Profinite commensurability of $S$-arithmetic groups
Volume 197 / 2021
Acta Arithmetica 197 (2021), 311-330
MSC: Primary 20G30; Secondary 20G25, 11F75.
DOI: 10.4064/aa200401-23-7
Published online: 11 November 2020
Abstract
Given an $S$-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places $S$ is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite commensurability class of a higher rank $S$-arithmetic group determines the number field up to arithmetical equivalence and the places in $S$ above unramified primes. We include applications to profiniteness questions of group invariants.
