Sur les �galit�s du miroir et certaines formes faibles de la conjecture de Greenberg

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 conjecture asserts that λ+ is always trivial. We study here a weak form of this conjecture, namely λ+=λ− if and only if λ+=λ−=0.