Notion de $\theta$-r\'egulateurs d'un nombre alg\'ebrique. Conjectures p-adiques

Abstract

Let K/Q be a Galois extension of degree n, of Galois group G, and let $\eta\in K^\times$. For all large enough prime p, we define, by use of the Frobenius theorem on group determinants, the family $(\Delta_p^\theta(\eta) \in \F_p)_\theta$ of local $\theta$-regulators of $\eta$, indexed by the Qp-irreducible characters $\theta$ of G. At each $\Delta_p^\theta (\eta)$ is associated a linear representation $L^\theta \simeq \delta V_\theta$, $0 \leq \delta \leq \varphi(1)$, which characterizes some properties of $\Delta_p^\theta (\eta)$, including its nullity equivalent to $\delta \geq 1$ (Th. 3.11). When $\eta \in \Q^\times$ and $\theta = 1$, $\Delta_p^1 (\eta)$ is the p-Fermat quotient of $\eta$. When $\eta$ is a "Minkowski unit", each $\Delta_p^\theta (\eta)$, $\theta \ne 1$, gives the residue modulo p of the $\theta$-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 ($\Delta_p^\theta(\eta) = 0$ and $L^\theta \simeq \delta V_\theta$) is $\frac{O(1)}{p^{f \delta^2}}$, 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, $\eta$ 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.