Modularity and Heights of CM cycles on Kuga-Sato varieties

TL;DR

This paper establishes a higher weight Gross--Zagier formula for CM cycles on Kuga--Sato varieties, linking modularity, automorphic representations, and arithmetic trace formulas to advance understanding of special cycles.

Contribution

It proves a higher weight Gross--Zagier formula and demonstrates the modularity of CM cycles, providing evidence for Beilinson--Bloch conjectures using arithmetic trace formulas and theta lifting.

Findings

Higher weight Gross--Zagier formula proved

Modularity of CM cycles established

Evidence for Beilinson--Bloch conjectures provided

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.