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

Gras, Georges

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

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 ${(\mathbb{Z}/2\mathbb{Z})}^t$ 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 $p\ge 2$ (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 $\mu$ 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 ${(\mathbb{Z}/2\mathbb{Z})}^t$ 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 $p\ge 2$ (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 $\mu$-invariants of some uniform pro-$p$-groups. All programs (in “verbatim”) can be used by the reader by simply copied and pasted.

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

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