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

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

22 Dec 2022arXiv:2212.11472links table onlyarchive 2025-07-28

Jacob Mayle, Tian Wang

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Let A = ∏_(1≤i≤n) Aᵢ be the product of principally polarized abelian varieties A₁, …, Aₙ of dimensions g₁, …, gₙ, respectively, each defined over a number field K, and pairwise nonisogenous over 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 Aᵢ, where c(A) is defined to be the smallest positive integer such that for any prime ℓ>c(A), the image of the ℓ-adic Galois representation of A is "as large as possible" in a suitable sense.

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