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 of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, by means of images in ${\mathcal T}_k$ of ideal norms from the layers $k_n$ of the cyclotomic tower (Theorem (5.2)). These images are obtained via the formal algorithm computing, by ``unscrewing'', the $p$-class group of~$k_n$. Conjecture (5.4) of equidistribution of these images would show that the number of steps $b_n$ of the algorithms is bounded as $n \to \infty$, so that Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, would hold true ``with probability $1$''. No assumption is made on $[k : \mathbb{Q}]$, nor on the decomposition of $p$ in $k/\mathbb{Q}$.