Sato-Tate Groups and Distributions of $y^\ell=x(x^\ell-1)$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908504100438016 |
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| author | Goodson, Heidi Hoque, Rezwan |
| author_facet | Goodson, Heidi Hoque, Rezwan |
| contents | Let $C_\ell/\mathbb Q$ denote the curve with affine model $y^\ell=x(x^\ell-1)$, where $\ell\geq 3$ is prime. In this paper we study the limiting distributions of the normalized $L$-polynomials of the curves by computing their Sato-Tate groups and distributions. We also provide results for the number of points on the curves over finite fields, including a formula in terms of Jacobi sums when the field $\mathbb F_q$ satisfies $q\equiv 1 \pmod{\ell^2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02522 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sato-Tate Groups and Distributions of $y^\ell=x(x^\ell-1)$ Goodson, Heidi Hoque, Rezwan Number Theory 11G10, 11G20, 14G10 Let $C_\ell/\mathbb Q$ denote the curve with affine model $y^\ell=x(x^\ell-1)$, where $\ell\geq 3$ is prime. In this paper we study the limiting distributions of the normalized $L$-polynomials of the curves by computing their Sato-Tate groups and distributions. We also provide results for the number of points on the curves over finite fields, including a formula in terms of Jacobi sums when the field $\mathbb F_q$ satisfies $q\equiv 1 \pmod{\ell^2}$. |
| title | Sato-Tate Groups and Distributions of $y^\ell=x(x^\ell-1)$ |
| topic | Number Theory 11G10, 11G20, 14G10 |
| url | https://arxiv.org/abs/2412.02522 |