prove a higher weight general Gross-Zagier formula for CM cycles on Kuga-Sato varieties over the modular curve X(N). To formulate and prove this result, we prove 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. 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.