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

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2018
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Elsevier
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Assuming 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.

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Abramovich, Dan and Várilly-Alvarado, Anthony. "Level structures on Abelian varieties, Kodaira dimensions, and Lang's conjecture." Advances in Mathematics, 329, (2018) Elsevier: 523-540. https://doi.org/10.1016/j.aim.2017.12.023.

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