On <span class="mathjax-formula">$p$</span>-rationality of number fields. Applications – PARI/GP programs

Georges Gras

doi:10.5802/pmb.35

On

p

-rationality of number fields. Applications – PARI/GP programs

Georges Gras ¹

¹ Villa la Gardette chemin Château Gagnière 38520 Le Bourg d’Oisans, France

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 29-51.

Abstract
Résumé

Let $K$ be a number field. We prove that its ray class group modulo $p^{2}$ (resp. $8$ ) if $p > 2$ (resp. $p = 2$ ) characterizes its $p$ -rationality. Then we give two short and very fast PARI Programs (Sections 3.1, 3.2) testing if $K$ (defined by an irreducible monic polynomial) is $p$ -rational or not. For quadratic fields we verify some densities of $3$ -rational fields related to Cohen–Lenstra–Martinet ones and analyse Greenberg’s conjecture on the existence of $p$ -rational fields with Galois groups ${(ℤ / 2 ℤ)}^{t}$ needed for the construction of some Galois representations with open image. We give examples for $p = 3$ , $t = 5$ and $t = 6$ (Sections 5.1, 5.2) and illustrate other approaches (Pitoun–Varescon, Barbulescu–Ray). We conclude about the existence of imaginary quadratic fields, $p$ -rational for all $p \geq 2$ (Angelakis–Stevenhagen study on the concept of “minimal absolute abelian Galois group”) which may enlighten a conjecture of $p$ -rationality (Hajir–Maire) giving large Iwasawa $μ$ -invariants of some uniform pro- $p$ -groups. All programs (in “verbatim”) can be used by the reader by simply copied and pasted.

Soit $K$ un corps de nombres. Nous montrons que son corps de classes de rayon modulo $p^{2}$ (resp. $8$ ) si $p > 2$ (resp. $p = 2$ ) caractérise sa $p$ -rationalité. Puis nous donnons deux programmes PARI (Sections 3.1, 3.2) très courts et rapides testant si $K$ (défini par un polynôme irréductible unitaire) est $p$ -rationnel ou non. Pour les corps quadratiques nous vérifions certaines densités de corps $3$ -rationnels en relation avec celles de Cohen–Lenstra–Martinet et nous analysons la conjecture de Greenberg sur l’existence de corps $p$ -rationnels de groupes de Galois ${(ℤ / 2 ℤ)}^{t}$ nécessaires pour la construction de certaines représentations galoisiennes d’image ouverte. Nous donnons des exemples pour $p = 3$ , $t = 5$ et $t = 6$ (Sections 5.1, 5.2) et illustrons d’autres approches (Pitoun–Varescon, Barbulescu–Ray). Nous concluons sur l’existence de corps quadratiques imaginaires $p$ -rationnels pour tout $p \geq 2$ (étude de Angelakis–Stevenhagen sur le concept de “groupe de Galois abélien absolu minimal”) qui peut éclairer une conjecture de $p$ -rationalité (Hajir–Maire) donnant de grands invariants $μ$ d’Iwasawa relatifs à certains pro- $p$ -groupes uniformes. Tous les programmes (en “verbatim”) sont utilisables par le lecteur par simple copié-collé.

Received: 2017-10-29
Published online: 2019-12-06

DOI: 10.5802/pmb.35

Classification: 11R04, 11R37, 11R11, 08-04
Mots-clés : $p$-rational fields, class field theory, abelian $p$-ramification, PARI/GP programs, quadratic fields, Greenberg’s conjecture on representations

Author's affiliations:

Georges Gras ¹

¹ Villa la Gardette chemin Château Gagnière 38520 Le Bourg d’Oisans, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{PMB_2019___2_29_0,
     author = {Georges Gras},
     title = {On $p$-rationality of number fields. {Applications} {\textendash} {PARI/GP} programs},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {29--51},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {2},
     year = {2019},
     doi = {10.5802/pmb.35},
     language = {en},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.35/}
}

TY  - JOUR
AU  - Georges Gras
TI  - On $p$-rationality of number fields. Applications – PARI/GP programs
JO  - Publications mathématiques de Besançon. Algèbre et théorie des nombres
PY  - 2019
SP  - 29
EP  - 51
IS  - 2
PB  - Presses universitaires de Franche-Comté
UR  - https://pmb.centre-mersenne.org/articles/10.5802/pmb.35/
DO  - 10.5802/pmb.35
LA  - en
ID  - PMB_2019___2_29_0
ER  -

%0 Journal Article
%A Georges Gras
%T On $p$-rationality of number fields. Applications – PARI/GP programs
%J Publications mathématiques de Besançon. Algèbre et théorie des nombres
%D 2019
%P 29-51
%N 2
%I Presses universitaires de Franche-Comté
%U https://pmb.centre-mersenne.org/articles/10.5802/pmb.35/
%R 10.5802/pmb.35
%G en
%F PMB_2019___2_29_0

Georges Gras. On $p$-rationality of number fields. Applications – PARI/GP programs. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2019), pp. 29-51. doi : 10.5802/pmb.35. https://pmb.centre-mersenne.org/articles/10.5802/pmb.35/

References
Cited by

[1] Athanasios Angelakis; Peter Stevenhagen Imaginary quadratic fields with isomorphic abelian Galois groups, ANTS X, Proceedings of the Tenth Algorithmic Number Theory Symposium (The Open Book Series), Volume 1 (2013), pp. 21-39 | DOI | MR | Zbl

[2] Razvan Barbulescu; Jishnu Ray Some remarks and experimentations on Greenberg’s $p$ -rationality conjecture (2017) (https://arxiv.org/abs/1706.04847)

[3] Karim Belabas; Jean-François Jaulent The logarithmic class group package in PARI/GP, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 2016 (2016), pp. 5-18 | Zbl

[4] Henri Cohen; Hendrik W. jun. Lenstra Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Lecture Notes in Mathematics), Volume 1068, Springer, 1984, pp. 33-62 | DOI | MR | Zbl

[5] Henri Cohen; Jacques Martinet Class groups of number fields: Numerical heuristics, Math. Comput., Volume 48 (1987), pp. 123-137 | DOI | MR | Zbl

[6] Christophe Delaunay; Frédéric Jouhet The Cohen–Lenstra heuristics, moments and $p^{j}$ -ranks of some groups, Acta Arith., Volume 164 (2014) no. 3, pp. 245-263 | DOI | MR | Zbl

[7] Ivan B. Fesenko; Sergei V. Vostokov Local Fields and Their Extensions, Translations of Mathematical Monographs, 121, American Mathematical Society, 2002 | MR | Zbl

[8] Étienne Fouvry; Jürgen Klüners Cohen–Lenstra heuristics of quadratic number fields, Algorithmic number theory (Lecture Notes in Computer Science), Volume 4076, Springer, 2006, pp. 40-55 | DOI | MR | Zbl

[9] Georges Gras Remarks on $K_{2}$ of number fields, J. Number Theory, Volume 23 (1986), pp. 322-335 | DOI | Zbl

[10] Georges Gras Class Field Theory: from theory to practice, Springer Monographs in Mathematics, Springer, 2005, xiii+507 pages | Zbl

[11] Georges Gras On the structure of the Galois group of the Abelian closure of a number field, J. Théor. Nombres Bordeaux, Volume 26 (2014) no. 3, pp. 635-654 | DOI | MR | Zbl

[12] Georges Gras Les $θ$ -régulateurs locaux d’un nombre algébrique : conjectures $p$ -adiques, Can. J. Math., Volume 68 (2016) no. 3, pp. 571-624 | DOI | Zbl

[13] Georges Gras The $p$ -adic Kummer–Leopoldt constant: normalized $p$ -adic regulator, Int. J. Number Theory, Volume 14 (2018) no. 2, pp. 329-337 | DOI | MR | Zbl

[14] Georges Gras Heuristics and conjectures in the direction of a $p$ -adic Brauer–Siegel theorem, Math. Comput., Volume 88 (2019) no. 318, pp. 1829-1965 | MR | Zbl

[15] Georges Gras Practice of incomplete p-ramification over a number field – History of abelian p-ramification (2019) (https://arxiv.org/abs/1904.10707v2, to appear in Commun. Adv. Math. Sci.)

[16] Georges Gras; Jean-François Jaulent Sur les corps de nombres réguliers, Math. Z., Volume 202 (1989), pp. 343-365 | DOI | Zbl

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

[18] Farshid Hajir; Christian Maire Prime decomposition and the Iwasawa mu-invariant, Math. Proc. Camb. Philos. Soc., Volume 166 (2019) no. 3, pp. 599-617 | DOI | MR | Zbl

[19] 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

[20] Jean-François Jaulent Théorie $ℓ$ -adique globale du corps de classes, J. Théor. Nombres Bordeaux, Volume 10 (1998) no. 2, pp. 355-397 | DOI | Zbl

[21] Jean-François Jaulent; Thong Nguyen Quang Do Corps $p$ -rationnels, corps $p$ -réguliers et ramification restreinte, J. Théor. Nombres Bordeaux, Volume 5 (1993) no. 2, pp. 343-363 | DOI | Zbl

[22] Ömer Küçüksakalli Class numbers of ray class fields of imaginary quadratic fields, Math. Comput., Volume 80 (2011) no. 274, pp. 1099-1122 | DOI | MR | Zbl

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

[24] 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, Springer, 1990 | DOI | Zbl

[25] Dawn C. Nelson A Variation on Leopoldt’s Conjecture: Some Local Units instead of All Local Units (2013) (https://arxiv.org/abs/1308.4637)

[26] Midori Onabe On the isomorphisms of the Galois groups of the maximal abelian extensions of imaginary quadratic fields, Natur. Sci. Rep. Ochanomizu Univ., Volume 27 (1976) no. 2, pp. 155-161 | MR | Zbl

[27] Carlo Pagano; Efthymios Sofos $4$ -ranks and the general model for statistics of ray class groups of imaginary quadratic fields (2017) (https://arxiv.org/abs/1710.07587)

[28] Frédéric Pitoun Calculs théoriques et explicites en théorie d’Iwasawa (2010) (Ph. D. Thesis)

[29] 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

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

[31] Lawrence C. Washington Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer, 1997 | MR | Zbl

Cited by Sources: