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

Abstract

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 \(X\) in \(\mathcal{A}_g^{[m]}\) are of general type for \(m > m_X\). We give variants of our main result, e.g., after additionally assuming Lang's geometric conjecture.