Les<i>θ</i>-régulateurs locaux d'un nombre algébrique : Conjectures<i>p</i>-adiques

Abstract

Abstract Let K/ℚ be Galois and let η K × be such that Reg ∞ (η)=0 .We define the local θ–regulator for the ℚ p –irreducible characters θ of G = Gal(Kℚ). Let V θ be the θ-irreducible representation. A linear representation is associated with whose nullity is equivalent to δ≥1. Each yields Reg θ p modulo p in the factorization of (normalized p –adic regulator). From Prob f ≥ 1 is a residue degree) and the Borel–Cantelli heuristic, we conjecture that for p large enough, Reg G p ( η ) is a p –adic unit (a single with f = δ=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.