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

@article{Abramovich2016CampanaPV, title={Campana points, Vojta's conjecture, and level structures on semistable abelian varieties}, author={Dan Abramovich and Anthony V{\'a}rilly-Alvarado}, journal={arXiv: Number Theory}, year={2016} }

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… Expand

#### 4 Citations

Campana points of bounded height on vector group compactifications

- Mathematics
- 2019

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana… Expand

Campana points and powerful values of norm forms

- Mathematics
- 2020

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we… Expand

Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

- Mathematics
- 2021

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational… Expand

#### References

SHOWING 1-9 OF 9 REFERENCES

Level structures on abelian varieties and Vojta’s conjecture

- Mathematics
- Compositio Mathematica
- 2017

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any… Expand

Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture

- Mathematics
- 2016

Assuming Lang's conjecture, we prove that for a fixed prime $p$, number field $K$, and positive integer $g$, there is an integer $r$ such that no principally polarized abelian variety $A/K$ of… Expand

Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques

- Mathematics
- 2005

We show the hyperbolic and complex function field versions of Lang’s conjecture for smooth projective surfaces S having a fibration f:S→C with orbifold base an orbifold curve of general type. The… Expand

Birational geometry for number theorists

- Mathematics
- 2007

Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006.
Does not include: rationality problems, techniques of the minimal model problem and much of… Expand

A more general abc conjecture

- Mathematics
- 1998

This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterl\'e and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new… Expand

Hom-stacks and restriction of scalars

- Mathematics
- 2006

Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack HomS(X ,Y) classifying morphisms between X and Y.… Expand

Degeneration of Abelian varieties

- Mathematics
- 1990

I. Preliminaries.- II. Degeneration of Polarized Abelian Varieties.- III. Mumford's Construction.- IV. Toroidal Compactification of Ag.- V. Modular Forms and the Minimal Compactification.- VI.… Expand

Diophantine geometry, volume 201 of Graduate Texts in Mathematics

- 2000

Diophantine geometry, volume 201 of Graduate Texts in Mathematics

- 2000