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

Let $A={\prod }_{1\le i\le n}{A}_i$ be the product of principally polarized abelian varieties $A_1,\dots ,A_n$ of dimensions $g_1,\dots ,g_n$, respectively, each defined over a number field $K$, and pairwise nonisogenous over $\bar{K}$. 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 $\mathrm{\ell }>c(A)$, the image of the $\mathrm{\ell }$-adic Galois representation of $A$ is “as large as possible” in a suitable sense.