  We introduce a qualitative conjecture, in the spirit of Campana, to the
effect that certain subsets of rational points on a variety over a number
field, or a Deligne-Mumford stack over a ring of S-integers, cannot be Zariski
dense. The conjecture interpolates, in a way that we make precise, between
Lang's conjecture for rational points on varieties of general type over number
fields, and the conjecture of Lang and Vojta that asserts that S-integral
points on a variety of logarithmic general type are not Zariski-dense. We show
our conjecture follows from Vojta's conjecture. Assuming our conjecture, we
prove the following theorem: Fix a number field K, a finite set S of places of
K containing the infinite places, and a positive integer g. Then there is an
integer m_0 such that, for any m > m_0, no principally polarized abelian
variety A/K of dimension g with semistable reduction outside of S has full
level-m structure.
