Notion de $θ$-régulateurs d'un nombre algébrique. Conjectures p-adiques

Abstract

Let K/Q be a Galois extension of degree n, of Galois group G, and let $η\in K^\times$. For all large enough prime p, we define, by use of the Frobenius theorem on group determinants, the family $(Δ_p^θ(η) \in \F_p)_θ$ of local $θ$-regulators of $η$, indexed by the Qp-irreducible characters $θ$ of G. At each $Δ_p^θ(η)$ is associated a linear representation $L^θ\simeq δV_θ$, $0 \leq δ\leq φ(1)$, which characterizes some properties of $Δ_p^θ(η)$, including its nullity equivalent to $δ\geq 1$ (Th. 3.11). When $η\in \Q^\times$ and $θ= 1$, $Δ_p^1 (η)$ is the p-Fermat quotient of $η$. When $η$ is a "Minkowski unit", each $Δ_p^θ(η)$, $θ\ne 1$, gives the residue modulo p of the $θ$-component of $p^{1-n} Reg_p (K)$, where Reg(K) is the classical p-adic regulator of K. We suggest that the "probability" of ($Δ_p^θ(η) = 0$ and $L^θ\simeq δV_θ$) is $\frac{O(1)}{p^{f δ^2}}$, where f is a suitable residue degree of p. We conjecture that $p^{1-n} Reg_p(K)$, which measures the order of the p-torsion group in Abelian p-ramification over K, is for p large enough a p-adic unit except perhaps for a set of prime numbers of zero density. For these cases said "of minimal p-divisibility" (Def. 3.17), it remains possible, $η$ being then a "partial local pth power" at p, to propose, in connection with the ABC conjecture, a stronger conjecture leading to the same conclusion for all large enough p (Section 7). Some other conjectural aspects on the Fermat quotient are discussed. We precise and verify these properties through numerical studies on various fields and publish the corresponding "PARI" programs.