Every finite abelian group is the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913708452610048 |
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| author | Marseglia, Stefano Springer, Caleb |
| author_facet | Marseglia, Stefano Springer, Caleb |
| contents | We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite field $\mathbb{F}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over $\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over $\mathbb{F}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_08125 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Every finite abelian group is the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$ Marseglia, Stefano Springer, Caleb Number Theory 14K15 (Primary) 14G15, 11G10 (Secondary) We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite field $\mathbb{F}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over $\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over $\mathbb{F}_2$. |
| title | Every finite abelian group is the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$ |
| topic | Number Theory 14K15 (Primary) 14G15, 11G10 (Secondary) |
| url | https://arxiv.org/abs/2105.08125 |