Cup products on curves over finite fields
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913499399061504 |
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| author | Bleher, Frauke M. Chinburg, Ted |
| author_facet | Bleher, Frauke M. Chinburg, Ted |
| contents | Suppose $k$ is a finite field, that $C$ is a smooth projective geometrically irreducible curve over $k$, and that $n$ is a positive integer not divisible by the characteristic of $k$. In this paper we compute cup products of elements of the étale cohomology groups $\mathrm{H}^1(C,\mathbb{Z}/n)$ and $\mathrm{H}^1(C,μ_n)$. Over the algebraic closure $\overline{k}$ of $k$, such cup products are connected to values of the Weil pairing on the $n$-torsion of the Jacobian of $\overline{C} = \overline{k} \otimes_k C$ by using a fixed isomorphism between $\mathbb{Z}/n$ and $μ_n$ over $\overline{C}$. Over $k$, such cup products are more subtle due to the fact that they take values in the group $\mathrm{H}^2(C,μ_n)=\mathrm{Pic}(C)/n\cdot \mathrm{Pic}(C)$ rather than in the group $\mathrm{H}^2(\overline{C},μ_n) = \mathbb{Z}/n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_00329 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cup products on curves over finite fields Bleher, Frauke M. Chinburg, Ted Algebraic Geometry Primary 14F20, 11G20, Secondary 14H45 Suppose $k$ is a finite field, that $C$ is a smooth projective geometrically irreducible curve over $k$, and that $n$ is a positive integer not divisible by the characteristic of $k$. In this paper we compute cup products of elements of the étale cohomology groups $\mathrm{H}^1(C,\mathbb{Z}/n)$ and $\mathrm{H}^1(C,μ_n)$. Over the algebraic closure $\overline{k}$ of $k$, such cup products are connected to values of the Weil pairing on the $n$-torsion of the Jacobian of $\overline{C} = \overline{k} \otimes_k C$ by using a fixed isomorphism between $\mathbb{Z}/n$ and $μ_n$ over $\overline{C}$. Over $k$, such cup products are more subtle due to the fact that they take values in the group $\mathrm{H}^2(C,μ_n)=\mathrm{Pic}(C)/n\cdot \mathrm{Pic}(C)$ rather than in the group $\mathrm{H}^2(\overline{C},μ_n) = \mathbb{Z}/n$. |
| title | Cup products on curves over finite fields |
| topic | Algebraic Geometry Primary 14F20, 11G20, Secondary 14H45 |
| url | https://arxiv.org/abs/2101.00329 |