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

@inproceedings{Abramovich2016LevelSO, title={Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture}, author={Dan Abramovich and Anthony V{\'a}rilly-Alvarado}, 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… CONTINUE READING

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