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

TL;DR

The paper proposes a new conjecture inspired by Campana's ideas, linking rational points' distribution on varieties to existing conjectures, and proves a finiteness result for level structures on semistable abelian varieties assuming this conjecture.

Contribution

It introduces a qualitative conjecture connecting rational points and level structures, and proves a finiteness theorem for abelian varieties assuming the conjecture.

Findings

Conjecture interpolates between Lang's and Vojta's conjectures.

Assuming the conjecture, finiteness of certain level structures is established.

The conjecture follows from Vojta's conjecture.

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.