Degree Three Unramified Cohomology Groups

TL;DR

This paper demonstrates that for certain groups of order p^9, the degree three unramified cohomology groups are non-trivial, extending Peyre's earlier results for groups of order p^{12}.

Contribution

It improves upon Peyre's work by showing non-trivial unramified cohomology for smaller groups of order p^9.

Findings

Non-trivial unramified cohomology for some groups of order p^9

Extension of Peyre's results to smaller group orders

Method applicable to groups of order p^9

Abstract

Let $p$ be an odd prime number. Peyre shows that there is a group $G$ of order $p^{12}$ such that $H_{nr}^3(\bm{C}(G), \bm{Q}/\bm{Z})$ is non-trivial. Using Peyre's method, we are able to prove that the same conclusion is true for some groups of order $p^9$.