On projective spaces over local fields

Let \(\mathcal{P}\) be the set of points of a finite-dimensional projective space over a local field \(F\), endowed with the topology \(\tau\) naturally induced from the canonical topology of \(F\). Intuitively, continuous incidence abelian group structures on \(\mathcal{P}\) are abelian group struc...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Authors: Cangiotti, Nicolò, Linzi, Alessandro
Format: Article
Language:English
Subjects:
Group theory
Isomorphism
Topology
Online Access:Get full text
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Summary:Let \(\mathcal{P}\) be the set of points of a finite-dimensional projective space over a local field \(F\), endowed with the topology \(\tau\) naturally induced from the canonical topology of \(F\). Intuitively, continuous incidence abelian group structures on \(\mathcal{P}\) are abelian group structures on \(\mathcal{P}\) preserving both the topology \(\tau\) and the incidence of lines with points. We show that the real projective line is the only finite-dimensional projective space over an Archimedean local field which admits a continuous incidence abelian group structure. The latter is unique up to isomorphism of topological groups. In contrast, in the non-Archimedean case we construct continuous incidence abelian group structures in any dimension \(n \in \mathbb{N}\). We show that if \(n>1\) and the characteristic of \(F\) does not divide \(n+1\), then there are finitely many possibilities up to topological isomorphism and, in any case, countably many.
ISSN:2331-8422