Galois Cohomology of Function Fields of Curves over Non-archimedean Local Fields

Abstract

Let $F$ be the function field of a curve over a non-archimedean local field. Let $m \geq 2$ be an integer coprime to the characteristic of the residue field of the local field. In this article, we show that every element in $H^{3}(F, \mu_{m}^{\otimes 2})$ is of the form $\chi \cup (f) \cup (g)$, where $\chi$ is in $H^{1}(F, \mathbb{Z}/m\mathbb{Z})$ and $(f)$, $(g)$ in $H^{1}(F, \mu_{m})$. This extends a result of Parimala and Suresh, where they show this when $m$ is prime and when $F$ contains $\mu_{m}$.