Frobenius nonclassicality of generalized Fermat curves with respect to conics
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917387569201152 |
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| author | Arakelian, Nazar Rodrigues, Leandro A. M. |
| author_facet | Arakelian, Nazar Rodrigues, Leandro A. M. |
| contents | The effective application of the Stöhr-Voloch theory for the linear system of plane curves of a fixed degree to bound the number of rational points of a family of plane curves defined over $\mathbb{F}_q$ requires the characterization of the $\mathbb{F}_q$-Frobenius nonclassical curves in the family. In this paper, we provide necessary and sufficient conditions for certain generalized Fermat curves $\mathcal{F}$ defined over $\mathbb{F}_q$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the Frobenius classical cases, we obtain nice bounds for the number $N_q(\mathcal{F})$ of rational points of $\mathcal{F}$ via Stöhr-Voloch theory, whereas in the Frobenius nonclassical cases, we derive explicit formulas for $N_q(\mathcal{F})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05092 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frobenius nonclassicality of generalized Fermat curves with respect to conics Arakelian, Nazar Rodrigues, Leandro A. M. Algebraic Geometry Number Theory The effective application of the Stöhr-Voloch theory for the linear system of plane curves of a fixed degree to bound the number of rational points of a family of plane curves defined over $\mathbb{F}_q$ requires the characterization of the $\mathbb{F}_q$-Frobenius nonclassical curves in the family. In this paper, we provide necessary and sufficient conditions for certain generalized Fermat curves $\mathcal{F}$ defined over $\mathbb{F}_q$ to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the Frobenius classical cases, we obtain nice bounds for the number $N_q(\mathcal{F})$ of rational points of $\mathcal{F}$ via Stöhr-Voloch theory, whereas in the Frobenius nonclassical cases, we derive explicit formulas for $N_q(\mathcal{F})$. |
| title | Frobenius nonclassicality of generalized Fermat curves with respect to conics |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2604.05092 |