The moduli space of representations of the modular group into $G_2$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908345406849024 |
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| author | Babei, Angelica Fiori, Andrew Franc, Cameron |
| author_facet | Babei, Angelica Fiori, Andrew Franc, Cameron |
| contents | In this paper we construct a large four-dimensional family of representations of the modular group into $G_2$. Precisely, this family is an etale cover of degree $96$ of an open subset of the moduli space of such representations. This moduli space has two main components, of dimensions one and four. The one-dimensional component consists of well-studied rigid representations, in the sense of Katz. We focus on the four-dimensional component which consists of representations that are not rigid. We also provide algebraic conditions to ensure that the specializations surject onto $G_2(\mathbf{F}_p)$ for primes $p\geq 5$. These representations give new examples of $ϕ$-congruence subgroups of the modular group as introduced in previous work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The moduli space of representations of the modular group into $G_2$ Babei, Angelica Fiori, Andrew Franc, Cameron Number Theory Algebraic Geometry 11F06, 14D20 In this paper we construct a large four-dimensional family of representations of the modular group into $G_2$. Precisely, this family is an etale cover of degree $96$ of an open subset of the moduli space of such representations. This moduli space has two main components, of dimensions one and four. The one-dimensional component consists of well-studied rigid representations, in the sense of Katz. We focus on the four-dimensional component which consists of representations that are not rigid. We also provide algebraic conditions to ensure that the specializations surject onto $G_2(\mathbf{F}_p)$ for primes $p\geq 5$. These representations give new examples of $ϕ$-congruence subgroups of the modular group as introduced in previous work. |
| title | The moduli space of representations of the modular group into $G_2$ |
| topic | Number Theory Algebraic Geometry 11F06, 14D20 |
| url | https://arxiv.org/abs/2501.17276 |