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Serre has shown that the family of $\ell$-adic Galois representations of an elliptic curve defined over a number field $K$ is almost independent. More explicitly:

let $E/K$ be an elliptic curve and for $\ell$ prime we have the following $\ell$-adic representation $\rho_{\ell}:Gal(\bar{K}/K)\longrightarrow Aut(T_{\ell}(E))\cong GL_2(\mathbb{Z}_{\ell})$ where $T_{\ell}(E)$ is the $\ell$-adic Tate module. We also have an induced representation $\rho:Gal(\bar{K}/K)\longrightarrow \prod_{\ell} GL_2(\mathbb{Z}_{\ell})$. The almost independence of the family of representations means that after passing to a finite extension (we assume that $K$ is already this finite extension) $\rho(Gal(\bar{K}/K)) = \prod_{\ell}\rho_{\ell}(Gal(\bar{K}/K))$.

We also have $\ell$-adic Galois representations attached to an elliptic curve $E'/F$ defined over a global function field $F$ of characteristic $p>0$ , $\rho'_{\ell}:Gal(F^{sep}/F)\longrightarrow Aut(T_{\ell}(E'))\cong GL_2(\mathbb{Z}_{\ell})$ and $\rho':Gal(F^{sep}/F)\longrightarrow \prod_{\ell\neq p} GL_2(\mathbb{Z}_{\ell})$.

My question is: is the family of $\ell$-adic Galois representations corresponding to the global function fields case independent? If so, where can I find a reference? Any related result would also be interesting.

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Funnily enough I was at a talk about this only last week. One doesn't get independence in the global function field case because there's an obstruction coming from the determinant; but there is a statement to the effect that if $K$ is a global function field with constant field $k$ of char $p$, then the image of $\operatorname{Gal}(K^{\mathrm{sep}} / K\overline{k})$ in $\prod_{\ell \ne p}SL_2(\mathbb{Q}_\ell)$ is open (I hope I've got that right!). There are now some much more general "almost independence" results due to Boeckle--Gajda--Petersen (for etale cohomology of arbitrary finite type schemes over finitely generated extensions of $\mathbb{F}_p$), see

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Thanks @David, I have recently seen papers on "almost independence" results due to Gajda and Petersen in the characteristic zero case (etale cohomology of finite type schemes over finitely generated extensions of $\mathbb{Q}$) but the one you referred to is the first one I saw for global function field. And it's very recent! – Andry Jul 15 '13 at 9:40

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