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arXiv · 2407.01205

The basis problem for modular forms for the Weil representation

Manuel K. -H. Müller
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Abstract

The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result on the basis problem for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups.

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Manuel K. -H. Müller. 2024-07-01. The basis problem for modular forms for the Weil representation. https://arxiv.org/abs/2407.01205

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