A note on Galois representations with big image

Abstract

Given an integer N ≥ 3, we will first construct reasonably “motivic” representations ρ : Gal(Q/Q(ζN ))→ GL(n,Q`) with open image, for any ` 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 Q when N is 3 or 4. This provides us with motivic galois representations of Gal(Q/Q) with open image in GL(n,Q`) for any even n ≥ 6 and any ` which is ≡ 1 mod 3 or mod 4.