On some twisted cohomology of the ring of integers

Abstract

As an analogy of Poincar´e series in the space of modular forms, T. Ono associated a module Mc/Pc for γ = [c] ∈ H1(G,R×) where finite group G is acting on a ring R. Mc/Pc is regarded as the 0-dimensional twisted Tate cohomology b H0(G,R+)γ. In the case that G is the Galois group of a Galois extension K of a number field k and R is the ring of integers of K, the vanishing properties of Mc/Pc are related to the ramification of K/k. We generalize this to arbitrary n-dimensional twisted cohomology of the ring of integers and obtain the extended version of theorems. Moreover, some explicit examples on quadratic and biquadratic number fields are given.