Greenberg's conjecture and capitulation in Zpd-extensions

AbstractLet p be an odd prime. Let k be an algebraic number field and let k˜ be the compositum of all the Zp-extensions of k, so that Gal(k˜/k)≃Zpd for some finite d. We shall consider fields k with Gal(k/Q)≃(Z/2Z)n. Building on known results for quadratic fields, we shall show that the Galois group of the maximal abelian unramified pro-p-extension of k˜ is pseudo-null for several such k's, thus confirming a conjecture of Greenberg. Moreover we shall see that pseudo-nullity can be achieved quite early, namely in a Zp2-extension, and explain the consequences of this on the capitulation of ideals in such extensions