4 problems

Somekawa's conjecture. The generalized Galois symbol $s_n$ is always injective.

Kahn's conjecture. If $r=s+1$, then the Somekawa K-group $K(k;G_1,\ldots,G_r)$ is torsion. If $r>s+1$ and $k$ does not admit an ordered field structure, then

Let $k$ be a field and let $G_1,\ldots,G_r$ be semi-abelian varieties over $k$. Let $K(k,G_1,\ldots,G_r)$ denote the Somekawa $K$-group, let $\mathcal{M}_k$ be a certain category o…

Kato-Somekawa conjecture. For every such collection of semi-abelian varieties and every positive integer $m$ prime to the characteristic of $k$, the Galois symbol map $h^q_m$ is in…