Degeneracy and Sato-Tate groups of $y^2=x^{p^2}-1$
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arXiv
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| Format: | Preprint |
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2025
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| author | Chen, Justin Goodson, Heidi Hoque, Rezwan Malikah, Sabeeha |
| author_facet | Chen, Justin Goodson, Heidi Hoque, Rezwan Malikah, Sabeeha |
| contents | We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves $C_{p^2}: y^2=x^{p^2}-1$, where $p$ is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_03299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degeneracy and Sato-Tate groups of $y^2=x^{p^2}-1$ Chen, Justin Goodson, Heidi Hoque, Rezwan Malikah, Sabeeha Number Theory Algebraic Geometry 11G10, 14C25, 14K22 We say that an abelian variety is degenerate if its Hodge ring is not generated by divisor classes. Degeneracy leads to some interesting challenges when computing Sato-Tate groups, and there are currently few examples and techniques presented in the literature. In this paper we focus on the Jacobians of the family of curves $C_{p^2}: y^2=x^{p^2}-1$, where $p$ is an odd prime. Using a construction developed by Shioda in the 1980s, we are able to characterize so-called indecomposable Hodge classes as well as the Sato-Tate groups of these Jacobian varieties. Our work is inspired by computation, and examples and methods are described throughout the paper. |
| title | Degeneracy and Sato-Tate groups of $y^2=x^{p^2}-1$ |
| topic | Number Theory Algebraic Geometry 11G10, 14C25, 14K22 |
| url | https://arxiv.org/abs/2512.03299 |