COMPUTING IWASAWA MODULES OF REAL QUADRATIC NUMBER-FIELDS

Let p be an odd prime and let X denote the projective limit of the p-p arts of the ideal class groups of the fields in the cyclotomic Z(p)-ex tension of a real quadratic number field F. We present a method to com pute the structure of X. As an illustration of the method we compute X for p = 3 and all real quadratic fields Q(root f) of conductor f < 10 000 and f not equal 1 (mod 3). For all fields we find that X is finite . In other words, Iwasawa's lambda-invariant is zero in these cases, w hich confirms a conjecture of Greenberg's.