Listing superspecial curves of genus three using Richelot isogeny graphs
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913942898475008 |
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| author | Ohashi, Ryo Onuki, Hiroshi Kudo, Momonari Yoshizumi, Ryo Nuida, Koji |
| author_facet | Ohashi, Ryo Onuki, Hiroshi Kudo, Momonari Yoshizumi, Ryo Nuida, Koji |
| contents | In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small $p$. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in $\mathbb{F}_{p^2}$. By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes $11 \leq p < 100$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_10500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Listing superspecial curves of genus three using Richelot isogeny graphs Ohashi, Ryo Onuki, Hiroshi Kudo, Momonari Yoshizumi, Ryo Nuida, Koji Algebraic Geometry 14K02, 14K25 (Primary) 14H45, 14Q05 (Secondary) In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small $p$. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in $\mathbb{F}_{p^2}$. By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes $11 \leq p < 100$. |
| title | Listing superspecial curves of genus three using Richelot isogeny graphs |
| topic | Algebraic Geometry 14K02, 14K25 (Primary) 14H45, 14Q05 (Secondary) |
| url | https://arxiv.org/abs/2401.10500 |