Cyclic Length in the Tame Brauer Group of the Function Field of a p-Adic Curve

TL;DR

This paper proves that for the function field of a p-adic curve, the n-torsion Brauer group is generated by cyclic classes, establishing the Brauer dimension as two and showing certain division algebras are decomposable.

Contribution

It demonstrates that the n-torsion subgroup of the Brauer group is generated by cyclic classes with length two, and confirms the Brauer dimension of the function field is two.

Findings

The n-torsion subgroup is generated by cyclic classes.

The Brauer dimension of the field is two.

Division algebras of period n and index n^2 are decomposable.

Abstract

Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,\mu_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.