## Duke Mathematical Journal

### A uniform open image theorem for $\ell$-adic representations, II

#### Abstract

Let $k$ be a field finitely generated over $\mathbb{Q}$, and let $X$ be a 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 any open subgroup of $\rho(\pi_{1}(X_{\overline{k}}))$ has finite abelianization. Let $G$ denote the image of $\rho$. Any closed point $x$ on $X$ induces a splitting $x:\Gamma_{\kappa(x)}:=\pi_{1}(\operatorname{Spec}(\kappa(x)))\rightarrow\pi_{1}(X_{\kappa(x)})$ of the restriction epimorphism $\pi_{1}(X_{\kappa(x)})\rightarrow \Gamma_{\kappa(x)}$ (here, $\kappa(x)$ denotes the residue field of $X$ at $x$) so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{\kappa(x)})\subset G$. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for any geometrically Lie perfect representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ and any integer $d\geq 1$, the set $X_{\rho,d}$ of all closed points $x\in X$ such that $G_{x}$ is not open in $G$ and $[\kappa(x):k]\leq d$ is finite and there exists an integer $B_{\rho,d}\geq 1$ such that $[G:G_{x}]\leqB_{\rho,d}$ for any closed point $x\in X\smallsetminus X_{\rho,d}$ with $[\kappa(x):k]\leq d$.

A key ingredient of our proof is that, for any integer $\gamma\geq 1$, there exists an integer $\nu=\nu(\gamma)\geq 1$ such that, given any projective system $\cdots\rightarrow Y_{n+1}\rightarrow Y_{n}\rightarrow\cdots\rightarrow Y_{0}$ of curves (over an algebraically closed field of characteristic $0$) with the same gonality $\gamma$ and with $Y_{n+1}\rightarrow Y_{n}$ a Galois cover of degree greater than $1$, one can construct a projective system of genus $0$ curves $\cdots\rightarrowB_{n+1}\rightarrow B_{n}\rightarrow \cdots\rightarrow B_{\nu}$ and degree $\gamma$ morphisms $f_{n}:Y_{n}\rightarrow B_{n}$, $n\geq \nu$, such that $Y_{n+1}$ is birational to $B_{n+1}\times_{B_{n},f_{n}}Y_{n}$, $n\geq \nu$. This, together with the case for $d=1$ (which is the main result of part I of this paper), gives the proof for general $d$.

Our method also yields the following unconditional variant of our main result. With the above assumptions on $k$ and $X$, for any $\ell$-adic representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ and integer $d\geq 1$, the set of all closed points $x\in X$ such that $G_{x}$ is of codimension at least $3$ in $G$ and $[\kappa(x):k]\leq d$ is finite.

#### Article information

Source
Duke Math. J., Volume 162, Number 12 (2013), 2301-2344.

Dates
First available in Project Euclid: 9 September 2013

https://projecteuclid.org/euclid.dmj/1378729689

Digital Object Identifier
doi:10.1215/00127094-2323013

Mathematical Reviews number (MathSciNet)
MR3102481

Zentralblatt MATH identifier
1279.14056

#### Citation

Cadoret, Anna; Tamagawa, Akio. A uniform open image theorem for $\ell$ -adic representations, II. Duke Math. J. 162 (2013), no. 12, 2301--2344. doi:10.1215/00127094-2323013. https://projecteuclid.org/euclid.dmj/1378729689

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