Greenberg's conjecture and units in multiple [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]p-extensions

Let A be the inverse limit of the p-part of the ideal class groups in a [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-extension K_∞/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra Λ (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of pth roots of unity, p has index of irregularity 1, satisfies Vandiver's conjecture, and satisfies a mild additional hypothesis on units. We 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-extension 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_∞ generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K_∞ contains all the p-power roots of unity.