Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866914381482164224 |
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| author | Giangreco-Maidana, Alejandro J. |
| author_facet | Giangreco-Maidana, Alejandro J. |
| contents | An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_01104 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$ Giangreco-Maidana, Alejandro J. Number Theory Algebraic Geometry 11G10, 14G15, 14K15 An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical. |
| title | Some arithmetic properties of Weil polynomials of the form $t^{2g}+at^g+q^g$ |
| topic | Number Theory Algebraic Geometry 11G10, 14G15, 14K15 |
| url | https://arxiv.org/abs/2001.01104 |