Level structures on abelian varieties and Vojta's conjecture

TL;DR

Under the assumption of Vojta's conjecture, the paper proves finiteness results for principally polarized abelian varieties with full level structures over a fixed number field, extending Vojta's conjecture to stacks.

Contribution

The authors develop a version of Vojta's conjecture for Deligne-Mumford stacks and use it to establish finiteness results for abelian varieties with large level structures.

Findings

Existence of a threshold m_0 beyond which no such abelian varieties exist

Development of Vojta's conjecture for stacks from schemes

Finiteness results conditional on Vojta's conjecture

Abstract

Assuming Vojta's conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and positive integer $g$, there is an integer $m_0$ such that for any $m > m_0$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta's conjecture for Deligne-Mumford stacks, which we deduce from Vojta's conjecture for schemes.