Abelian surfaces over finite fields containing no curves of genus $3$ or less
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arXiv
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| Format: | Preprint |
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2024
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| author | Berardini, Elena Maidana, Alejandro Giangreco Marseglia, Stefano |
| author_facet | Berardini, Elena Maidana, Alejandro Giangreco Marseglia, Stefano |
| contents | We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to $2$ initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus $3$ is equivalent to admitting a polarisation of degree $4$. Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus $2$ curves have a polarisation of degree $4$. Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus $\leq 2$, containing no abelian surface with a polarisation of degree $4$. Finally, we describe the absolutely irreducible genus $3$ curves lying on abelian surfaces containing no curves of genus less than or equal to $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Abelian surfaces over finite fields containing no curves of genus $3$ or less Berardini, Elena Maidana, Alejandro Giangreco Marseglia, Stefano Algebraic Geometry Number Theory Primary: 14K15, 11G20, Secondary: 14G15, 11G10 We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to $2$ initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus $3$ is equivalent to admitting a polarisation of degree $4$. Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus $2$ curves have a polarisation of degree $4$. Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus $\leq 2$, containing no abelian surface with a polarisation of degree $4$. Finally, we describe the absolutely irreducible genus $3$ curves lying on abelian surfaces containing no curves of genus less than or equal to $2$. |
| title | Abelian surfaces over finite fields containing no curves of genus $3$ or less |
| topic | Algebraic Geometry Number Theory Primary: 14K15, 11G20, Secondary: 14G15, 11G10 |
| url | https://arxiv.org/abs/2408.02493 |