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


Published:20160328
Printed: Jun 2016
Georges Gras,
Villa la Gardette , chemin Ch\^ateau Gagni\`ere , F38520 Le Bourg d'Oisans, France
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
BorelCantelli 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, LeopoldtJaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$ramification, probabilistic number theory
$p$adic regulators, LeopoldtJaulent 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
