$\mathbf{F}_p$-representations over $p$-fields

TL;DR

This paper classifies irreducible $F_p$-representations of certain finite and profinite groups related to local fields using the method of little groups, providing a comprehensive understanding of their structure.

Contribution

It introduces a classification of irreducible $F_p$-representations over $p$-fields, extending to the automorphism groups of maximal Galois extensions of local fields.

Findings

Classification of irreducible $F_p$-representations of $G=T imes_q Sigma$

Complete description of irreducible continuous $F_p$-representations of Galois groups of local fields

Application of the method of little groups to local field automorphism groups

Abstract

Let $p$ be a prime, $k$ a finite extension of $\mathbf{F}_p$ of cardinal $q$, $l$ a finite extension of $k$ of group $\Sigma=\mathrm{Gal}(l|k)$, and $T$ a subgroup of $l^\times$. Using the method of "little groups", we classify irreducible $\mathbf{F}_p$-representations of the group $G=T\times_q\Sigma$, the twisted product of $\Sigma$ with the $\Sigma$-module $T$. We then use these results to classify irreducible continuous $\mathbf{F}_p$-representations of the profinite group of automorphisms $\mathrm{Gal}(\tilde K|K)$ of the maximal galoisian extension $\tilde K$ of a local field $K$ with residue field $k$.