Mild pro-p-groups with 4 generators

Rédei symbols and arithmetical mild pro-2-groups

Generalizing results of Morishita and Vogel, an explicit description of the triple Massey product for the Galois group $G_S(2)$ of the maximal 2-extension of $\mathbb{Q}$ unramified outside a finite set of prime numbers $S$ containing 2 is given in terms of Rédei symbols. We show that mild pro-2-groups with Zassenhaus invariant 3 occur as Galois groups of the form $G_S(2)$. Furthermore, a non-analytic mild fab pro-2-group having only 3 generators is constructed .

Generators for Cubic Surfaces with two Skew Lines over Finite Fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.

温和的p-群与有序独异点

Mild Pro-$p$ Groups and Ordered Monoids

本文提出一个有限呈现实现p-群温和性的判据，推广了Schmidt和Gärtner的结果，并涵盖了Labute的非奇异电路判据，同时与Quadrelli等人关于p-右角Artin群温和性的三角条件相联系。

Mild pro-p groups and the Koszulity conjectures

Let $p$ be a prime, and $\mathbb{F}_p$ the field with $p$ elements. We prove that if $G$ is a mild pro-$p$ group with quadratic $\mathbb{F}_p$-cohomology algebra $H^\bullet(G,\mathbb{F}_p)$, then the algebras $H^\bullet(G,\mathbb{F}_p)$ and $\mathrm{gr}\mathbb{F}_p[\![G]\!]$ - the latter being induced by the quotients of consecutive terms of the $p$-Zassenhaus filtration of $G$ - are both Koszul, and they are quadratically dual to each other. Consequently, if the maximal pro-$p$ Galois group of a field is mild, then Positselski's and Weigel's Koszulity conjectures hold true for such a field.

Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group

Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of places of $k$ distinct from $p$-adic places. We give sufficient conditions for the Galois group $G_S$, of the maximal pro-$p$-extension of $k$ which is unramified outside $S$, to be \textit{mild}, hence of cohomological dimension $2$.