Rationality and arithmetic of the moduli of abelian varieties
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913755095367680 |
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| author | Loughran, Daniel Sankaran, Gregory |
| author_facet | Loughran, Daniel Sankaran, Gregory |
| contents | We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian threefold over $\mathbb{F}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri-Lang conjecture we also show that this is not the case for abelian varieties of dimension at least seven. About moduli spaces, we show that $\mathcal{A}_g$ is unirational over $\mathbb{Q}$ for $g \leq 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01244 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rationality and arithmetic of the moduli of abelian varieties Loughran, Daniel Sankaran, Gregory Algebraic Geometry Number Theory (14K10), 14K15, 14H40, 14D23, 14M20, 14G12, 11G10 We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian threefold over $\mathbb{F}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri-Lang conjecture we also show that this is not the case for abelian varieties of dimension at least seven. About moduli spaces, we show that $\mathcal{A}_g$ is unirational over $\mathbb{Q}$ for $g \leq 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians. |
| title | Rationality and arithmetic of the moduli of abelian varieties |
| topic | Algebraic Geometry Number Theory (14K10), 14K15, 14H40, 14D23, 14M20, 14G12, 11G10 |
| url | https://arxiv.org/abs/2310.01244 |