Let
$K/\mathbb{Q}$
be Galois and let
$\eta \,\in \,{{K}^{\times }}$
be such that
$\text{Re}{{\text{g}}_{\infty }}\left( \eta \right)\,\ne \,0$
. We define the local
$\theta $
-regulator
$\Delta _{p}^{\theta }\left( \eta \right)\,\in \,{{\mathbb{F}}_{p}}$
for the
${{\mathbb{Q}}_{p}}$
-irreducible characters
$\theta $
of
$G=\,\text{Gal}\left( K/\mathbb{Q} \right)$
. Let
${{V}_{\theta }}$
be the
$\theta $
-irreducible representation. A linear representation
${{\mathfrak{L}}^{\theta }}\,\simeq \,\delta \,{{V}_{\theta }}$
is associated with
$\Delta _{p}^{\theta }\left( \eta \right)$
whose nullity is equivalent to
$\delta \,\ge \,1$
. Each
$\Delta _{p}^{\theta }\left( \eta \right)$
yields
$\text{R}eg_{p}^{\theta }\left( \eta \right)$
modulo
$p$
in the factorization
${{\Pi }_{\theta }}{{\left( \text{Reg}_{p}^{\theta }\left( \eta \right) \right)}^{\phi \left( 1 \right)}}$
of
$\text{Reg}_{p}^{G}\,\left( \eta \right)\,:=\frac{\text{Re}{{\text{g}}_{p}}\left( \eta \right)}{_{p}[K\,:\,\mathbb{Q}]}$
(normalized
$p$
-adic regulator). From Prob
$\left( \Delta _{p}^{\theta }\left( \eta \right)=0\,\text{and}\,{{\mathfrak{L}}^{\theta }}\simeq \delta {{V}_{\theta }} \right)\,\le {{p}^{-f{{\delta }^{2}}}}$
(
$f\,\ge \,1$
is a residue degree) and the Borel-Cantelli heuristic, we conjecture that for
$p$
large enough,
$\text{Reg}_{p}^{G}\left( \eta \right)$
is a
$p$
-adic unit or
${{p}^{\phi \left( 1 \right)}}\,||\,\text{Reg}_{p}^{G}\left( \eta \right)$
(a single
$\theta $
with
$f\,=\,\delta \,=\,1$
); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups
${{C}_{3,}}\,{{C}_{5}},\,{{D}_{6}}$
) is 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.