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Algorithmic complexity of Greenberg’s conjecture. (English) Zbl 1469.11420

Summary: 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”.

MSC:

11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants
11R37 Class field theory
11Y40 Algebraic number theory computations
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