| An idea of Poincare about automorphic functions can be applied to an arbitrary (G, R) with a group G acting on a ring R. For a 1-cocycle c on the unit group Rx, we can define a module Mc/Pc, which is an invariant depending only on the cohomology class [c]. We are mainly interested in the case where G = Gal(K/ k) is the Galois group of number field extension and R = OK is the ring of integers. We determine Mc/ Pc, completely for the case K = Q( m ) the quadratic number field over Q. For a nontrivial cocycle, the index depends on the parity of the coefficient v of the fundamental unit epsilon = u + vo of OxK , and it is related to the central element of the continued fraction expansion of m . We may generalize this computation using Hilbert 90. |
