CM theory, maximal hyperelliptic curves, and Chebyshev polynomials
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arXiv
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| Format: | Preprint |
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2025
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| author | Tafazolia, Saeed Top, Jaap |
| author_facet | Tafazolia, Saeed Top, Jaap |
| contents | This paper studies hyperelliptic curves $\cH_d$ corresponding to $y^2=φ_d(x)$ over finite fields, with $φ_d(x)$ a Chebyshev polynomial. Starting from the case where $d=\ell$ is an odd prime number, new cases $(d,q)$ are presented where $\cH_d$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. In addition, new conditions ruling out the possibility that $\cH_d/\FF_{q^2}$ is maximal for given $(d,q)$, are presented. The arguments involve a mix of results on slopes of Frobenius, explicit descriptions of abelian subvarieties of the jacobian of $\cH_d$ with complex multiplication,
and a technique from the theory of $2$-descent on jacobians of hyperelliptic curves. In particular, the method used here to prove maximality in characteristics $p\equiv 1\bmod 4$ for $d\equiv 1\bmod 4$ a prime number, deserves attention, as it differs from earlier maximality arguments for other curves. Using the new results as well as extensive calculations with Magma, we pose some questions. A positive answer would completely classify the pairs $(q,d)$ resulting in maximality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_00273 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | CM theory, maximal hyperelliptic curves, and Chebyshev polynomials Tafazolia, Saeed Top, Jaap Number Theory Algebraic Geometry This paper studies hyperelliptic curves $\cH_d$ corresponding to $y^2=φ_d(x)$ over finite fields, with $φ_d(x)$ a Chebyshev polynomial. Starting from the case where $d=\ell$ is an odd prime number, new cases $(d,q)$ are presented where $\cH_d$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. In addition, new conditions ruling out the possibility that $\cH_d/\FF_{q^2}$ is maximal for given $(d,q)$, are presented. The arguments involve a mix of results on slopes of Frobenius, explicit descriptions of abelian subvarieties of the jacobian of $\cH_d$ with complex multiplication, and a technique from the theory of $2$-descent on jacobians of hyperelliptic curves. In particular, the method used here to prove maximality in characteristics $p\equiv 1\bmod 4$ for $d\equiv 1\bmod 4$ a prime number, deserves attention, as it differs from earlier maximality arguments for other curves. Using the new results as well as extensive calculations with Magma, we pose some questions. A positive answer would completely classify the pairs $(q,d)$ resulting in maximality. |
| title | CM theory, maximal hyperelliptic curves, and Chebyshev polynomials |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2509.00273 |