SUR LA CONJECTURE FAIBLE DE GREENBERG DANS LE CAS ABÉLIEN p-DÉCOMPOSÉ

Abstract

Let p be an odd prime. For any CM number field K containing a primitive pth root of unity, class field theory and Kummer theory put together yield the well known reflection inequality λ+ ≤ λ- between the "plus" and "minus" parts of the λ-invariant of K. Greenberg's classical conjecture predicts the vanishing of λ+. We propose a weak form of this conjecture: λ+ = λ- if and only if λ+ = λ- = 0, and we prove it when K+ is abelian, p is totally split in K+, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case).