Let p be a fixed prime number. Let K be a totally real number field of discriminant DK and let TK be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt’s conjecture). We conjecture the existence of a constant Cp > 0 such that log(#TK) ≤ Cp · log( √ DK) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer–Siegel Theorem, wearing here on the valuation of the residue at s = 1 (essentially equal to #TK) of the p-adic ζ-function ζp(s) of K. We shall use a different definition that of Washington, given in the 1980’s, and approach this question via the arithmetical study of TK since p-adic analysis seems to fail because of possible abundant “Siegel zeros” of ζp(s), contrary to the classical framework. We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. Such a conjecture (if exact) reinforces our conjecture that any fixed number field K is p-rational (i.e., TK = 1) for all p ≫ 0. 1. Abelian p-ramification – Main definitions and notations Let K be a totally real number field of degree d, and let p ≥ 2 be a prime number fulfilling the Leopoldt conjecture in K. We denote by ClK the p-class group of K (ordinary sense) and by EK the group of p-principal global units ε ≡ 1 (mod ∏ p|p p) of K. Let’s recall from [9, 12] the diagram of the so called abelian p-ramification theory, in which K = KQ is the cyclotomic Zp-extension of K (as compositum with that of Q), HK the p-Hilbert class field and H pr K the maximal abelian p-ramified (i.e., unramified outside p) pro-p-extension of K.

pdf/heuristics-and-conjectures-in-the-direction-of-a-p-adic-4lzf3rutn3.pdf

Heuristics and conjectures in the direction of a $p$-adic Brauer–Siegel Theorem

For $\Gamma=\mathbb{Z}_p$, Iwasawa was the first to construct $\Gamma$-extensions over number fields with arbitrarily large $\mu$-invariants. In this work, we investigate other uniform pro-$p$ groups which are realizable as Galois groups of towers of number fields with arbitrarily large $\mu$-invariant. For instance, we prove that this is the case if $p$ is a regular prime and $\Gamma$ is a uniform pro-$p$ group admitting a fixed-point-free automorphism of odd order dividing $p-1$. Both in Iwasawa's work, and in the present one, the size of the $\mu$-invariant appears to be intimately related to the existence of primes that split completely in the tower.

Prime Decomposition and the Iwasawa mu-invariant

— LetK be a number field. We prove that its ray class group modulo p2 (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/2Z) 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 ≥ 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. Résumé. — (Sur la p-rationalité des corps de nombres. Applications – Programmes PARI/GP) Soit K un corps de nombres. Nous montrons que son corps de classes de rayon modulo p2 (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 prationnels de groupes de Galois (Z/2Z) 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 ≥ 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. 2010 Mathematics Subject Classification. — 11R04, 11R37, 11R11, 08-04.

/pdf/on-p-rationality-of-number-fields-applications-pari-gp-19b601ni4d.pdf

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

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete p-ramification'' (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S $\le$ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H\_(K,S) of any field K given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the ``story'' of abelian S-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion T\_(K,S) of Gal(H\_(K,S)/K) (even if S=P) is a fundamental obstruction in many problems. All relationships involving S-ramification, as Iwasawa's theory, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text.

/pdf/practice-of-incomplete-p-ramification-over-a-number-field-cx7rglhn2c.pdf

Practice of incomplete p-ramification over a number field -- History of abelian p-ramification.

Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre etranger a p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiee en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous etudions la φ-composante X φ de X pour les caracteres Qp-irreductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractere non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons egalement une etude numerique sur la liberte de Xφ dans des extensions cycliques K{Qde degre 3 et de degre 4 (`a partir de familles de polynomes donnees par Balady, Lecacheux et plus recemment par Balady et Washington),et ainsi que dans des extensions diedrales de degre 6 de Q. Les resultats de nos calculs renforcent une recente conjecture de Gras

/pdf/composantes-isotypiques-de-pro-p-extensions-de-corps-de-4r3owads25.pdf

Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité

We fix a prime number $p$ and $\\K$ a number field, we denote by $M$ the maximal abelian $p$-extension of $\\Ko$ unramified outside $p$. The aim of this paper is to study the $\\Z_p$-module $\\gal(M/\\Ko)$ and to give a method to effectively compute its structure as a $\\Z_p$-module. Then we give numerical results, for real quadratic fields, together with interpretations via Cohen-Lenstra's heuristics.

Computing the torsion of the $p$-ramified module of a number field

We fix a prime number $p$ and $\K$ a number field, we denote by $M$ the maximal abelian $p$-extension of $\Ko$ unramified outside $p$. The aim of this paper is to study the $\Z_p$-module $\gal(M/\Ko)$ and to give a method to effectively compute its structure as a $\Z_p$-module. Then we give numerical results, for real quadratic fields, together with interpretations via Cohen-Lenstra's heuristics.

Computing the torsion of the $p$-ramified module

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Computing the torsion of the $p$-ramified module of a number field

Computing the torsion of the $p$-ramified module