Conjectures de Greenberg et extensions¶pro-p-libres d'un corps de nombres

Abstract:

For an algebraic number field k and a prime number p (if p=2, we assume that μ₄⊂k), we study the maximal rank ρ _p of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ _k =r ₂(k)+1 if and only if k is a so-called p-rational field.