In this [MO answer](https://mathoverflow.net/questions/83421/voevodskys-counterexample-to-the-existence-of-a-motivic-t-structure?rq=1) of M. Bondarko, he says: 



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> "the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."
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and in [Remarks on Grothendieck's standard conjectures](http://arxiv.org/pdf/1006.1116v2.pdf) A. Beilinson says: 



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> "We show that Grothendieck’s standard conjectures (over a field of characteristic zero) follow from either of two other motivic conjectures, namely, that of existence of the motivic t-structure and (a weak version of) Suslin’s Lawson homology conjecture".
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**My question is:** What about with standard conjectures in positive characteristic?



Standard conjectures on positive characteristic
Over finite fields, the Tate conjecture by itself doesn't imply the standard conjectures, but together with the Hodge conjecture for CM abelian varieties it does (Milne 2002, 2009).


In a little more detail: Milne showed that the Hodge conjecture for CM abelian varieties implies the Tate conjecture for all abelian varieties over the algebraically closure k of a finite field in a 1999 paper (Compositio), and that it implies the Hodge standard conjecture for AVs/k in a 2002 Annals paper. If you assume the Tate conjecture, then the category of motives over k is generated by abelian varieties (mentioned in Milne 1994, Seattle I). In terms of motives, the Hodge standard conjecture says that the pairings defined by ample line bundles are positive for a polarization on the Tannakian category of motives. 


