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

## TL;DR

This paper characterizes reduced norms of certain division algebras over complete discrete valuation fields with global residue fields of positive characteristic, extending known results to the $p$-torsion case.

## Contribution

It proves that the subgroup of reduced norms equals the kernel of a specific cup product map for division algebras of $p$-power degree under certain conditions, extending previous prime-to-$p$ results.

## Key findings

- Reduced norms form the kernel of a cup product map in the $p$-torsion case.
- The result applies to tamely ramified or period-$p$ division algebras.
- Extends the theorem of Parimala, Preeti, and Surech to $p$-torsion division algebras.

## Abstract

Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of positive characteristic $p$. Let $D$ 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 $\lambda\in F^*\mapsto (D)\cup(\lambda)\in H^3(F,\,\mathbb{Q}_{p}/\Z_{p}(2))$, if either $D$ is tamely ramified or of period $p$. This gives a $p$-torsion counterpart of a recent theorem of Parimala, Preeti and Surech, where the same result is proved for division algebras of prime-to-$p$ degree.