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In this MO answer of M. Bondarko, he says:

"the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."

and in Remarks on Grothendieck's standard conjectures A. Beilinson says:

"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".

My question is: What about with standard conjectures in positive characteristic?

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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.

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  • $\begingroup$ Could you give more precise references? $\endgroup$
    – abx
    Commented Aug 24, 2014 at 13:34
  • $\begingroup$ What about the Lefschetz type conjecture, and the comparison between numerical and homological equivalence? $\endgroup$
    – abx
    Commented Aug 24, 2014 at 17:15
  • $\begingroup$ By the Tate conjecture, I mean the equivalent statements of 2.9 of Tate's Seattle article (the order of the pole of the zeta function is equal to the rank of a certain group of algebraic classes). This implies that numerical equivalence equals homological equivalence (ibid.). The Lefschetz standard conjecture says that a certain map of spaces of algebraic cycles is an isomorphism. The Tate conjecture (in the above form) implies that it becomes an isomorphism when tensored with $\mathbb{Q}_l$. $\endgroup$
    – anon
    Commented Aug 27, 2014 at 14:57

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