An effective open image theorem for products of principally polarized abelian varieties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913594783825920 |
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| author | Mayle, Jacob Wang, Tian |
| author_facet | Mayle, Jacob Wang, Tian |
| contents | Let $A = \prod_{1\leq i\leq n} A_i$ be the product of principally polarized abelian varieties $A_1, \ldots, A_n$ of dimensions $g_1, \ldots, g_n$, respectively, each defined over a number field $K$, and pairwise nonisogenous over $\overline{K}$. We make effective an open image theorem for $A$ due to Hindry and Ratazzi. More specifically, we give an explicit bound of the constant $c(A)$ under GRH, in terms of standard invariants of $K$ and each $A_i$, where $c(A)$ is defined to be the smallest positive integer such that for any prime $\ell>c(A)$, the image of the $\ell$-adic Galois representation of $A$ is "as large as possible" in a suitable sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11472 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An effective open image theorem for products of principally polarized abelian varieties Mayle, Jacob Wang, Tian Number Theory Primary 11F80, Secondary 11G05, 11G10 Let $A = \prod_{1\leq i\leq n} A_i$ be the product of principally polarized abelian varieties $A_1, \ldots, A_n$ of dimensions $g_1, \ldots, g_n$, respectively, each defined over a number field $K$, and pairwise nonisogenous over $\overline{K}$. We make effective an open image theorem for $A$ due to Hindry and Ratazzi. More specifically, we give an explicit bound of the constant $c(A)$ under GRH, in terms of standard invariants of $K$ and each $A_i$, where $c(A)$ is defined to be the smallest positive integer such that for any prime $\ell>c(A)$, the image of the $\ell$-adic Galois representation of $A$ is "as large as possible" in a suitable sense. |
| title | An effective open image theorem for products of principally polarized abelian varieties |
| topic | Number Theory Primary 11F80, Secondary 11G05, 11G10 |
| url | https://arxiv.org/abs/2212.11472 |