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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.19623 |
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| _version_ | 1866929538251882496 |
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| author | Franklin, Jesse |
| author_facet | Franklin, Jesse |
| contents | We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $Γ\leq \GL_2(\bbF_q[T]).$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve for $Γ_2$ and the algebra of Drinfeld modular forms for $Γ_2,$ where $Γ_2$ is the subgroup of square-determinant matrices in $Γ.$ This allows one to compute the latter ring by geometric invariants using the techniques of Voight, Zureick-Brown and O'Dorney. We also show how to decompose the algebra of modular forms for $Γ_2$ into a direct sum of two algebras of modular forms for $Γ$ and generalize this result to a larger class of congruence subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19623 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Geometry of Drinfeld Modular Forms Franklin, Jesse Number Theory Algebraic Geometry We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $Γ\leq \GL_2(\bbF_q[T]).$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve for $Γ_2$ and the algebra of Drinfeld modular forms for $Γ_2,$ where $Γ_2$ is the subgroup of square-determinant matrices in $Γ.$ This allows one to compute the latter ring by geometric invariants using the techniques of Voight, Zureick-Brown and O'Dorney. We also show how to decompose the algebra of modular forms for $Γ_2$ into a direct sum of two algebras of modular forms for $Γ$ and generalize this result to a larger class of congruence subgroups. |
| title | The Geometry of Drinfeld Modular Forms |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2310.19623 |