ON SOME TWISTED COHOMOLOGY OF THE RING OF INTEGERS

As an analogy of Poincar´e series in the space of modular forms, T. Ono associated a module M c /P c for γ = [c] ∈ H 1 (G, R × ) where finite group G is acting on a ring R. M c /P c is re-garded as the 0-dimensional twisted Tate cohomology b H 0 (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 M c /P c 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. More-over, some explicit examples on quadratic and biquadratic number fields are given.