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

Georges Gras

doi:10.5802/pmb.35

p

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

p

-rationalité des corps de nombres. Applications – Programmes PARI/GP]

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.

Résumé
Abstract

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.

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.

Reçu le : 2017-10-29
Publié le : 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

Affiliations des auteurs :

Georges Gras ¹

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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/

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Algèbre et Théorie des Nombres