Campana points, Vojta's conjecture, and level structures on semistable abelian varieties

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.

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