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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2401.17445 |
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| _version_ | 1866918199505715200 |
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| author | Agugliaro, Thomas |
| author_facet | Agugliaro, Thomas |
| contents | For each prime number $p$ and each integer $g \geqslant 5$, we construct infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_p$ satisfying the standard conjecture of Hodge type. The main tool is a recent theorem of Ancona on certain rank $2$ motives. These varieties are constructed explicitly through Honda-Tate theory. Moreover, they have Tate classes that are not generated by divisors nor liftable to characteristic zero. Also, we prove a result towards a classification of simple abelian varieties for which the result of Ancona can be applied to. Along the way, we prove results of independent interest about Honda-Tate theory and about multiplicative relations between algebraic integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Examples for the standard conjecture of Hodge type Agugliaro, Thomas Algebraic Geometry Number Theory For each prime number $p$ and each integer $g \geqslant 5$, we construct infinitely many abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_p$ satisfying the standard conjecture of Hodge type. The main tool is a recent theorem of Ancona on certain rank $2$ motives. These varieties are constructed explicitly through Honda-Tate theory. Moreover, they have Tate classes that are not generated by divisors nor liftable to characteristic zero. Also, we prove a result towards a classification of simple abelian varieties for which the result of Ancona can be applied to. Along the way, we prove results of independent interest about Honda-Tate theory and about multiplicative relations between algebraic integers. |
| title | Examples for the standard conjecture of Hodge type |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2401.17445 |