Let $\Pi$ be the fundamental group of a smooth variety over a finite field. Let $\lambda$ be a non-Archimedean place of the field of algebraic numbers. The philosophy of motives predicts that the $\lambda$-adic pro-semisimple completion of $\Pi$ "does not depend" on $\lambda$. I will try to explain the prediction (which includes a kind of "reciprocity law" involving a sum over all $\ell$-adic cohomology theories). I will also try to explain what can actually be proved.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/6826