An explicit open image theorem for products of elliptic curves

Abstract

Let $K$ be a number field and $E_1,\dots ,E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute Galois group of $K$ naturally attached to $E_1\times \cdots \times E_n$. The main result is an explicit bound for the index of this image in $\left\{(x_1,\dots ,x_n)\in {\mathrm{GL}}_2(\hat{\mathbb{Z}}{)}^n|\det x_i=\det x_j\forall i,j\right\}$.

arXiv comment18 pages. v2: improved exposition