Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
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| Format: | Preprint |
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2024
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| _version_ | 1866916822869082112 |
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| author | Gallese, Andrea Goodson, Heidi Lombardo, Davide |
| author_facet | Gallese, Andrea Goodson, Heidi Lombardo, Davide |
| contents | Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. We compute several interesting arithmetic invariants of $J_m$: its decomposition up to isogeny into simple abelian varieties, the minimal field $\mathbb{Q}(\operatorname{End}(J_m))$ over which its endomorphisms are defined, and its connected monodromy field $\mathbb{Q}(\varepsilon_{J_m})$. Currently, there is no general algorithm that computes the last invariant. For large enough values of $m$, the abelian varieties $J_m$ provide non-trivial examples of high-dimensional phenomena, such as degeneracy and the non-triviality of the extension $\mathbb{Q}(\varepsilon_{J_m})/\mathbb{Q}(\operatorname{End}(J_m))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians Gallese, Andrea Goodson, Heidi Lombardo, Davide Number Theory Algebraic Geometry 11F80, 11G10, 14C25 Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. We compute several interesting arithmetic invariants of $J_m$: its decomposition up to isogeny into simple abelian varieties, the minimal field $\mathbb{Q}(\operatorname{End}(J_m))$ over which its endomorphisms are defined, and its connected monodromy field $\mathbb{Q}(\varepsilon_{J_m})$. Currently, there is no general algorithm that computes the last invariant. For large enough values of $m$, the abelian varieties $J_m$ provide non-trivial examples of high-dimensional phenomena, such as degeneracy and the non-triviality of the extension $\mathbb{Q}(\varepsilon_{J_m})/\mathbb{Q}(\operatorname{End}(J_m))$. |
| title | Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians |
| topic | Number Theory Algebraic Geometry 11F80, 11G10, 14C25 |
| url | https://arxiv.org/abs/2405.20394 |