Theta characteristics and noncongruence modular forms
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911973297356800 |
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| author | Oh, Gyujin |
| author_facet | Oh, Gyujin |
| contents | The Hodge bundle $ω$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta characteristic $ν$ (or any odd power of it) different from $ω$ is a noncongruence modular form. On the other hand, we show how $ν\neω$ gives rise to a ``twisted'' analogue of the diagonal period map to a Siegel threefold, whose difference attributes to the stackiness of the moduli of abelian surfaces $\mathcal{A}_{2}$. Some questions on the Brill--Noether theory of the modular curves are answered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Theta characteristics and noncongruence modular forms Oh, Gyujin Number Theory Algebraic Geometry The Hodge bundle $ω$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta characteristic $ν$ (or any odd power of it) different from $ω$ is a noncongruence modular form. On the other hand, we show how $ν\neω$ gives rise to a ``twisted'' analogue of the diagonal period map to a Siegel threefold, whose difference attributes to the stackiness of the moduli of abelian surfaces $\mathcal{A}_{2}$. Some questions on the Brill--Noether theory of the modular curves are answered. |
| title | Theta characteristics and noncongruence modular forms |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2407.18429 |