Reduced norms of division algebras over complete discrete valuation fields of local-global type

Let F be a complete discrete valuation field whose residue field k is a global field of positive characteristic p. et D be a central division F-algebra of p-power degree. We prove that the subgroup of F - consisting of reduced norms of D is exactly the kernel of the cup product map λ F - ↦(D) ∪ (λ) 3(F, ℚ p/ &p(2)), if either D is tamely ramified or of period p. This gives a p-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-p degree. 2020 World Scientific Publishing Company.