Invariants and coinvariants of class groups in Zp-extensions and Greenberg's Conjecture

Let K/k be a Z_p-extension of a number field k, k_n its n-th layer and A_n the p-class group of k_n. In this paper we give two criteria, both based on the group of invariants B_n of A_n, which imply the finiteness of the Iwasawa module X(K/k) and we discuss some of their consequences. The first criterion deals with stabilization and capitulation of the B_n, while the second one uses the nilpotency of the Galois group Gal(L(K)/k), where L(K) is the maximal unramified abelian pro-p-extension of K.