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arXiv · 1902.06534

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

Yong Hu
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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.

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Yong Hu. 2019-02-18. Reduced norms of division algebras over complete discrete valuation fields of local-global type. https://doi.org/10.1142/s0219498820502175

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