On Congruence Theorem for valued division algebras

Abstract: Let $K$ be a field equipped with a Henselian valuation, and let $D$ be a tame central division algebra over the field $K$. Denote by $\mathrm{TK}_1(D)$ the torsion subgroup of the Whitehead group ${\rm K}_1(D) = D^*/D'$, where $D^*$ is the multiplicative group of $D$ and $D'$ is its derived subgroup. Let ${\bf G}$ be the subgroup of $D^*$ such that $\mathrm{TK}_1(D) = {\bf G}/D'$. In this note, we prove that either $(1 + M_D) \cap {\bf G} \subseteq D'$, or the residue field $\overline{K}$ has characteristic $p > 0$ and the group ${\bf H} := ((1 + M_D) \cap {\bf G})D'/D'$ is a $p$-group. Additionally, we provide examples of valued division algebras with non-trivial ${\bf H}$. This illustrates that, in contrast to the reduced Whitehead group ({\rm SK}_1(D)), a complete analogue of the Congruence Theorem does not hold for ({\rm TK}_1(D)).