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Assuming the standard conjectures (and whatever is needed in addition), is there a nice proof of the Weil-conjectures written completely in the language of motives?

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See Theorem 5.6 in Kleiman's article "The Standard Conjectures", in the Motives volume (PSPM 55.1). I'm not quite sure what you mean by "written completely in the language of motives", so it might not be exactly what you are looking for.

I initially posted this as a comment, but it really should be an answer, so I'm re-posting it as one.

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