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

@article{Abramovich2016LevelSO,
  title={Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture},
  author={Dan Abramovich and Anthony V{\'a}rilly-Alvarado},
  journal={arXiv: Algebraic Geometry},
  year={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 dimension $g$ has full level $p^r$ structure. To this end, we use a result of Zuo to prove that for each closed subvariety $X$ in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$, there exists a level $m_X$ such that the irreducible components of the preimage of… Expand
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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 anyExpand
ABELIAN $n$ -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$ . In 2008, Skorobogatov and Zarhin showed that the Brauer groupExpand
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