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Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given anon-Archimedean place $\lambda$ of the field of algebraic numbers which isprime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ asan object of the groupoid whose objects are pro-semisimple groups and whosemorphisms are isomorphisms up to conjugation by elements of the neutralconnected component. We prove that this object does not depend on $\lambda$. Ifdim X=1 we also prove a crystalline generalization of this fact.  We deduce this from the Langlands conjecture for function fields (proved byL. Lafforgue) and its crystalline analog (proved by T. Abe) using areconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky.  We also formulate two related conjectures, each of which is a "reciprocitylaw" involving a sum over all $l$-adic cohomology theories (including thecrystalline theory for $l=p$).

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