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The l-adic pro-semisimple completion of a profinite group packages its l-adic representations valued in (not necessarily connected) semisimple groups. When the profinite group considered is the ├ętale fundamental group of a smooth variety over a finite field k, Drinfeld has proven an "independence-of-l" result for these pro-semisimple completions. This talk will describe the result precisely, and explain how Drinfeld ultimately reduces it to the existence of compatible systems containing a given l-adic local system on X. This existence result in turn follows from the Langlands correspondence when X is a curve, and in general from an earlier result of Drinfeld reducing the general case to the case of curves.

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