Exponential generation and largeness for compact \(p\)-adic Lie groups
type: Item
instance of: scholarly article
title: Exponential generation and largeness for compact \(p\)-adic Lie groups
zbMATH Open document ID: 1219.22011
published in: Algebra \& Number Theory
publication date: 2011-04-19T00:00:00Z
review text: Let \(\mathcal G\) be a closed subgroup of the compact group \(\text{GL}_n(\mathbb Z_p)\). Here \(p\) is a prime that is sufficiently large in relation to the positive integer \(n\). Let \(G\) be the Zariski closure of \(\mathcal G\) in \({\text{GL}_n}_{/{\mathbb Q_p}}\). In the article under review, the author uses scheme-theoretic techniques to obtain a criterion (when the radical of \(G\) is unipotent) in terms of the reduction of \(\mathcal G\) modulo \(p\) that implies that the index \([\text{GL}_n(\mathbb Z_p)\cap G(\mathbb Q_p) : \mathcal G]\) is bounded by a constant depending only on \(n\). More precisely, the author introduces the Nori dimension of \(\mathcal G\). If \(\Gamma\) is a subgroup of \(\text{GL}_n(\mathbb F_p)\) [\textit{M. V. Nori}, Invent. Math. 88, 257--275 (1987; Zbl 0632.20030)], defined the algebraic group \(G_{N(\Gamma)}\) generated by the one-parameter subgroups \(t\mapsto \text{exp}(t \log u)\), where \(u\) ranges over all elements of order \(p\) in \(\Gamma\). The Nori dimension of \(\Gamma\) is defined to be \(\dim G_{N(\Gamma)}\). The author defines the Nori dimension of \(\mathcal G\) to be that of the image of \(\mathcal G\) in \(\text{GL}_n(\mathbb F_p)\). The main result of this paper is that for \(p\) sufficiently large, the Nori dimension of \(\mathcal G\) is at most \(\dim G\), and in the case of equality, (1) \(\mathcal G\) is open in \(G(\mathbb Q_p)\), (2) \(G/G^\circ\) has order prime to \(p\) and has a normal abelian subgroup of index bounded by a constant depending only on \(n\), (3) if the radical of \(G\) is unipotent, \([\text{GL}_n(\mathbb Z_p)\cap G(\mathbb Q_p) : \mathcal G]\) is bounded by a constant depending only on \(n\). This mirrors a bounded-index result due to Nori for subgroups of \(\text{GL}_n(\mathbb F_p)\) generated by elements of order \(p\). It should have applications to monodromy of projective varieties over function fields of positive characteristic.
reviewed by: Joshua Lansky
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5880072
zbMATH Keywords: exponentially generated
zbMATH Keywords: Nori's theorem
zbMATH Keywords: p-adic Lie group
Wikidata QID: Q115231083
author: Michael Larsen
MaRDI profile type: Publication
full work available at URL: ant.2010.4.1029
OpenAlex ID: W1990158188
DOI: 10.2140/ANT.2010.4.1029
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description: scientific article; zbMATH DE number 5880072