Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s p-rationality conjecture
Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 159-177.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1115
Classification: 11R29, 11Y40
Keywords: class number, Cohen–Lenstra heuristic, $p$-rational number fields, $p$-adic regulator
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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JTNB_2020__32_1_159_0,
     author = {Razvan Barbulescu and Jishnu Ray},
     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},
     pages = {159--177},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1115},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1115/}
}
TY  - JOUR
AU  - Razvan Barbulescu
AU  - Jishnu Ray
TI  - Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 159
EP  - 177
VL  - 32
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1115/
DO  - 10.5802/jtnb.1115
LA  - en
ID  - JTNB_2020__32_1_159_0
ER  - 
%0 Journal Article
%A Razvan Barbulescu
%A Jishnu Ray
%T Numerical verification of the Cohen–Lenstra–Martinet heuristics and of Greenberg’s $p$-rationality conjecture
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 159-177
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1115/
%R 10.5802/jtnb.1115
%G en
%F JTNB_2020__32_1_159_0
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, Volume 32 (2020) no. 1, pp. 159-177. doi : 10.5802/jtnb.1115. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1115/

[1] Miho Aoki; Takashi Fukuda An Algorithm for Computing p-Class Groups of Abelian Number Fields, Algorithmic Number Theory – ANTS VII (Lecture Notes in Computer Science), Volume 4076 (2006) | DOI | MR | Zbl

[2] Razvan Barbulescu; Jishnu Ray Electronic manuscript of computations of “Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg’s p-rationality conjecture”, 2017 (available online at https://webusers.imj-prg.fr/~razvan.barbaud/pRational/pRational.html)

[3] Jens Bauch; Daniel J. Bernstein; Henry de Valence; Tanja Lange; Christine Van Vredendaal Short generators without quantum computers: the case of multiquadratics, Advances in cryptology – EUROCRYPT 2017 (Lecture Notes in Computer Science), Volume 10210 (2017), pp. 27-59 | DOI | MR | Zbl

[4] Henri Cohen A course in computational algebraic number theory, Graduate Texts in Mathematics, 138, Springer, 2013 | Zbl

[5] Henri Cohen; Hendrik W. Lenstra Heuristics on class groups, Number theory (New York, 1982) (Lecture Notes in Mathematics), Volume 1052, Springer, 1984, pp. 26-36 | DOI | MR

[6] Henri Cohen; Jacques Martinet Class groups of number fields: numerical heuristics, Math. Comput., Volume 48 (1987) no. 177, pp. 123-137 | DOI | MR | Zbl

[7] Claus Fieker; Yinan Zhang An application of the p-adic analytic class number formula, LMS J. Comput. Math., Volume 19 (2016) no. 1, pp. 217-228 | DOI | MR | Zbl

[8] Georges Gras Class Field Theory: from theory to practice, Springer monographs of mathematics, Springer, 2013 | Zbl

[9] Marie-Nicole Gras Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q, J. Reine Angew. Math., Volume 277 (1975) no. 89, 116 pages | MR | Zbl

[10] Ralph Greenberg Galois representations with open image, Ann. Math. Qué., Volume 40 (2016) no. 1, pp. 83-119 | DOI | MR | Zbl

[11] Tuomas Hakkarainen On the computation of class numbers of real abelian fields, Math. Comput., Volume 78 (2009) no. 265, pp. 555-573 | DOI | MR | Zbl

[12] Paul Hartung Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by 3, J. Number Theory, Volume 6 (1974) no. 4, pp. 276-278 | DOI | MR | Zbl

[13] Tommy Hofmann; Yinan Zhang Valuations of p-adic regulators of cyclic cubic fields, J. Number Theory, Volume 169 (2016), pp. 86-102 | DOI | MR | Zbl

[14] Sigekatu Kuroda Über die Klassenzahlen algebraischer Zahlkörper, Nagoya Math. J., Volume 1 (1950), pp. 1-10 | DOI | MR | Zbl

[15] Franciscus Jozef van der Linden Class number computations of real abelian number fields, Math. Comput., Volume 39 (1982) no. 160, pp. 693-707 | DOI | MR | Zbl

[16] Stéphane Louboutin L-functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field, Math. Comput., Volume 59 (1992) no. 199, pp. 213-230 | MR | Zbl

[17] Stéphane Louboutin Majorations explicites du résidu au point 1 des fonctions zêta de certains corps de nombres, J. Math. Soc. Japan, Volume 50 (1998) no. 1, pp. 57-69 | DOI | Zbl

[18] Gunter Malle Cohen-Lenstra heuristic and roots of unity, J. Number Theory, Volume 128 (2008) no. 10, pp. 2823-2835 | DOI | MR | Zbl

[19] Abbas Movahhedi Sur les p-extensions des corps p-rationnels, Ph. D. Thesis, Université Paris VII (France) (1988)

[20] Abbas Movahhedi Sur les p-extensions des corps p-rationnels, Math. Nachr., Volume 149 (1990), pp. 163-176 | DOI | MR | Zbl

[21] Abbas Movahhedi; Thong Nguyen Quang Do Sur l’arithmétique des corps de nombres p-rationnels, Séminaire de Théorie des Nombres, Paris 1987–88 (Progress in Mathematics), Volume 81, Birkhäuser, 1990, pp. 155-200 | DOI | MR | Zbl

[22] Frédéric Pitoun; Firmin Varescon Computing the torsion of the p-ramified module of a number field, Math. Comput., Volume 84 (2015) no. 291, pp. 371-383 | DOI | MR | Zbl

[23] Oliver Schirokauer Discrete logarithms and local units, Philosophical Transactions of the Royal Society of London A: Math., Phys. and Eng. Sci., Volume 345 (1993) no. 1676, pp. 409-423 | MR | Zbl

[24] The PARI Group PARI/GP version 2.9.0, 2016 (available from http://pari.math.u-bordeaux.fr/)

[25] The Sage Developers SageMath, the Sage Mathematics Software System (Version 7.5.1), 2016 (http://www.sagemath.org/)

[26] Lawrence C. Washington Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer, 1997, xiv+487 pages | DOI | MR | Zbl

Cited by Sources: