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arXiv · 2307.15809

The Kudla-Millson lift of Siegel cusp forms

Paul Kiefer·Riccardo Zuffetti
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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.

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Paul Kiefer, Riccardo Zuffetti. 2025-11-05. The Kudla-Millson lift of Siegel cusp forms. https://arxiv.org/abs/2307.15809

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