On the spread of mild ownership mild over imaginary quadratic extension of Q

In this work, we are interested in the pro-p groups , 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 . In favourable conditions, we show by computation that the group we study often satisfies a weak version () of Labute-Schmidt's criterion. Then, a theoretical criterion is established for proving mildness of some groups to which the () criterion does not apply. This theoretical criterion is finally illustrated by examples for 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).