The kernel n-unipotent conjecture for maximal pro-p Galois groups

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The Kernel n-Unipotent Conjecture for maximal pro-p Galois groups

   Let $F$ be a field containing a primitive $p$-th root of unity, and let $G=G_F(p)$ be the maximal pro-$p$ Galois group of $F$. For a pro-$p$ group $G$, the kernel $n$-unipotent property means that its $p$-Zassenhaus term satisfies

   $$G_{(n)}=\bigcap\ker\bigl(\rho\colon G\to {\mathbb U}_n({\mathbb F}_p)\bigr),$$

   where the intersection runs over all continuous homomorphisms. Let $n\geq 3$ be an integer. Kernel n-Unipotent Conjecture. The group $G$ has the kernel $n$-unipotent property. The text notes that this is known for $n=3$, for $n=3,4$ in the Demushkin case, and for all $n$ for rank-two Demushkin groups, but presents the general assertion as a conjecture.

   source: Jan Minac, Nguyen Duy Tan and Ido Efrat, “The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields”, arXiv:1312.2655 (2014).