A note on Poincaré sums for finite groups

Abstract

A simple and beautiful idea of Poincaré on Poincaré series in automorphic functions can be applied to an arbitrary ring $R$ acted by a group $G$. When $G$ is finite, the key is to look at the 0-dimensional Tate cohomology of $(G, R)$ twisted by the 1-cohomology class of the group of units of $R$. As a simplest case, we examine when $R$ is the ring of integers of a quadratic field.