On Poincaré sums for number fields

Abstract

Let $G$ be a finite group acting on a ring $R$. To know the twisted Tate cohomology ${\hat{H}}^0(G,R^{+})_{\gamma}$ parametrized by $\gamma=[c]\in H^1(G,R^{\times})$ is a basic theme inspired by Poincaré. We shall consider this when $G$ is the Galois group of a Galois extension $K/k$ of number fields and $R$ is the ring of integers of $K$.