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Les $\theta$-régulateurs locaux d'un nombre algébrique : Conjectures $p$-adiques

  Published:2016-03-28
 Printed: Jun 2016
  • Georges Gras,
    Villa la Gardette , chemin Ch\^ateau Gagni\`ere , F-38520 Le Bourg d'Oisans, France
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Abstract

Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that $\operatorname{Reg}_\infty (\eta) \ne 0$. We define the local $\theta$-regulators $\Delta_p^\theta(\eta) \in \mathbb{F}_p$ for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of $G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \, V_\theta$ is associated with $\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta \geq 1$. Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$ modulo $p$ in the factorization $\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of $\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]} }$ (normalized $p$-adic regulator). From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal L}^\theta \simeq \delta \, V_\theta\big ) \leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$ is a $p$-adic unit or that $p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$ with $f=\delta=1$); this obstruction may be lifted assuming the existence of a binomial probability law confirmed through numerical studies (groups $C_3$, $C_5$, $D_6$). This conjecture would imply that, for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational. We recall some deep cohomological results that may strengthen such conjectures.
Keywords: $p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory $p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory
MSC Classifications: 11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40 show english descriptions $p$-adic theory, local fields [See also 14G20, 22E50]
Algebraic numbers; rings of algebraic integers
Ordinary representations and characters
Matrices, determinants [See also 15B36]
Class field theory
Units and factorization
Algebraic number theory computations
11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]
11R04 - Algebraic numbers; rings of algebraic integers
20C15 - Ordinary representations and characters
11C20 - Matrices, determinants [See also 15B36]
11R37 - Class field theory
11R27 - Units and factorization
11Y40 - Algebraic number theory computations
 

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