Profinite commensurability of S-arithmetic groups

TL;DR

This paper investigates how the profinite completion of S-arithmetic groups encodes information about the underlying algebraic group, number field, and set of places, revealing that certain key features are determined by the profinite class.

Contribution

It demonstrates that the profinite commensurability class of higher rank S-arithmetic groups determines the number field up to arithmetical equivalence and identifies places above unramified primes.

Findings

Profinite class determines the number field up to arithmetical equivalence.

Profinite class encodes information about places above unramified primes.

Applications to group invariants and profiniteness questions.

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.