Greenberg's conjecture and units in multiple Z(p)-extensions

Let Abe the inverse limit of the p-part of the ideal class groups in a Z(p) (r)-extension K-infinity/K. Greenberg conjectures that if r is maximal, the n A is pseudo-null as a module over the Iwasawa algebra Lambda (that is, ha s codimension at least 2). We prove this conjecture in the case that K is t he field of pth roots of unity, p has index of irregularity 1, satisfies Va ndiver's conjecture, and satisfies a mild additional hypothesis on units. W e also show that if K is the field of pth roots of unity and r is maximal, Greenberg's conjecture for K implies that the maximal p-ramified pro-p-exte nsion of K cannot have a free pro-p quotient of rank r unless p is regular. Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K-infinity generated by p-power roots of p-units. we show that the Galois group of this extension is torsion-fre e as a Lambda -module if there is only one prime of K above p and K-infinit y contains all the p-power roots of unity.