Campana points, Vojta's conjecture, and level structures on semistable abelian varieties
Abstract
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 DeligneMumford stack over a ring of Sintegers, 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 Sintegral points on a variety of logarithmic general type are not Zariskidense. 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 levelm structure.
 Publication:

arXiv eprints
 Pub Date:
 August 2016
 arXiv:
 arXiv:1608.05651
 Bibcode:
 2016arXiv160805651A
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 Primary 11J97;
 14K10;
 Secondary 14K15;
 11G18
 EPrint:
 7 pages