Over which fields (of positive characteristic) is the Beilinson-Soulé vanishing conjecture known to hold?

Question

Let $k$ be a field, and denote by $K_p(k)^{(n)}$ the weight $n$ eigenspace of the Adams operations on the $p$-th $K$-group of $k$.

The Beilinson-Soulé (BS) vanishing conjecture predicts that $$ K_{2q-p}(k)^{(q)}=0 $$ for $p \leq 0$ and $q>0$ (cf. Levine, "Tate motives and vanishing conjectures...").

If this conjecture holds for the field $k$ then (as demonstrated by Levine in ibid.) there exists a rigid abelian tensor category $MTM(k)$ of mixed Tate motives over $k$.

For $k$ a number field, the conjecture is true by results of Borel. Moreover, it is stated in Deligne-Goncharov's "Groupes Fondamentaux motiviques de Tate mixtes" at the beginning of section 1.6. that

"The only fields in characteristic zero for which the Beilinson-Soulé vanishing conjectures are proved are number fields, function fields of genus 0 curves over number fields and inductive limits of such fields."

My Question: What about fields of positive characteristic? Are there any for which BS vanishing is known (maybe the first thing to look at would be finite extensions of $\Bbb F_q(t)$ in keeping with the usual analogy with number fields)?

If not, are there at least partial (possibly negative) results in this direction? Or does the matter seem altogether hopeless?

Also, it would be nice to obtain a reference for the above claim in Deligne-Goncharov (but probably that's better off as a separate question).

Answer 1

Maybe let me begin with the remark that the Adams eigenspace decomposition concerns the rationalized algebraic K-theory $K_i(k)\otimes_{\mathbb{Z}}\mathbb{Q}$. Consequently, the Beilinson-Soulé conjectures as well as the abelian category of motives of Levine are rational objects.

Now for what's known in positive characteristic. Quillen has computed K-theory of finite fields in

D. Quillen. On the cohomology and K-theory of the general linear groups over a finite field. Ann. of Math. (2) 96 (1972), 552--586.

The well-known result is that the K-groups are torsion, so the rationalized K-theory is trivial and hence satisfies the Beilinson-Soulé vanishing.

Similarly, Harder has computed the group cohomology of S-arithmetic groups in positive characteristic in

G. Harder. Die Kohomologie $S$-arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 42 (1977), 135--175.

The (maybe less well-known) result is that for an $S$-arithmetic group $G$ coming from a simple Chevalley group of rank $r$, the only non-trivial rational cohomology groups of $G$ are in degrees $0$ and $r|S|$. In particular, taking the colimit over all finite subsets of places, the rational cohomology of $SL_n(K)$ vanishes (in positive degrees) when $K$ is a global field of positive characteristic. Rational homotopy theory again tells you that the rational K-groups are trivial except for $K_0$ and $K_1$, hence Beilinson-Soulé holds for global fields of positive characteristic as well.

It is worth pointing out that the associated abelian categories of motives with rational coefficients is very simple in the case of finite fields: the vanishing of rational K-theory implies that the abelian category of motives is equivalent to the category of graded finite-dimensional $\mathbb{Q}$-vector spaces. Another way to say this is that the motivic Galois group is $\mathbb{G}_m$.

I am not sure if I can help with the genus $0$ function field statement in Deligne-Goncharov. I would expect that this follows from the Gersten resolution $$ 0\to K_i(F)\to K_i(F(T))\to \bigoplus_{x\in (\mathbb{A}^1_F)^{(1)}}K_{i-1}(\kappa(x))\to 0 $$ where $x$ runs through the closed points of $\mathbb{A}^1_F$ and $\kappa(x)$ is the corresponding residue field. If this resolution is somehow compatible with the Adams operations (like inducing exact sequences of eigenspaces) it would be possible to deduce statements about the Adams eigenspaces. Hopefully an expert can correct me or back me up on this.