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arXiv · 2212.11472

An effective open image theorem for products of principally polarized abelian varieties

Jacob Mayle·Tian Wang
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Abstract

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.

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Jacob Mayle, Tian Wang. 2022-12-22. An effective open image theorem for products of principally polarized abelian varieties. https://arxiv.org/abs/2212.11472

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