Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911718796427264 |
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| author | Gazaki, Evangelia Love, Jonathan R. |
| author_facet | Gazaki, Evangelia Love, Jonathan R. |
| contents | Let $A$ be an abelian surface over an algebraically closed field $\overline{k}$ with an embedding $\overline{k}\hookrightarrow\mathbb{C}$. When $A$ is isogenous to a product of elliptic curves, we describe a large collection of pairwise non-isomorphic hyperelliptic curves mapping birationally into $A$. For infinitely many integers $g\geq 2$, this collection has infinitely many curves of genus $g$, and no two curves in the collection have the same image under any isogeny from $A$. Using these hyperelliptic curves, we find many rational equivalences in the Chow group of zero-cycles $\text{CH}_0(A)$. We use these results to give some progress towards Beilinson's conjecture for zero-cycles, which predicts that for a smooth projective variety $X$ over $\overline{\mathbb{Q}}$ the kernel of the Albanese map of $X$ is zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06361 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles Gazaki, Evangelia Love, Jonathan R. Algebraic Geometry Let $A$ be an abelian surface over an algebraically closed field $\overline{k}$ with an embedding $\overline{k}\hookrightarrow\mathbb{C}$. When $A$ is isogenous to a product of elliptic curves, we describe a large collection of pairwise non-isomorphic hyperelliptic curves mapping birationally into $A$. For infinitely many integers $g\geq 2$, this collection has infinitely many curves of genus $g$, and no two curves in the collection have the same image under any isogeny from $A$. Using these hyperelliptic curves, we find many rational equivalences in the Chow group of zero-cycles $\text{CH}_0(A)$. We use these results to give some progress towards Beilinson's conjecture for zero-cycles, which predicts that for a smooth projective variety $X$ over $\overline{\mathbb{Q}}$ the kernel of the Albanese map of $X$ is zero. |
| title | Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2309.06361 |