Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture

Razvan Barbulescu; Jishnu Ray

doi:10.5802/jtnb.1115

Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s

p

-rationality conjecture

Razvan Barbulescu ¹ ; Jishnu Ray ²

¹ UMR 5251, CNRS, INP Université de Bordeaux, 351, cours de la Libération 33400 Talence, France
² Department of Mathematics The University of British Columbia Room 121, 1984 Mathematics Road V6T 1Z2, Vancouver, BC, Canada

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 159-177.

Résumé
Abstract

Dans cet article, nous entreprenons une expérimentation numérique pour donner des arguments en faveur de la conjecture de $p$ -rationalité de Greenberg. Nous donnons une famille de corps biquadratiques $p$ -rationnels et trouvons de nouveaux exemples numériques de corps $p$ -rationnels multiquadratiques. Dans le cas des corps multiquadratiques et multicubiques, on montre que la conjecture de Greenberg est une conséquence de l’heuristique de Cohen–Lenstra–Martinet et d’une conjecture de Hofmann et Zhang sur le régulateur $p$ -adique. Nous apportons de nouveaux résultats numériques en faveur de ces conjectures. Nous comparons les outils algorithmiques existants et proposons certaines améliorations.

In this paper we make a series of numerical experiments to support Greenberg’s $p$ -rationality conjecture, we present a family of $p$ -rational biquadratic fields and we find new examples of $p$ -rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen–Lenstra–Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$ -adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.

Reçu le : 2019-03-20
Révisé le : 2019-10-16
Accepté le : 2019-11-29
Publié le : 2020-08-21

DOI : 10.5802/jtnb.1115

Classification : 11R29, 11Y40
Mots clés : class number, Cohen–Lenstra heuristic, $p$-rational number fields, $p$-adic regulator

Affiliations des auteurs :

Razvan Barbulescu ¹ ; Jishnu Ray ²

¹ UMR 5251, CNRS, INP Université de Bordeaux, 351, cours de la Libération 33400 Talence, France
² Department of Mathematics The University of British Columbia Room 121, 1984 Mathematics Road V6T 1Z2, Vancouver, BC, Canada

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Numerical verification of the {Cohen{\textendash}Lenstra{\textendash}Martinet} heuristics and of {Greenberg{\textquoteright}s} $p$-rationality conjecture},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Razvan Barbulescu; Jishnu Ray. Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 159-177. doi : 10.5802/jtnb.1115. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1115/

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