8 problems

Let $P$ be a pro-$p$ group, and write … Let $\varphi^2:H_{\operatorname{cont}}^{2}(P)\to H_{\operatorname{disc}}^{2}(P)$ be the comparison map. Prasad's conjecture. For every $p$-a…

Let $K$ be a number field and let $L/K$ be an everywhere unramified Galois pro-$p$ extension. A pro-$p$ group is uniform if it is a uniform pro-$p$ group in the sense of Lazard. Un…

Let $K$ be a number field. An unramified pro-$p$-extension is a Galois pro-$p$ extension of $K$ unramified at every place. Weak Fontaine–Mazur conjecture. Every unramified pro-$p$-…

Classification conjecture. $G$ is strongly hereditarily self-similar of index $p$ if and only if $G$ is isomorphic to

Index invariance conjecture. If $H\simeq K$, then

Self-similarity conjecture. For all primes $p\geqslant 5$, no open subgroup of $SL_1^1(\Delta_p)$ is self-similar. For all primes $p\geqslant 3$, no open subgroup of…

Let $\mathcal{G}=(G,I)$ be a uniformly powerful pro-$p$ group that is full as a profinite group. For each $n$, let $P_n(G)$ denote the corresponding term in the lower $p$-series, a…

Let $p\geq 3$ and let $G$ be a torsion-free pro-$p$ group of finite rank. Write $d(G)$ for the minimal number of topological generators of $G$, and let $\dim(G)$ denote its dimensi…