The Kudla-Millson lift of Siegel cusp forms
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908630239936512 |
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| author | Kiefer, Paul Zuffetti, Riccardo |
| author_facet | Kiefer, Paul Zuffetti, Riccardo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15809 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Kudla-Millson lift of Siegel cusp forms Kiefer, Paul Zuffetti, Riccardo Number Theory Algebraic Geometry 11F27, 11F37, 11F55, 14G35 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. |
| title | The Kudla-Millson lift of Siegel cusp forms |
| topic | Number Theory Algebraic Geometry 11F27, 11F37, 11F55, 14G35 |
| url | https://arxiv.org/abs/2307.15809 |