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Arithmetic Siegel-Weil formula on $\mathcal{X}_0(N)$: singular terms

Baiqing Zhu

TL;DR

The paper establishes an arithmetic Siegel-Weil formula on the modular curve $\mathcal{X}_0(N)$, relating the generating series of codimension-2 arithmetic cycles to derivatives of a genus-2 incoherent Eisenstein series. It develops a term-by-term analytic-geometric dictionary: singular Fourier coefficients of genus-2 Eisenstein series are expressed via genus-1 data through local-density difference formulas, while the geometric side computes heights and intersection numbers for the modified Hodge line bundle and cuspidal components. The work leverages explicit reduction of cusps, an explicit section $\Delta_N$ of the Hodge bundle, and the extended arithmetic Chow framework to prove that the arithmetic degrees of $\widehat{\mathcal{Z}}(T,\mathsf{y})$ match $\frac{\psi(N)}{24}$ times $\partial\textup{Eis}_T(\mathsf{z},\Delta(N)^2)$ for all $T$, including singular ranks. This extends prior square-free cases and connects local density theory, Green functions, and vertical components into a cohesive arithmetic Siegel-Weil correspondence on $\mathcal{X}_0(N)$ with explicit divisor and height formulas."

Abstract

For arbitrary level $N$, we relate the generating series of codimension 2 special cycles on $\mathcal{X}_{0}(N)$ to the derivatives of a genus 2 Eisenstein series, especially the singular terms of both sides. On the analytic side, we use difference formulas of local densities to relate the singular Fourier coefficients of the genus 2 Eisenstein series to the nonsingular Fourier coefficients of a genus 1 Eisenstein series. On the geometric side, we study the reduction of cusps to compute the divisor class of the Hodge bundle and the heights of special divisors. When $N$ is square-free, this gives a different proof of the main results in the works of Du, Yang and Sankaran, Shi, and Yang.

Arithmetic Siegel-Weil formula on $\mathcal{X}_0(N)$: singular terms

TL;DR

The paper establishes an arithmetic Siegel-Weil formula on the modular curve , relating the generating series of codimension-2 arithmetic cycles to derivatives of a genus-2 incoherent Eisenstein series. It develops a term-by-term analytic-geometric dictionary: singular Fourier coefficients of genus-2 Eisenstein series are expressed via genus-1 data through local-density difference formulas, while the geometric side computes heights and intersection numbers for the modified Hodge line bundle and cuspidal components. The work leverages explicit reduction of cusps, an explicit section of the Hodge bundle, and the extended arithmetic Chow framework to prove that the arithmetic degrees of match times for all , including singular ranks. This extends prior square-free cases and connects local density theory, Green functions, and vertical components into a cohesive arithmetic Siegel-Weil correspondence on with explicit divisor and height formulas."

Abstract

For arbitrary level , we relate the generating series of codimension 2 special cycles on to the derivatives of a genus 2 Eisenstein series, especially the singular terms of both sides. On the analytic side, we use difference formulas of local densities to relate the singular Fourier coefficients of the genus 2 Eisenstein series to the nonsingular Fourier coefficients of a genus 1 Eisenstein series. On the geometric side, we study the reduction of cusps to compute the divisor class of the Hodge bundle and the heights of special divisors. When is square-free, this gives a different proof of the main results in the works of Du, Yang and Sankaran, Shi, and Yang.
Paper Structure (49 sections, 48 theorems, 249 equations)

This paper contains 49 sections, 48 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of the main results
  4. Special cycles of codimension 1
  5. Special cycles of codimension 2
  6. Strategy of the proof of main results
  7. The singular coefficients of Eisenstein series
  8. Vertical components of the Hodge line bundle
  9. Acknowledgement
  10. The Analytic Side
  11. Local densities and difference formulas
  12. Notations on quadratic lattices
  13. Local densities of quadratic lattices
  14. Eisenstein series and Whittaker function
  15. Incoherent Eisenstein series
  16. ...and 34 more sections

Key Result

Theorem 1.2.1

Let $N$ be a positive integer. The generating series $\widehat{\phi}_1$ is a nonholomorphic Siegel modular form of genus 1 and weight $\frac{3}{2}$ with values in $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$.

Theorems & Definitions (115)

  • Theorem 1.2.1: Theorem \ref{['global-modularity-2']}
  • Remark 1.2.2
  • Theorem 1.2.3: Theorem \ref{['mainglobal']}
  • Remark 1.2.4
  • Proposition 1.3.1: Proposition \ref{['2to1mei']}
  • Remark 1.3.2
  • Example 2.1.1
  • Definition 2.2.1
  • Definition 2.2.2
  • Example 2.2.3
  • ...and 105 more