Notion de $\theta$-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 $\eta \in K^{\times }$. 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 $\theta$-regulators of $\eta$, indexed by the Qp-irreducible characters $\theta$ of G. At each ${\Delta }_p^{\theta }(\eta )$ is associated a linear representation $L^{\theta }\simeq \delta V_{\theta }$, $0\le \delta \le \phi (1)$, which characterizes some properties of ${\Delta }_p^{\theta }(\eta )$, including its nullity equivalent to $\delta \ge 1$ (Th. 3.11). When $η\in \Q^\times$ 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 $p^{1-n}Re{g}_p(K)$, 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 $p^{1-n}Re{g}_p(K)$, 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.

arXiv commentAddition of 4.6. Improvement of some technical proofs (e.g. 3.2.1), minor corrections of statements in conjectures of 7.3, in French