What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
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2$\begingroup$ In this Article of Wikipedia says: The Hodge conjecture implies the Lefschetz conjecture and conjecture D (num = hom) for varieties over fields of characteristic zero. Likewise for fields of finite characteristic the Tate conjectures in ℓ-adic cohomology imply the Lefschetz conjecture. $\endgroup$– user56558Commented Jul 29, 2014 at 14:56
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2 Answers
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I think the list on Wikipedia is incomplete:
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as conjecture D (for singular cohomology) over fields of characteristic 0.
The Tate conjecture also implies Lefschetz, Kunneth, and conjecture D (for etale cohomology) over all fields (not just characteristic $p$).
The reason is pretty simple. All of these conjectures are about the existence of an algebraic cycle: For the Lefschetz conjecture, a cycle inducing the Lefschetz operator. For the Kunneth conjecture, a cycle inducing the projector. For conjecture D, a cycle pairing nontrivially with a fixed homologically nontrivial cycle. In each case, it is easy to check that there exists a cohomology class with the specified properties which is a Hodge class / Tate class. Hence assuming the Hodge conjecture / Tate conjecture, there is an algebraic cycle as well.
In terms of the reverse implication, as far as I know there is none.
answered Jul 29, 2014 at 16:04
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$\begingroup$ This may be silly, but I don't understand your argument for conjecture D. If you have a rational cohomology class pairing nontrivially with an algebraic class, its $(p,p)$ component has the same property, but why should it be rational? In char. 0 this is no problem because the Lefschetz type conjecture + the Hodge index theorem imply conjecture D. But I doubt that the Tate conjecture implies conjecture D. $\endgroup$– abxCommented Jul 30, 2014 at 4:09
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$\begingroup$ $H^{2i}(X,\mathbb Q_\ell)$ and $H^{2d-2i}(X,\mathbb Q_\ell)$ are dual Galois representations. If we have a cycle class in the first one that gives a cyclotomic subrepresentation, so we get a cyclotomic quotient representation of the other one. I think the Tate conjecture is usually taken to include semisimplicity (or it implies that?), so this gives you a cyclotomic subrepresentation that is dual to the first one. $\endgroup$ Commented Jul 30, 2014 at 4:30
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$\begingroup$ OK. I have never seen the semi-simplicity included in the statement of the Tate conjecture, but I am not an expert. $\endgroup$– abxCommented Jul 30, 2014 at 5:59
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$\begingroup$ The Tate conjecture furnishes $\mathbf Q_\ell$-linear combinations of cycles. Is it enough? $\endgroup$– ACLCommented Aug 4, 2014 at 14:09
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$\begingroup$ @ACL: For the Kunneth conjecture, the cycles we are searching for are idempotents in a certain ring. If upon tensoring with $\mathbf Q_\ell$ an algebra has a certain idempotent, then I believe that before tensoring it has corresponding elements whose eigenvalues are in $\mathbf Q_\ell$, that is, they are in an algebraic number field that splits at $\ell$. Combining all $\ell$ I think we get the desired statement. So maybe one needs Tate at each $\ell$. $\endgroup$ Commented Aug 4, 2014 at 21:19
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The period and Hodge conjectures imply the standard conjectures. Yves andré (Une introduction aux motifs)
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2$\begingroup$ What relation with the standard conjectures? $\endgroup$– abxCommented Aug 4, 2014 at 7:28