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Deformations of Theta Integrals and A Conjecture of Gross-Zagier

Jan H. Bruinier, Yingkun Li, Tonghai Yang

TL;DR

This work completes the Gross–Zagier conjecture by constructing an analogue of incoherent Eisenstein series over a real quadratic field via the Doi–Naganuma lift of a deformed theta integral. Leveraging Kudla's program, exceptional isogenies, and a detailed global–local matching framework, it expresses the archimedean contributions to CM values as logarithms of algebraic numbers with explicit factorization. The paper introduces the deformed theta integral tilde{𝒪} and the associated lift tilde{𝓘}, derives their Fourier expansions and rationality properties, and proves exact factorization formulas for principal higher Green functions at CM points for arbitrary level K. These results establish algebraicity and precise ideal factorization in biquadratic CM settings, linking theta lifts, Eisenstein series, and arithmetic intersection theory, and suggesting extensions to broader CM algebras and non-degeneracy phenomena.

Abstract

In this paper, we complete the proof of the conjecture of Gross and Zagier concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic space.

Deformations of Theta Integrals and A Conjecture of Gross-Zagier

TL;DR

This work completes the Gross–Zagier conjecture by constructing an analogue of incoherent Eisenstein series over a real quadratic field via the Doi–Naganuma lift of a deformed theta integral. Leveraging Kudla's program, exceptional isogenies, and a detailed global–local matching framework, it expresses the archimedean contributions to CM values as logarithms of algebraic numbers with explicit factorization. The paper introduces the deformed theta integral tilde{𝒪} and the associated lift tilde{𝓘}, derives their Fourier expansions and rationality properties, and proves exact factorization formulas for principal higher Green functions at CM points for arbitrary level K. These results establish algebraicity and precise ideal factorization in biquadratic CM settings, linking theta lifts, Eisenstein series, and arithmetic intersection theory, and suggesting extensions to broader CM algebras and non-degeneracy phenomena.

Abstract

In this paper, we complete the proof of the conjecture of Gross and Zagier concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic space.
Paper Structure (26 sections, 31 theorems, 353 equations)

This paper contains 26 sections, 31 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. A Problem posed by Gross and Zagier
  3. Comparison to Previous Works
  4. Proof Strategy
  5. Outlook and Organization
  6. Preliminaries
  7. Differential Operators
  8. Quadratic Space associated to Real Quadratic Field
  9. The Weil Representation and Theta Functions
  10. CM Points and Higher Green Functions
  11. Product of Modular Curves as a Shimura Variety
  12. Eisenstein Series
  13. Hecke's Cusp Form
  14. The Deformed Theta Integral
  15. Doi-Naganuma Lift of Hecke's Cusp Form
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $\Phi_{f}^r$ be a rational, principal higher Green function on $X_K$. Suppose that $E/\mathbb{Q}$ is a biquadratic CM number field with the real quadratic field $F = \mathbb{Q}(\sqrt{D})$, and $Z(W) \cap Z_f = \emptyset$. Then there exists a positive integer $\kappa$ and $a_1, a_2 \in F^\times$ For any prime $\mathfrak{p}$ of $F$, the value $\kappa^{-1} \operatorname{ord}_\mathfrak{p}(a_j)$ i

Theorems & Definitions (73)

  • Conjecture 1.1
  • Theorem 1.2: Algebraicity and Factorization
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 63 more