Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Kudla-Millson lift of Siegel cusp forms

Paul Kiefer, Riccardo Zuffetti

TL;DR

This work establishes the injectivity of the genus-2 Kudla–Millson lift for genus-2 Siegel cusp forms valued in the Weil representation of an even lattice $L$ under the hypothesis that $L$ splits off two hyperbolic planes and has sufficiently large rank. The authors develop vector-valued indefinite genus-2 Siegel theta functions and introduce Jacobi-type theta functions to facilitate a double unfolding: first to Jacobi-type integrals and then via Jacobi-Siegel theta theory to Fourier coefficients of the input cusp form. The main contributions include an explicit Fourier–Jacobi expansion of the lift, a second unfolding to Jacobi inner products, and an injectivity theorem for $L$ with the required hyperbolic-splitting, with concrete cohomological applications to orthogonal Shimura varieties and moduli spaces of quasi-polarized K3 surfaces. The results yield a cohomology-injection, providing lower bounds on $ ext{dim}\,H^4(X, obreak \,oldsymbol{C})$ and linking automorphic data to geometric cycles via Kudla’s generating series of special cycles. The work also enriches the theory of higher-genus theta lifts by introducing genus-2 Jacobi Siegel theta functions and detailed reduction formulas to sublattices, enabling precise inner-product computations that underpin the injectivity proof.

Abstract

We study the injectivity of the Kudla-Millson lift of genus 2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of sufficiently large rank, then the lift is injective. As an application, we deduce that the image of the lift in the degree 4 cohomology of the associated orthogonal Shimura variety has the same dimension as the lifted space of cusp forms. Our results also cover the case of moduli spaces of quasi-polarized K3 surfaces. To prove the injectivity, we introduce vector-valued indefinite Siegel theta functions of genus 2 and of Jacobi type attached to L. We describe their behavior with respect to the split of a hyperbolic plane in L. This generalizes results of Borcherds to genus higher than 1.

The Kudla-Millson lift of Siegel cusp forms

TL;DR

This work establishes the injectivity of the genus-2 Kudla–Millson lift for genus-2 Siegel cusp forms valued in the Weil representation of an even lattice under the hypothesis that splits off two hyperbolic planes and has sufficiently large rank. The authors develop vector-valued indefinite genus-2 Siegel theta functions and introduce Jacobi-type theta functions to facilitate a double unfolding: first to Jacobi-type integrals and then via Jacobi-Siegel theta theory to Fourier coefficients of the input cusp form. The main contributions include an explicit Fourier–Jacobi expansion of the lift, a second unfolding to Jacobi inner products, and an injectivity theorem for with the required hyperbolic-splitting, with concrete cohomological applications to orthogonal Shimura varieties and moduli spaces of quasi-polarized K3 surfaces. The results yield a cohomology-injection, providing lower bounds on and linking automorphic data to geometric cycles via Kudla’s generating series of special cycles. The work also enriches the theory of higher-genus theta lifts by introducing genus-2 Jacobi Siegel theta functions and detailed reduction formulas to sublattices, enabling precise inner-product computations that underpin the injectivity proof.

Abstract

We study the injectivity of the Kudla-Millson lift of genus 2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of sufficiently large rank, then the lift is injective. As an application, we deduce that the image of the lift in the degree 4 cohomology of the associated orthogonal Shimura variety has the same dimension as the lifted space of cusp forms. Our results also cover the case of moduli spaces of quasi-polarized K3 surfaces. To prove the injectivity, we introduce vector-valued indefinite Siegel theta functions of genus 2 and of Jacobi type attached to L. We describe their behavior with respect to the split of a hyperbolic plane in L. This generalizes results of Borcherds to genus higher than 1.
Paper Structure (31 sections, 36 theorems, 283 equations)

This paper contains 31 sections, 36 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. The genus 2 Kudla--Millson lift in terms of Siegel theta functions
  3. The double unfolding of the lift and its injectivity
  4. Some applications
  5. Outline of the paper
  6. Lattices and orthogonal Shimura varieties
  7. Lattices and local-global principles
  8. Orthogonal Shimura varieties
  9. Vector-valued Siegel and Jacobi modular forms
  10. Very homogeneous polynomials and differential operators
  11. The Weil representation and theta functions
  12. Siegel modular forms of genus 2
  13. Siegel theta functions of genus 2 à la Borcherds
  14. Jacobi forms and Jacobi Siegel theta functions
  15. Reduction of Siegel theta functions to sublattices
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $L$ be an even lattice of signature $(b, 2)$ that splits off two orthogonal hyperbolic planes, and let $L^+$ be the orthogonal complement of such hyperbolic planes. If $L_p^+\coloneqq L^+\otimes\mathbb{Z}_p$ splits off two hyperbolic planes over $\mathbb{Z}_p$ for every prime $p$, then the Kudla

Theorems & Definitions (93)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 83 more