On the Rank of Jacobian Varieties of the Curves $y^s=ax^r+b$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908622456356864 |
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| author | Salami, Sajad |
| author_facet | Salami, Sajad |
| contents | We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties of general type, we prove that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Rank of Jacobian Varieties of the Curves $y^s=ax^r+b$ Salami, Sajad Number Theory 1G30, 14H40, 14G05, 11G50 We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties of general type, we prove that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus. |
| title | On the Rank of Jacobian Varieties of the Curves $y^s=ax^r+b$ |
| topic | Number Theory 1G30, 14H40, 14G05, 11G50 |
| url | https://arxiv.org/abs/2510.27109 |