## Duke Mathematical Journal

### A uniform open image theorem for $\ell$-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 $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ is said to be geometrically Lie perfect if the Lie algebra of $\rho(\pi_{1}(X_{\overline{k}}))$ is perfect. Typical examples of such representations are those arising from the action of $\pi_{1}(X)$ on the generic $\ell$-adic Tate module $T_{\ell}(A_{\eta})$ of an abelian scheme $A$ over $X$ or, more generally, from the action of $\pi_{1}(X)$ on the $\ell$-adic étale cohomology groups $\operatorname{H}^{i}_{\mathrm{\acute{e}t}}(Y_{\overline{\eta}},\mathbb{Q}_{\ell})$, $i\geq 0$, 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 $x:\Gamma_{k}:=\pi_{1}(\operatorname{Spec}(k))\rightarrow\pi_{1}(X)$ of the canonical restriction epimorphism $\pi_{1}(X)\rightarrow\Gamma _{k}$ so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{k})\subset G$. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$, 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 $B_{\rho}\geq1$ such that $[G:G_{x}]\leq B_{\rho}$ for every $x\inX(k)\smallsetminus X_{\rho}$.

#### Article information

Source
Duke Math. J., Volume 161, Number 13 (2012), 2605-2634.

Dates
First available in Project Euclid: 11 October 2012

https://projecteuclid.org/euclid.dmj/1349960278

Digital Object Identifier
doi:10.1215/00127094-1812954

Mathematical Reviews number (MathSciNet)
MR2988904

Zentralblatt MATH identifier
1305.14016

#### Citation

Cadoret, Anna; Tamagawa, Akio. A uniform open image theorem for $\ell$ -adic representations, I. Duke Math. J. 161 (2012), no. 13, 2605--2634. doi:10.1215/00127094-1812954. https://projecteuclid.org/euclid.dmj/1349960278

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