On $\ell$-torsion in degree $\ell$ superelliptic Jacobians over $\mathbf{F}_q$
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| Format: | Preprint |
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2024
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| _version_ | 1866913629210673152 |
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| author | Li, Wanlin Love, Jonathan Stubley, Eric |
| author_facet | Li, Wanlin Love, Jonathan Stubley, Eric |
| contents | We study the $\ell$-torsion subgroup in Jacobians of curves of the form $y^{\ell} = f(x)$ for irreducible $f(x)$ over a finite field $\mathbf{F}_{q}$ of characteristic $p \neq \ell$. This is a function field analogue of the study of $\ell$-torsion subgroups of ideal class groups of number fields $\mathbf{Q}(\sqrt[\ell]{N})$. We establish an upper bound, lower bound, and parity constraint on the rank of the $\ell$-torsion which depend only on the parameters $\ell$, $q$, and $\text{deg}\, f$. Using tools from class field theory, we show that additional criteria depending on congruence conditions involving the polynomial $f(x)$ can be used to refine the upper and lower bounds. For certain values of the parameters $\ell,q,\text{deg}\, f$, we determine the $\ell$-torsion of the Jacobian for all curves with the given parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_20719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $\ell$-torsion in degree $\ell$ superelliptic Jacobians over $\mathbf{F}_q$ Li, Wanlin Love, Jonathan Stubley, Eric Number Theory 11R29, 11R58 (Primary), 11R34, 11R37, 11G20, 11G45 (Secondary) We study the $\ell$-torsion subgroup in Jacobians of curves of the form $y^{\ell} = f(x)$ for irreducible $f(x)$ over a finite field $\mathbf{F}_{q}$ of characteristic $p \neq \ell$. This is a function field analogue of the study of $\ell$-torsion subgroups of ideal class groups of number fields $\mathbf{Q}(\sqrt[\ell]{N})$. We establish an upper bound, lower bound, and parity constraint on the rank of the $\ell$-torsion which depend only on the parameters $\ell$, $q$, and $\text{deg}\, f$. Using tools from class field theory, we show that additional criteria depending on congruence conditions involving the polynomial $f(x)$ can be used to refine the upper and lower bounds. For certain values of the parameters $\ell,q,\text{deg}\, f$, we determine the $\ell$-torsion of the Jacobian for all curves with the given parameters. |
| title | On $\ell$-torsion in degree $\ell$ superelliptic Jacobians over $\mathbf{F}_q$ |
| topic | Number Theory 11R29, 11R58 (Primary), 11R34, 11R37, 11G20, 11G45 (Secondary) |
| url | https://arxiv.org/abs/2412.20719 |