## 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 ${H}_{\mathrm{\xe9t}}^{i}({Y}_{\overline{\eta}},{\mathbb{Q}}_{\ell})$, $i\ge 0$, 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 $x\in X\left(k\right)\setminus {X}_{\rho}$.

## Citation

Anna Cadoret. Akio Tamagawa. "A uniform open image theorem for $\ell $-adic representations, I." Duke Math. J. 161 (13) 2605 - 2634, 1 October 2012. https://doi.org/10.1215/00127094-1812954

## Information