  We use the theta lift to study the multiplicity with which certain
automorphic representations of cohomological type occur in a family of
congruence covers of an arithmetic manifold. When the family of covers is a
so-called `p-adic congruence tower' we obtain sharp asymptotics for the number
of representations which occur as lifts. When combined with theorems on the
surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson and
Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and
Xue.
