ON SOME TWISTED COHOMOLOGY OF THE RING OF INTEGERS

Abstract

As an analogy of Poincar(cid:19)e series in the space of modular forms, T. Ono associated a module Mc=Pc for fl = [c] 2 H 1(G; R(cid:2)) where finite group G is acting on a ring R. Mc=Pc is regarded as the 0-dimensional twisted Tate cohomology bH 0(G; R+)fl. 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.

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