Bounding the symbol length in the Galois cohomology of function fields of $p$-adic curves

Abstract

Let $K$ be a function field of a $p$-adic curve and $l$ a prime not equal to $p$. Assume that $K$ contains a primitive $l$th root of unity. We show that every element in the $l$-torsion subgroup of the Brauer group of $K$ is a tensor product of two cyclic algebras over $K$.