A uniform open image theorem for \(\ell\)-adic representations. I
Let \(k\) be a field finitely generated over \(\mathbb{Q}\), \(X/k\) be a smooth separated and geometrically connected curve. Let \(\rho: \pi_1(X)\to {\text{GL}}_{m}(\mathbb{Z}_l)\) be an \(\ell\)-adic representation, \(G:=\rho(\pi_1(X))\). The representation \(\rho\) is called geometrically Lie perfect if the image \(G^{\text{geo}}:=\rho(\pi_1(X_{\bar{k}}))\) is a Lie perfect \(\ell\)-adic Lie group, i.e. the abelianization of \text{Lie}(\(G^{\text{geo}}\)) is trivial, where \(\pi_1(X_{\bar{k}})\) is regarded as a subgroup of \(\pi_1(X)\) as in the fundamental exact sequence \[ 1 \to \pi_1(X_{\bar{k}})\to \pi_1(X)\to{\text{Gal} }(\bar{k}/k)\to 1.\tag{\(*\)} \] Let \(x\in X(k)\) be a \(k\)-rational point, then we get a splitting \(s: {\text{Gal}}(\bar{k}/k)\to \pi_1(X)\) of \((*)\). Strictly speaking, the splitting is up to an inner automorphism of \(\pi_1(X)\) by an element in \(\pi_1(X_{\bar{k}})\) because of the choice of the base point, but it doesn't play any role here. Let \(G_x:= \rho\circ s({\text{Gal}}(\bar{k}/k))\). The main purpose of this paper is to show that when \(\rho\) is a geometrically Lie perfect representation the set \(X_{\rho}\) of all \(x\in X(k)\) such that \(G_x\) is not open in \(G\) is finite and there exists an integer \(B_{\rho}\geq 1\) such that \([G:G_x]\leq B_{\rho}\) for every \(x\in X(k)\setminus X_{\rho}\).
scientific article; zbMATH DE number 6109716
scientific article