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arXiv · 2306.17660

A converse theorem for Borcherds products and the injectivity of the Kudla-Millson theta lift

Oliver Stein
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Abstract

We prove a converse theorem for the multiplicative Borcherds lift for lattices of square-free level whose associated discriminant group is anisotropic. This can be seen as generalization of Bruinier's results in \cite{Br2}, which provides a converse theorem for lattices of prime level. The surjectivity of the Borcherds lift in our case follows from the injectivity of the Kudla-Millson theta lift. We generalize the corresponding results in \cite{BF1} to the aforementioned lattices and thereby in particular to lattices which are not unimodular and not of type $(p,2)$. Along the way, we compute the contribution of both, the non-Archimedean and Archimedean places of the $L^2$-norm of the Kudla-Millson theta lift. As an application we refine a theorem of Scheithauer on the non-existence of reflective automorphic products.

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Oliver Stein. 2023-06-30. A converse theorem for Borcherds products and the injectivity of the Kudla-Millson theta lift. https://arxiv.org/abs/2306.17660

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