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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.23930 |
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| _version_ | 1866917383547912192 |
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| author | Chen, Haojie Hu, Chuangqiang |
| author_facet | Chen, Haojie Hu, Chuangqiang |
| contents | A longstanding and important problem in algebraic geometry is the characterization of algebraic function fields. In this paper, we focus on the characterization problem for cyclotomic function field $L(Λ_M)$, which is an important class of explicit function fields with applications in number theory and coding theory. Motivated by Arakelian and Quoos' classification of $L(Λ_M)$ with an irreducible quadratic modulus, we provide a complete characterization of the cyclotomic function field $L(Λ_M)$ with modulus $M = x^2$. More precisely, we prove that a function field $\mathcal{F}$ over $\mathbb{F}_q$ is $\mathbb{F}_q$-isomorphic to $L(Λ_{x^2})$ if and only if it satisfies the following three conditions: (i) $\mathcal{F}$ has a subgroup $G$ isomorphic to the direct product $(\mathbb{F}_q,+) \times \mathbb{F}_q^*$; (ii) its genus is $g(\mathcal{F}) = 1 + q(q-3)/2$; and (iii) the cardinality of $\mathbb{F}_q$-rational places is exactly $q+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23930 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Note on Cyclotomic Function Fields with Quadratic Modulus Chen, Haojie Hu, Chuangqiang Number Theory Algebraic Geometry A longstanding and important problem in algebraic geometry is the characterization of algebraic function fields. In this paper, we focus on the characterization problem for cyclotomic function field $L(Λ_M)$, which is an important class of explicit function fields with applications in number theory and coding theory. Motivated by Arakelian and Quoos' classification of $L(Λ_M)$ with an irreducible quadratic modulus, we provide a complete characterization of the cyclotomic function field $L(Λ_M)$ with modulus $M = x^2$. More precisely, we prove that a function field $\mathcal{F}$ over $\mathbb{F}_q$ is $\mathbb{F}_q$-isomorphic to $L(Λ_{x^2})$ if and only if it satisfies the following three conditions: (i) $\mathcal{F}$ has a subgroup $G$ isomorphic to the direct product $(\mathbb{F}_q,+) \times \mathbb{F}_q^*$; (ii) its genus is $g(\mathcal{F}) = 1 + q(q-3)/2$; and (iii) the cardinality of $\mathbb{F}_q$-rational places is exactly $q+1$. |
| title | A Note on Cyclotomic Function Fields with Quadratic Modulus |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2603.23930 |