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

Barbulescu, Razvan; Ray, Jishnu

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

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.

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

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