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

TL;DR

This paper extends the understanding of Galois cohomology for function fields of curves over non-archimedean local fields, showing that certain third cohomology classes can be expressed as cup products of lower-degree classes.

Contribution

It generalizes previous results by demonstrating that all elements in a specific third cohomology group can be decomposed into cup products without the restriction that the field contains roots of unity.

Findings

All elements in H^3(F, μ_m^{⊗2}) are cup products of elements in H^1(F, Z/mZ) and H^1(F, μ_m)

Extension of Parimala and Suresh's result to cases where m is not necessarily prime and F may not contain μ_m

Broader applicability in Galois cohomology of function fields 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 $\mu_{m}$.