Modularity and Heights of CM cycles on Kuga-Sato varieties

2021/05/26 by Qiu, Congling
#11F12 #11F27 #11F72 #11G40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 11G18

paper · doi:10.48550/arxiv.2105.12561

Abstract

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.

Related

• Heights of CM cycles and derivatives of L-series
• CM cycles on Kuga-Sato varieties over Shimura curves and Selmer groups
• Modular Curves, Modular Surfaces, and Modular Fourfolds
• Arithmetic Intersection on a Hilbert Modular Surface and the Faltings Height
• Derived special cycles on Shimura varieties