# A note on Galois representations with big image

### Nicholas M. Katz

Princeton University, USA

## Abstract

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

$ρ:Gal(Q /Q(ζ_{N}))→GL(n,Q_{ℓ})$

with open image, for any $ℓ$ 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 $GL(n,Q_{ℓ})$ for any even $n≥6$ and any $ℓ$ which is $≡1$ mod 3 or mod 4.

## Cite this article

Nicholas M. Katz, A note on Galois representations with big image. Enseign. Math. 65 (2019), no. 3/4, pp. 271–301

DOI 10.4171/LEM/65-3/4-1