3 problems

Positselski's pro-$\ell$ Bogomolov conjecture. The maximal pro-$\ell$ Galois group of $\sqrt[\ell^\infty]{\mathbb{K}}$ is a free pro-$\ell$ group.

Let $\ell$ be a prime, and let $K$ and $L$ be any fields as in Theorem 2 of the source paper. Write $\mathcal{G}_{K}(\ell)$ for the relevant pro-$\ell$ Galois group. Pro-$\ell$ Gal…

Let $F$ be a field of characteristic different from $l$, set … and let $E$ be a finite extension of $F$. Denote by $K^{\mathrm M}_2(E)/l$ and $K^{\mathrm M}_2(KE)/l$ the degree-two…