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A truncated Siegel-Weil formula and Borcherds forms

Armando Gutierrez Terradillos

TL;DR

This work analyzes the integral of the logarithm of Borcherds forms over the truncated modular curve $X^{mod,\hat{T}}$ by applying the regularized Siegel-Weil formula of Gan–Qiu–Takeda in the modular-curve setting. The authors factor the Siegel theta function into convergent and divergent parts using the mixed Weil model, connect the convergent piece to Eisenstein-series data, and handle divergent contributions via a truncated Rankin–Selberg framework, yielding an explicit asymptotic expansion. The main result gives a precise formula for the truncated integral in terms of Fourier coefficients $c_{\mu_j}(n)$ and the limiting values $\kappa_{\mu_j}(m)$ of Eisenstein-series coefficients, including a possible $\log(\hat{T})$ divergence when $c_{\mu_0}(0)\neq 0$ and a convergent special case when $c_{\mu_0}(0)=0$. Overall, the paper extends Kudla’s integral framework to a limiting, truncated context on the modular curve, clarifying the interaction between Borcherds forms, regularized theta lifts, and Eisenstein-series data with concrete geometric implications for arithmetic degrees and zeta/L-value connections.

Abstract

In this paper we use the regularized Siegel-Weil formula of Gan-Qiu-Takeda to obtain an expression of the integral of the theta function over the truncated modular curve. We apply this result to express the integral over the truncated modular curve of the logarithm of the Borcherds form and we describe explicitly its asymptotic behaviour, and in particular the convergent and divergent contributions. The result provides a complement to the work of Kudla on integrals of Borcherds forms in a limiting case which falls out the range of applications.

A truncated Siegel-Weil formula and Borcherds forms

TL;DR

This work analyzes the integral of the logarithm of Borcherds forms over the truncated modular curve by applying the regularized Siegel-Weil formula of Gan–Qiu–Takeda in the modular-curve setting. The authors factor the Siegel theta function into convergent and divergent parts using the mixed Weil model, connect the convergent piece to Eisenstein-series data, and handle divergent contributions via a truncated Rankin–Selberg framework, yielding an explicit asymptotic expansion. The main result gives a precise formula for the truncated integral in terms of Fourier coefficients and the limiting values of Eisenstein-series coefficients, including a possible divergence when and a convergent special case when . Overall, the paper extends Kudla’s integral framework to a limiting, truncated context on the modular curve, clarifying the interaction between Borcherds forms, regularized theta lifts, and Eisenstein-series data with concrete geometric implications for arithmetic degrees and zeta/L-value connections.

Abstract

In this paper we use the regularized Siegel-Weil formula of Gan-Qiu-Takeda to obtain an expression of the integral of the theta function over the truncated modular curve. We apply this result to express the integral over the truncated modular curve of the logarithm of the Borcherds form and we describe explicitly its asymptotic behaviour, and in particular the convergent and divergent contributions. The result provides a complement to the work of Kudla on integrals of Borcherds forms in a limiting case which falls out the range of applications.
Paper Structure (26 sections, 56 theorems, 296 equations)

This paper contains 26 sections, 56 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgements
  4. Borcherds forms and the Siegel-Weil formula
  5. GSpin Shimura varieties
  6. The modular curve without level
  7. Borcherds forms and the theta correspondence
  8. Borcherds forms
  9. Automorphic theta kernel and its relation with the singular theta lift
  10. Regularized Siegel-Weil formula
  11. Eisenstein series
  12. The regularized Siegel-Weil formula
  13. Relevant complex variable Eisenstein series
  14. Truncated Siegel-Weil formula
  15. Factorization of the theta function
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\varphi^{\infty}_{z_0,\mu_j}$ be the Schwartz function defined in schgauss and let $\mathrm{Conv}(g_{\tau},h,\varphi^{\infty}_{z_0,\mu_j})$ be the absolutely convergent part of $\theta(g_{\tau},h,\varphi^{\infty}_{z_0,\mu_j})$. Then, there exists a $\tilde{\varphi}_j\in\mathcal{S}(V(\mathbb{A}) where $c\in\mathbb{C}$ does not depend on $g_{\tau}$.

Theorems & Definitions (133)

  • Theorem : \ref{['swcor']}
  • Theorem : \ref{['mainresult']} & \ref{['MainCorollary']}
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 123 more