3
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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 http://arxiv.org/abs/1302.6597.

$\endgroup$
1
  • $\begingroup$ 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! $\endgroup$
    – Andry
    Jul 15, 2013 at 9:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.