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The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.

A variety $X$ over a finite field $k$ is liftable if there exists a number field $K$ and a variety $\mathcal X$ over $K$ such that $X$ is the reduction of $\mathcal X$ (with respect to some model) at some maximal ideal of $O_K$.

Suppose RH holds for liftable varieties (forgetting Deligne proved this). Can we deduce RH for all varieties from this?

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1  
Since all affines are liftable, an appropriate version of the RH for sufficiently bad (e.g. non-proper) liftable varieties, plus a good theory of weights, would imply the RH in general. But a "good theory of weights" is a large part of the content of the RH, so this is a bit lame. –  Daniel Litt Jan 8 '13 at 19:49
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There's a recent paper by Scholl explaining how to deduce general RH from the corresponding statement for smooth projective hypersurfaces in projective space (which are clearly liftable). His argument does not use Lefschetz pencils or the Fourier transform (but it does uses a theorem of Deligne from Weil II on local monodromy). –  anon Jan 9 '13 at 5:21
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