The Chabauty space of ℚp×

Bourquin, Antoine; Valette, Alain

doi:10.2140/involve.2021.14.89

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The Chabauty space of $\mathbb{Q}_p^\times$

Antoine Bourquin and Alain Valette

Vol. 14 (2021), No. 1, 89–102

DOI: 10.2140/involve.2021.14.89

Abstract

Let $𝒞 (G)$ denote the Chabauty space of closed subgroups of the locally compact group $G$ . We first prove that $𝒞 (ℚ_{p}^{\times})$ is a proper compactification of $ℕ$ , identified with the set $N$ of open subgroups with finite index. Then we identify the space $𝒞 (ℚ_{p}^{\times}) \ N$ up to homeomorphism; e.g., for $p = 2$ , it is the Cantor space on which two copies of $\bar{N}$ (the 1-point compactification of $ℕ$ ) are glued.

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Keywords

Chabauty space, locally compact group, $p$-adic group, topological space, proper compactification

Mathematical Subject Classification 2010

Primary: 22B05, 54H11

Milestones

Received: 4 November 2019

Revised: 11 July 2020

Accepted: 15 September 2020

Published: 4 March 2021

Communicated by Kenneth S. Berenhaut

Authors

Antoine Bourquin
Institut de Mathématiques Université de Neuchâtel Neuchâtel Switzerland

Alain Valette
Faculté des Sciences Institut de Mathématiques Université de Neuchâtel Neuchâtel Switzerland

Vol. 14, No. 1, 2021