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Cutting towers of number fields

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Abstract

Given a prime p, a number field \({K}\) and a finite set of places S of \({K}\), let \({K}_S\) be the maximal pro-p extension of \({K}\) unramified outside S. Using the Golod–Shafarevich criterion one can often show that \({K}_S/{K}\) is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod–Shafarevich criterion. In the tame setting we are able to produce infinite asymptotically good extensions in which infinitely many primes split completely, and in which every prime has Frobenius of finite order, a phenomenon that had been expected by Ihara. We also achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases.

Résumé

Etant donnés un nombre premier p, un corps de nombres \({K}\) et un ensemble fini S de places de \({K}\), soit \({K}_S\) la pro-p extension maximale de \({K}\) non-ramifiée en dehors de S. À l’aide du critère de Golod–Shafarevich, on parvient souvent à montrer que l’extension \({K}_S/{K}\) est infinie. Dans les cas modéré et sauvage, nous construisons des sous-extensions infinies où la ramification est bornée, et ce à l’aide d’un raffinement du critère de Golod–Shafarevich. Dans le cas modéré, nous exhibons des extensions infinies et asymptotiquement bonnes pour lesquelles l’ensemble des premiers totalement décomposés est infini et où l’élément de Frobenius de chaque premier est fini, une possibilité qui avait été suggérée par Ihara. Nous établissons également de nouveaux records pour les constantes de Martinet (bornes sur le discriminant normalisé) dans les extensions totalement réelles ainsi que dans les extensions totalement complexes.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Farshid Hajir

  2. FEMTO-ST Institute, University Bourgogne Franche-Comté, CNRS, 15B avenue des Montboucons, 25000, Besancon, France

    Christian Maire

  3. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Ravi Ramakrishna

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  1. Farshid Hajir
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  2. Christian Maire
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  3. Ravi Ramakrishna
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Correspondence to Ravi Ramakrishna.

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This work has been done during a visiting scholar position for the second author at Cornell University for the academic year 2017–2018, and funded by the program “Mobilité sortante” of the Région Bourgogne Franche-Comté; CM thanks the Department of Mathematics at Cornell University for providing a beautiful research atmosphere. Christian Maire was also partially supported by the ANR project FLAIR (ANR-17-CE40-0012) and by the EIPHI Graduate School (ANR-17-EURE-0002). Ravi Ramakrishna was supported by Simons Collaboration Grant 524863.

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Hajir, F., Maire, C. & Ramakrishna, R. Cutting towers of number fields. Ann. Math. Québec 45, 321–345 (2021). https://doi.org/10.1007/s40316-021-00156-8

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