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It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the Riemann Hypothesis in "Weil II"? In particular, can the Standard Conjectures be "souped up" to get a version that does imply Weil II RH?

It seems that the main difference, in a nutshell, is that Weil II releases many of the assumptions on smoothness/properness/projectivity in the main results and considers cohomology with coefficients on a much more general class of sheaves. In particular, this implies that while the eigenvalues of Frobenius remain Weil numbers, they no longer necessarily have the same modulus for a given cohomology space. As a secondary question, are there any other major issues moving from Weil I to Weil II that I am missing here?

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Since Deligne's main results don't even require the coefficient sheaf to be "of geometric origin", just to satisfy fibral bounded on Frobenius eigenvalues, it does seem unlikely that anything like the standard conjectures could imply results with the strength of Weil II even if one assumes whatever form of resolution of singularities one wants. (I would be interested to learn otherwise.) Weil II seems to be a good illustration of the inadequacy of the Standard Conjectures. – BCnrd Nov 2 '10 at 20:20
The standard conjectures do imply the existence of a weight filtration and motivic t-structure on the category of $DM_{gm}(S)$ of geometric motives over $S$ the spectrum of a field (see Beilinson's paper on arxiv). Is it true for more general $S$? – AFK Nov 2 '10 at 22:39
YBL: I would be very surprised if this were true, even when $S$ is spec of a field. Also in Beilinson's paper, it seems the implication goes the other way. – Donu Arapura Nov 8 '10 at 22:09
duplicate…? – David Corwin Jul 30 '13 at 19:35
@David. No, they are different questions. – Joël Jan 3 '14 at 22:18

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