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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.08422 |
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| _version_ | 1866917073502863360 |
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| author | Tafazolian, Saeed Top, Jaap |
| author_facet | Tafazolian, Saeed Top, Jaap |
| contents | We construct explicit families of hyperelliptic curves over $\QQ$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by \[ v^2 = (u+2)\,φ_d(u), \quad d = 2^e \text{ or } d=p \geq 3 \text{ prime}, \] where $φ_d(x)$ is the Chebyshev polynomial of degree $d$. We prove that the Jacobians are simple and determine the associated CM-fields explicitly. Our approach exploits the interplay between Chebyshev polynomials and Galois coverings, providing concrete examples of abelian varieties with CM and explicit criteria for their construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Families of Hyperelliptic Curves with CM Jacobians Tafazolian, Saeed Top, Jaap Algebraic Geometry We construct explicit families of hyperelliptic curves over $\QQ$ whose Jacobians admit complex multiplication (CM). Each curve in these families is defined by \[ v^2 = (u+2)\,φ_d(u), \quad d = 2^e \text{ or } d=p \geq 3 \text{ prime}, \] where $φ_d(x)$ is the Chebyshev polynomial of degree $d$. We prove that the Jacobians are simple and determine the associated CM-fields explicitly. Our approach exploits the interplay between Chebyshev polynomials and Galois coverings, providing concrete examples of abelian varieties with CM and explicit criteria for their construction. |
| title | Explicit Families of Hyperelliptic Curves with CM Jacobians |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2511.08422 |