  Arithmetic quotients are quotients of bounded symmetric domains by arithmetic
groups, and modular subvarieties of arithmetic quotients are themselves
arithmetic quotients of lower dimension which live on arithmetic quotients, by
an embedding induced from an inclusion of groups of hermitian type. We show the
existence of such modular subvarieties, drawing on earlier work of the author.
If $\Gamma$ is a fixed arithmetic subgroup, maximal in some sense, then we
introduce the notion of ``$\Gamma$-integral symmetric'' subgroups, which in
turn defines a notion of ``integral modular subvarieties'', and we show that
there are finitely many such on an (isotropic, i.e, non-compact) arithmetic
variety.
