A uniform open image theorem for ℓ-adic representations, I

Abstract

Let $k$ be a field finitely generated over $\mathbb{Q}$, and let $X$ be a smooth, separated, and geometrically connected curve over $k$. Fix a prime $\ell$. A representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ is said to be geometrically Lie perfect if the Lie algebra of $\rho \left({\pi }_1\right(X_{\overline{k}}\left)\right)$ is perfect. Typical examples of such representations are those arising from the action of ${\pi }_1\left(X\right)$ on the generic $\ell$-adic Tate module $T_{\ell }\left(A_{\eta }\right)$ of an abelian scheme $A$ over $X$ or, more generally, from the action of ${\pi }_1\left(X\right)$ on the $\ell$-adic étale cohomology groups $H_{\mathrm{ét}}^i(Y_{\overline{\eta }},{\mathbb{Q}}_{\ell })$, $i\ge 0$, of the geometric generic fiber of a smooth proper scheme $Y$ over $X$. Let $G$ denote the image of $\rho$. Any $k$-rational point $x$ on $X$ induces a splitting $x:{\Gamma }_k:={\pi }_1(Spec(k\left)\right)\to {\pi }_1\left(X\right)$ of the canonical restriction epimorphism ${\pi }_1\left(X\right)\to {\Gamma }_k$ so one can define the closed subgroup $G_x:=\rho \circ x\left({\Gamma }_k\right)\subset G$. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$, the set $X_{\rho }$ of all $x\in X\left(k\right)$ such that $G_x$ is not open in $G$ is finite and there exists an integer $B_{\rho }\ge 1$ such that $[G:G_x]\le B_{\rho }$ for every $x\in X\left(k\right)\setminus X_{\rho }$.