# A note on Galois representations with big image

### Nicholas M. Katz

Princeton University, USA

## 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)

$\rho: Gal(\overline{\mathbb Q}/\mathbb Q(\zeta_N)) \rightarrow GL(n,\mathbb Q_\ell)$

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.

## 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