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 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2016
- DOI:
- 10.48550/arXiv.1608.05651
- arXiv:
- arXiv:1608.05651
- Bibcode:
- 2016arXiv160805651A
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Algebraic Geometry;
- Primary 11J97;
- 14K10;
- Secondary 14K15;
- 11G18
- E-Print:
- 7 pages