# Standard conjectures on positive characteristic

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?

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.

• Could you give more precise references?
– abx
Aug 24, 2014 at 13:34
• What about the Lefschetz type conjecture, and the comparison between numerical and homological equivalence?
– abx
Aug 24, 2014 at 17:15
• 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 $\mathbb{Q}_l$.
– anon
Aug 27, 2014 at 14:57