A note on Galois representations with big image (Q2024036)

In the present paper, the author constructs motivic representations \(\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\zeta_N)) \rightarrow \mathrm{GL}(n, \mathbb{Q}_{\ell})\) with open image for any \(\ell\) which is \(1\bmod N\) and for certain \(n\), where \(N \geq 3\) is an integer. He does this in three different ways. The third of them has a descent to \(\mathbb{Q}\) when \(N\) is \(3\) or \(4\). This gives motivic Galois representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) with open image in \(\mathrm{GL}(n, \mathbb{Q}_{\ell})\) for any even \(n \geq 6\) and any \(\ell\) which is \(\equiv 1\bmod 3\) or \(\bmod 4\).