Abramovich, DanVárilly-Alvarado, Anthony2018-06-282018-06-282018Abramovich, Dan and Várilly-Alvarado, Anthony. "Level structures on Abelian varieties, Kodaira dimensions, and Lang's conjecture." <i>Advances in Mathematics,</i> 329, (2018) Elsevier: 523-540. https://doi.org/10.1016/j.aim.2017.12.023.https://hdl.handle.net/1911/102314Assuming Lang's conjecture, we prove that for a prime p, number field K, and positive integer g, there is an integer r such that no principally polarized abelian variety A/K has full level-pr structure. To this end, we use a result of Zuo to prove that for each closed subvariety X in the moduli space Ag of principally polarized abelian varieties of dimension g, there exists a level mX such that the irreducible components of the preimage of X in Ag[m] are of general type for m>mX.engThis is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier.Journal articleAbelian varietiesModuli spacesBirational geometryRational pointshttps://doi.org/10.1016/j.aim.2017.12.023