Galois cohomology of function fields of curves over non-archimedean local fields

Let F F be the function field of a curve over a non-archimedean local field. Let m ≥ 2 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 , μ m ⊗ 2 ) H^{3}(F, \mu _{m}^{\otimes 2}) is of the form χ ∪ ( f ) ∪ ( g ) \chi \cup (f) \cup (g) , where χ \chi is in H 1 ( F , Z / m Z ) H^{1}(F, \mathbb {Z}/m\mathbb {Z}) and ( f ) (f) , ( g ) (g) in H 1 ( F , μ m ) H^{1}(F, \mu _{m}) . This extends a result of Parimala and Suresh [Ann. of Math. (2) 172 (2010), pp. 1391–1405], where they show this when m m is prime and when F F contains μ m \mu _{m} .