Algorithmic complexity of Greenberg’s conjecture

Abstract

Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( $$\lambda = \mu = 0$$ ) is governed (under Leopoldt’s conjecture) by the finite torsion group $${{\mathscr {T}}}_k$$ of the Galois group of the maximal abelian p-ramified pro-p-extension of k, by means of images, in $${{\mathscr {T}}}_k$$ , of ideal norms from the layers $$k_n$$ of the cyclotomic tower (Theorem 4.2). These images are obtained via the algorithm computing, by “unscrewing”, the p-class group of $$k_n$$ . Conjecture 4.3 of equidistribution of these images would show that the number of steps $$b_n$$ of the algorithms is bounded as $$n \rightarrow \infty $$ , so that (Theorem 3.3) Greenberg’s conjecture, hopeless within the sole framework of Iwasawa’s theory, would hold true “with probability 1”.