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Modularity and Heights of CM cycles on Kuga-Sato varieties

Congling Qiu

TL;DR

The paper advances the Gross--Zagier program to higher weight by establishing a general Gross--Zagier-type formula for CM cycles on Kuga--Sato varieties over modular curves of arbitrary level, linking height pairings to derivatives of base-change L-functions L'(1/2, πK ⊗ Ω). Central to the approach are two modularity results for CM cycles: (i) a semisimplicity statement for the Hecke action on CM cycles whose irreducible components correspond to weight 2k holomorphic cuspidal automorphic representations, and (ii) a stronger modularity result for the action of the full Hecke algebra via arithmetic relative trace formulas and arithmetic theta lifting. The main identity is proven by an arithmetic relative trace formula comparison, pairing height derivatives with automorphic distributions arising from a mixed Siegel–Weil framework, with local-global matching across split/non-split primes and the infinite place. These results provide evidence for Beilinson–Bloch-type conjectures in higher dimensions and extend the weight-2 Gross--Zagier paradigm to higher weight via CM cycles on Kuga--Sato varieties. The work thus synthesizes CM cycle modularity, height machinery, and relative trace formulas to produce a comprehensive higher-weight Gross--Zagier theory with explicit local–global data and consequences for special values and derivatives of automorphic L-functions.

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.

Modularity and Heights of CM cycles on Kuga-Sato varieties

TL;DR

The paper advances the Gross--Zagier program to higher weight by establishing a general Gross--Zagier-type formula for CM cycles on Kuga--Sato varieties over modular curves of arbitrary level, linking height pairings to derivatives of base-change L-functions L'(1/2, πK ⊗ Ω). Central to the approach are two modularity results for CM cycles: (i) a semisimplicity statement for the Hecke action on CM cycles whose irreducible components correspond to weight 2k holomorphic cuspidal automorphic representations, and (ii) a stronger modularity result for the action of the full Hecke algebra via arithmetic relative trace formulas and arithmetic theta lifting. The main identity is proven by an arithmetic relative trace formula comparison, pairing height derivatives with automorphic distributions arising from a mixed Siegel–Weil framework, with local-global matching across split/non-split primes and the infinite place. These results provide evidence for Beilinson–Bloch-type conjectures in higher dimensions and extend the weight-2 Gross--Zagier paradigm to higher weight via CM cycles on Kuga--Sato varieties. The work thus synthesizes CM cycle modularity, height machinery, and relative trace formulas to produce a comprehensive higher-weight Gross--Zagier theory with explicit local–global data and consequences for special values and derivatives of automorphic L-functions.

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.

Paper Structure

This paper contains 94 sections, 64 theorems, 249 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Gross--Zagier formula and generalizations
  3. Results
  4. Legacy from previous works
  5. Relative trace formulas
  6. Generalizations
  7. Structure of the paper
  8. Some notations and conventions
  9. CM cycles on Kuga--Sato varieties
  10. Kuga--Sato varieties and Hecke operators
  11. Kuga--Sato varieties
  12. Group and semi-group action
  13. Hecke action on cohomological Chow cycles
  14. Hecke action on homological Chow cycles
  15. Relation
  16. ...and 79 more sections

Key Result

Theorem 1.2.1

Assume that $S$ has cardinality at least three, and contains the prime 2 and all finite places of ${\mathbb {Q}}$ ramified in $K$. The ${\mathcal{H}}^S$-module $\overline{CM}$ is semi-simple whose irreducible components are the ${\mathcal{H}}^S$-modules associated to weight $2k$ holomorphic cuspidal

Theorems & Definitions (126)

  • Theorem 1.2.1: Theorem \ref{['strongmodularity']}
  • Theorem 1.2.2
  • Remark 2.1.1
  • Remark 2.1.2
  • Remark 2.2.1
  • Remark 2.2.2
  • Lemma 2.2.3
  • proof
  • Conjecture 2.3.1
  • Remark 2.3.2
  • ...and 116 more