A note on Galois representations with big image

Abstract

Given an integer $N\ge 3$, we will first construct motivic representations (i.e., built out of pieces of the cohomology of projective smooth varieties, in fact curves)

with open image, for any $\ell$ which is $1$ mod $N$ and for certain $n$. We will do this in three different ways. The third of them has a descent to $\mathbb{Q}$ when $N$ is 3 or 4. This provides us with motivic Galois representations of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with open image in $GL(n,{\mathbb{Q}}_{\ell })$ for any even $n\ge 6$ and any $\ell$ which is $\equiv 1$ mod 3 or mod 4.