By arithmeticity and superrigidity, a commensurability class of lattices in a higher
rank Lie group is defined by a unique algebraic group over a unique number subfield
of
or
. We
prove an adelic version of superrigidity which implies that two such commensurability
classes define the same profinite commensurability class if and only if the algebraic
groups are adelically isomorphic. We discuss noteworthy consequences on profinite
rigidity questions.