A program to test the p-rationality of any number field

Abstract

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, very fast PARI Programs (§§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 related to Cohen-Lenstra-Martinet ones and analyse Greenberg's conjecture on the existence of p-rational fields with Galois groups (Z/2Z) t needed for the construction of some Galois representations with open image. We give examples for p = 3, t = 5 and t = 6 (§§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 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 prop groups. R{é}sum{é} 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) caract{é}rise sa p-rationalit{é}. Puis nous donnons deux courts programmes PARI (§§3.1, 3.2) trs rapides testant si K (d{é}fini par un polyn{ô}me irr{é}ductible uni-taire) est p-rationnel ou non. Pour les corps quadratiques nous v{é}rifions certaines densit{é}s 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/2Z) t n{é}cessaires pour la construction de certaines repr{é}sentations galoisiennes d'image ouverte. Nous donnons des exemples pour p = 3, t = 5 et t = 6 (§§5.1, 5.2) et il-lustrons d'autres approches (Pitoun-Varescon, Barbulescu-Ray). Nous concluons sur l'existence de corps quadratiques imaginaires p-rationnels pour tout p $\ge$ 2 (Angelakis-Stevenhagen sur le concept de "groupe de Galois ab{é}lien absolu minimal") qui peut{é}clairerpeut{é}clairer une conjecture de p-rationalit{é} (Hajir-Maire) donnant de grands invariants $\mu$ d'Iwasawa relatifs {à} certains prop -groupes uniformes.

To appear in Publ. Math. Fac. Sci. Besançon (2019)New Sections and new applications

References in corpus (4)

• The p-adic Kummer-Leopoldt constant - Normalized p-adic regulator
• Valuations of $p$-adic regulators of cyclic cubic fields
• 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields
• The Cohen-Lenstra heuristics, moments and $p^j$-ranks of some groups

Cited by in corpus (3)

• Weber's class number problem and $p$-rationality in the cyclotomic $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$
• Test of Vandiver's conjecture with Gauss sums -- Heuristics
• Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties