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 $m > m_X$.

17 pages. References to new work of Brunebarbe added; discussion of implications arising from Lang's geometric conjecture suppressed in light of Brunebarbe's new results. Section 4 recast in more general terms; see Proposition 4.3 and Theorem 1.13