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According to Weil, the Weil conjecture should follow once one has a sufficiently powerful cohomology machine. And it is proved using one of them, namely étale cohomology.

My question is, has there been any attempt, after its proof using étale cohomology, to prove it using other Weil cohomology theories? i.e. a cohomology theory which has finiteness, allow Poincaré duality, Künneth formula, cycle map, weak and strong Lefschetz. After all it is a motivic thing.

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Yes. See Kedlaya's Fourier transforms and p-adic Weil II. This is a proof using Berthelot's rigid cohomology.

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