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 $m > m_X$.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1601.02483
 Bibcode:
 2016arXiv160102483A
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 Primary 14K10;
 14K15;
 Secondary 11G18
 EPrint:
 17 pages. References to new work of Brunebarbe added