Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new converse theorem for Borcherds products

Ingmar Metzler

TL;DR

The paper proves a new converse theorem for Borcherds products by leveraging a duality with the Kudla–Millson theta lift. It establishes injectivity of the Kudla–Millson lift in the orthogonal setting with minimal hyperbolic splitting, enabling a direct path to the converse: any meromorphic orthogonal form with special divisors arises as a Borcherds product. The approach centers on cycle-integrals and their relation to $L$-values, refining previous injectivity results and removing extra splitting hypotheses. Together, these results enhance our understanding of the Borcherds–Kudla–Millson correspondence and its arithmetic applications on orthogonal Shimura varieties.

Abstract

We establish a new converse theorem for Borcherds products. Moreover, the injectivity of the Kudla-Millson theta lift is demonstrated in the O$(n,2)$ case in greater generality than is currently available in the literature. Both results are derived under the assumption of a single hyperbolic split of the base lattice.

A new converse theorem for Borcherds products

TL;DR

The paper proves a new converse theorem for Borcherds products by leveraging a duality with the Kudla–Millson theta lift. It establishes injectivity of the Kudla–Millson lift in the orthogonal setting with minimal hyperbolic splitting, enabling a direct path to the converse: any meromorphic orthogonal form with special divisors arises as a Borcherds product. The approach centers on cycle-integrals and their relation to -values, refining previous injectivity results and removing extra splitting hypotheses. Together, these results enhance our understanding of the Borcherds–Kudla–Millson correspondence and its arithmetic applications on orthogonal Shimura varieties.

Abstract

We establish a new converse theorem for Borcherds products. Moreover, the injectivity of the Kudla-Millson theta lift is demonstrated in the O case in greater generality than is currently available in the literature. Both results are derived under the assumption of a single hyperbolic split of the base lattice.
Paper Structure (15 sections, 16 theorems, 80 equations)

This paper contains 15 sections, 16 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Vector-valued modular forms
  3. The classical setting
  4. Eisenstein series
  5. Theta series
  6. Orthogonal setting
  7. Special divisors
  8. Siegel-Weil formula
  9. Siegel--Weil formula
  10. Kudla--Millson Schwartz form
  11. A geometric integral
  12. Theta lifts
  13. The Borcherds lift
  14. The Kudla--Millson lift
  15. Results on theta lifts

Key Result

Theorem 1.1

Assume that $L$ splits a hyperbolic plane and ${m^{+}} > 3$. Then every meromorphic modular form $F$ with respect to $\Gamma(L)$ whose divisor is a linear combination of special divisors $Z(\mu,n)$ is (up to a non-zero constant factor) the Borcherds lift $\Psi(z,f)$ of a weakly holomorphic modular f

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 34 more