  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.
