Algorithmic complexity of Greenberg's conjecture

Abstract

Let k be a totally real number field. We show that the ``complexity'' of Greenberg's conjecture (lambda = mu = 0) is of p-adic nature and is governed by the torsion group T_k of the Galois group of the maximal abelian p-ramified pro-p-extension of k, by means of images, in T_k, of ideal norms along the layers k_n/k of the cyclotomic tower; these images are obtained via the algorithm computing, by ``unscrewing'', the p-class group of k_n (Theorem 5.2). Conjecture 5.4 of equidistribution of these images, finite in number, would show that Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, holds true ``with probability 1''. No assumption is made on [k : Q], nor on the decomposition of p in k/Q.