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

TL;DR

Under Lang's conjecture, the paper proves a uniform bound on the level structure of principally polarized abelian varieties over number fields, linking geometric properties of moduli spaces to arithmetic constraints.

Contribution

It establishes a uniform bound on the level structure of abelian varieties assuming Lang's conjecture, using geometric properties of moduli spaces and a result by Zuo.

Findings

Existence of a uniform bound on level structures under Lang's conjecture.

Irreducible components of preimages in moduli spaces are of general type for large levels.

Connection between geometric properties of moduli spaces and arithmetic restrictions.

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 $m > m_X$.