On direct summands of products of Jacobians over arbitrary fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912497198432256 |
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| author | Scavia, Federico Suzuki, Fumiaki |
| author_facet | Scavia, Federico Suzuki, Fumiaki |
| contents | We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both algebraic over $k$. This extends results of Beckmann--de Gaay Fortman and Voisin over the complex numbers to arbitrary fields, and refines an obstruction to the direct summand property over $\mathbb{Q}$ due to Petrov--Skorobogatov. We also give applications to the integral Tate conjecture for divisors and for $1$-cycles on abelian varieties over finitely generated fields; our results also address a $p$-adic version of the integral Tate conjecture over finite fields of characteristic $p$, for the first time beyond the case of divisors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On direct summands of products of Jacobians over arbitrary fields Scavia, Federico Suzuki, Fumiaki Algebraic Geometry 14C25 (Primary) 14K15, 14H40, 14G15 (Secondary) We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both algebraic over $k$. This extends results of Beckmann--de Gaay Fortman and Voisin over the complex numbers to arbitrary fields, and refines an obstruction to the direct summand property over $\mathbb{Q}$ due to Petrov--Skorobogatov. We also give applications to the integral Tate conjecture for divisors and for $1$-cycles on abelian varieties over finitely generated fields; our results also address a $p$-adic version of the integral Tate conjecture over finite fields of characteristic $p$, for the first time beyond the case of divisors. |
| title | On direct summands of products of Jacobians over arbitrary fields |
| topic | Algebraic Geometry 14C25 (Primary) 14K15, 14H40, 14G15 (Secondary) |
| url | https://arxiv.org/abs/2507.09821 |