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On the central derivatives of l-functions and modularity of heenger cycles

Tuoping Du, Zhifeng Peng

Abstract

This paper establishes an arithmetic intersection formula for central L-derivatives in higher weights.We prove that for a general cusp form (extending the previous result for newforms), the derivative is represented by the global height pairing between higher Heegner cycles. This result provides a framework for the Gross-Zagier-Zhang formula and its generalizations.Furthermore, we investigate the modularity of the generating series of Heegner cycles,proving a weak version of the conjecture and reducing the full modularity to a vanishing conjecture,for which we provide supporting evidence.

On the central derivatives of l-functions and modularity of heenger cycles

Abstract

This paper establishes an arithmetic intersection formula for central L-derivatives in higher weights.We prove that for a general cusp form (extending the previous result for newforms), the derivative is represented by the global height pairing between higher Heegner cycles. This result provides a framework for the Gross-Zagier-Zhang formula and its generalizations.Furthermore, we investigate the modularity of the generating series of Heegner cycles,proving a weak version of the conjecture and reducing the full modularity to a vanishing conjecture,for which we provide supporting evidence.
Paper Structure (23 sections, 24 theorems, 139 equations)

This paper contains 23 sections, 24 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Plan of the Proof
  3. Preliminaries
  4. Modular Forms
  5. Eisenstein Series
  6. Heegner Divisors
  7. Regularized theta lift
  8. Theta Functions
  9. Regularized Theta Lift
  10. Non-holomorphic Hejhal--Poincaré Series
  11. Heegner cycles and height pairing
  12. Heegner Cycles
  13. Moduli Stacks
  14. Height Pairing
  15. Local intersections of Heegner cycles
  16. ...and 8 more sections

Key Result

Theorem 3

If the Chow group $\operatorname{CH}^{{\kappa}}(\mathcal{Y})$ is replaced by $\mathrm{Heeg_{{\kappa}}^{alg}(\operatorname{Cl} (\mathrm{X}))}$, then Conjecture confirst holds.

Theorems & Definitions (46)

  • Conjecture 1: Modularity
  • Remark 2
  • Theorem 3
  • Conjecture 4
  • Proposition 5
  • Theorem 6
  • Remark 7
  • Corollary 8
  • Proposition 9
  • Theorem 10: Gross-Zagier-Zhang formula in Zhang
  • ...and 36 more