Some explicit arithmetic on curves of genus three and their applications
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913451021959168 |
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| author | Moriya, Tomoki Kudo, Momonari |
| author_facet | Moriya, Tomoki Kudo, Momonari |
| contents | A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny.
In this paper, we develop some explicit arithmetic on curves of genus $3$, including algorithms to compute the codomain of a decomposed Richelot isogeny. As solutions to compute the domain of a decomposed Richelot isogeny, explicit formulae of defining equations for Howe curves of genus $3$ are also given. Using the formulae, we shall construct an algorithm with complexity $\tilde{O}(p^3)$ (resp. $\tilde{O}(p^4)$) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02926 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some explicit arithmetic on curves of genus three and their applications Moriya, Tomoki Kudo, Momonari Algebraic Geometry Symbolic Computation Number Theory A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny. In this paper, we develop some explicit arithmetic on curves of genus $3$, including algorithms to compute the codomain of a decomposed Richelot isogeny. As solutions to compute the domain of a decomposed Richelot isogeny, explicit formulae of defining equations for Howe curves of genus $3$ are also given. Using the formulae, we shall construct an algorithm with complexity $\tilde{O}(p^3)$ (resp. $\tilde{O}(p^4)$) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus $3$. |
| title | Some explicit arithmetic on curves of genus three and their applications |
| topic | Algebraic Geometry Symbolic Computation Number Theory |
| url | https://arxiv.org/abs/2209.02926 |