A uniform open image theorem for $\ell$-adic representations (joint work with Akio Tamagawa - R.I.M.S.)

Duration: 1 hour 2 mins 12 secs

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Description:	Cadoret, A (Bordeaux 1) Friday 28 August 2009, 14:00-15:00

Created:

2009-09-04 14:48

Collection:

Non-Abelian Fundamental Groups in Arithmetic Geometry

Publisher:

Isaac Newton Institute

Cadoret, A (Bordeaux 1)

Language:

eng (English)

Credits:

Author:

Cadoret, A (Bordeaux 1)


Abstract:	In this talk, we extend some of the results presented in Tamagawa's talk to more general $\ell$-adic representations.\\ \indent Let $k$ be a finitely generated field of characteristic $0$, $X$ a smooth, separated, geometrically connected curve over $k$ with generic point $\eta$. A $\ell$-adic representation $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ is said to be geometrically strictly rationnally perfect (GSRP for short) if $\hbox{\rm Lie}(\rho(\pi_{1}(X_{\overline{k}})))^{ab}=0$. 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 etale cohomology groups $H^{i}(Y_{\overline{\eta}},\mathbb{Q}_{\ell})$, $i\geq 0$ of the geometric generic fiber of a smooth proper scheme $Y$ over $X$. Let $G$ denote the image of $\rho$. Any closed point $x$ on $X$ induces a splitting $x:\Gamma_{\kappa(x)}:=\pi_{1}(\hbox{\rm Spec}(\kappa(x))) \rightarrow\pi_{1}(X_{\kappa(x)})$ of the canonical restriction epimorphism $\pi_{1}(X_{\kappa(x)})\rightarrow \Gamma_{\kappa(x)}$ (here, $\kappa(x)$ denotes the field of definition of $x$) so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{\kappa(x)})\subset G$ (up to inner automorphisms).\\ \indent The main result I am going to discuss is the following uniform open image theorem. \textit{Under the above assumptions, for any representation $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ and any integer $d\geq 1$, the set $X_{\rho, d,\geq 3}$ of all closed points $x\in X$ such that $G_{x}$ has codimension $\geq 3$ in $G$ and $[\kappa(x):k]\leq d$ is finite. Furthermore, if $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ is GSRP then the set $X_{\rho, d,\geq 1}$ of all closed points $x\in X$ such that $G_{x}$ has codimension $\geq 1$ in $G$ and $[\kappa(x):k]\leq d$ is finite and there exists an integer $B_{\rho,d}\geq 1$ such that $[G:G_{x}]\leq B_{\rho,d}$ for any closed point $x\in X\smallsetminus X_{\rho,d,\geq 1}$ such that $[\kappa(x):k]\leq d$.}\\

Abstract:

In this talk, we extend some of the results presented in Tamagawa's talk to more general $\ell$-adic representations.\\ \indent Let $k$ be a finitely generated field of characteristic $0$, $X$ a smooth, separated, geometrically connected curve over $k$ with generic point $\eta$. A $\ell$-adic representation $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ is said to be geometrically strictly rationnally perfect (GSRP for short) if $\hbox{\rm Lie}(\rho(\pi_{1}(X_{\overline{k}})))^{ab}=0$. 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 etale cohomology groups $H^{i}(Y_{\overline{\eta}},\mathbb{Q}_{\ell})$, $i\geq 0$ of the geometric generic fiber of a smooth proper scheme $Y$ over $X$. Let $G$ denote the image of $\rho$. Any closed point $x$ on $X$ induces a splitting $x:\Gamma_{\kappa(x)}:=\pi_{1}(\hbox{\rm Spec}(\kappa(x))) \rightarrow\pi_{1}(X_{\kappa(x)})$ of the canonical restriction epimorphism $\pi_{1}(X_{\kappa(x)})\rightarrow \Gamma_{\kappa(x)}$ (here, $\kappa(x)$ denotes the field of definition of $x$) so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{\kappa(x)})\subset G$ (up to inner automorphisms).\\ \indent The main result I am going to discuss is the following uniform open image theorem. \textit{Under the above assumptions, for any representation $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ and any integer $d\geq 1$, the set $X_{\rho, d,\geq 3}$ of all closed points $x\in X$ such that $G_{x}$ has codimension $\geq 3$ in $G$ and $[\kappa(x):k]\leq d$ is finite. Furthermore, if $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ is GSRP then the set $X_{\rho, d,\geq 1}$ of all closed points $x\in X$ such that $G_{x}$ has codimension $\geq 1$ in $G$ and $[\kappa(x):k]\leq d$ is finite and there exists an integer $B_{\rho,d}\geq 1$ such that $[G:G_{x}]\leq B_{\rho,d}$ for any closed point $x\in X\smallsetminus X_{\rho,d,\geq 1}$ such that $[\kappa(x):k]\leq d$.}\\

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