Filtrations, arithmetic and explicit examples in an equivariant context

Abstract: Pro-$p$ groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences $(a_n)$ and $(c_n)$, closely related to a special filtration of a finitely generated pro-p group $G$, called the Zassenhaus filtration. These sequences give the cardinality of $G$, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer $n$ such that $a_n=0$ (or $c_n$ has a polynomial growth) if and only if $G$ is a Lie group over $p$-adic fields.

In 2016, Minac, Rogelstad and Tan inferred an explicit relation between $a_n$ and $c_n$. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an equivariant context: when the automorphism group of $G$ admits a subgroup of order a prime $q$ dividing $p-1$.

In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.

combinatoricsnumber theory

Audience: researchers in the topic

Lethbridge number theory and combinatorics seminar