A note on Poincaré sums of Galois representations (Q1187007)

[This is a joint review for parts I--III.] The motivation for these three notes is a plan to build some sort of non- abelian Kummer theory. Let \(K/k\) be a finite Galois extension with Galois group \(G\), let \(\rho:\;G\to GL_ r(k)\) be a \(k\)-representation and let \(\chi\) be its character. Let for each \(x\in K\) the Poincaré sum \(P_ \chi(x)\) be defined to be \(\sum_{s\in G}x^ s\chi(s)\). These are viewed as generalizations of Kummer generators. In these three notes some elementary properties of these numbers \(P_ \chi(x)\) are established.