A truncated Siegel-Weil formula and Borcherds forms

TL;DR

This paper utilizes the regularized Siegel-Weil formula to analyze integrals of theta functions and Borcherds forms over truncated modular curves, revealing their asymptotic behavior and contributions.

Contribution

It introduces a new expression for integrals of Borcherds forms over truncated modular curves using the regularized Siegel-Weil formula, extending previous work.

Findings

Explicit asymptotic behavior of Borcherds form integrals

Identification of convergent and divergent contributions

Extension of Kudla's work to new limiting cases

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.