Sur la propagation de la propriété mild au-dessus d’une extension quadratique imaginaire de $$\varvec{\mathbb {Q}}$$ Q

Abstract

In this work, we are interested in the pro-p groups \(G_S\), which are Galois groups of maximal pro-p extensions of number fields unramified outside a finite set S of primes not dividing p. We focus on whether the mildness property is preserved over imaginary quadratic extensions. Our starting point is Labute-Schmidt’s criterion (Schmidt, Doc Math 12:441–471, 2007), based on the study of the cup-product on the first cohomology group \(H^1(G_S,\mathbb {F}_p)\). In favourable conditions, we show by computation that the group we study often satisfies a weak version (\(LS_f\)) of Labute-Schmidt’s criterion. Then, a theoretical criterion is established for proving mildness of some groups to which the (\(LS_f\)) criterion does not apply. This theoretical criterion is finally illustrated by examples for \(p=3\) and compared to Labute and Vogel’s works (Labute, J Reine Angew Math 596:155–182, 2006 et Vogel, Circular sets of primes of imaginary quadratic number fields, 2006).