An explicit open image theorem for products of elliptic curves

Let K   be a number field and let E1,…,EnE1,…,En be elliptic curves over K  , pairwise non-isogenous over K‾ and without complex multiplication over K‾. We study the image G∞G∞ of the adelic representation of Gal(K‾/K) naturally attached to E1×⋯×EnE1×⋯×En. The main result is an explicit bound for the index of G∞G∞ in {(x1,…,xn)∈GL2(Zˆ)n|det⁡xi=det⁡xj∀i,j}.