# On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field

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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.