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

摘要

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-p(r) structure. To this end, we use a result of Zuo to prove that for each closed subvariety X in the moduli space 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 X in A(g)([m]) are of general type for m m(X). (C) 2018 Elsevier Inc. All rights reserved.