The basis problem for modular forms for the Weil representation
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913555717029888 |
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| author | Müller, Manuel K. -H. |
| author_facet | Müller, Manuel K. -H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The basis problem for modular forms for the Weil representation Müller, Manuel K. -H. Number Theory 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. |
| title | The basis problem for modular forms for the Weil representation |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.01205 |