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Algorithmic complexity of Greenberg’s conjecture

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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”.

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Notes

  1. For more information on the main pioneering works about the practice of this theory, see “History of abelian p-ramification” in [10, Appendix] (e.g., Gras: “Crelle’s Journal” (1982/83), Jaulent: “Ann. Inst. Fourier” (1984) [12], Nguyen Quang Do: “Ann. Inst. Fourier” (1986) [20], Movahhedi “Thèse” (1988) and others). For convenience, we mostly refer to our book (2003/2005), which contains all the needed results in the most general statements. For more broad context about the base field k and the set S, see [19] and its bibliography.

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Acknowledgements

I warmly thank the referee for a careful reading of the paper, for some corrections, and a question which is the subject of Remark 3.5.

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  1. In retirement from Franche-Comté University, 4, chemin de Château Gagnière, Villa la Gardette, 38520, Le Bourg d’Oisans, France

    Georges Gras

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Gras, G. Algorithmic complexity of Greenberg’s conjecture. Arch. Math. 117, 277–289 (2021). https://doi.org/10.1007/s00013-021-01618-9

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