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Level structures on abelian varieties and Vojta’s conjecture
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 2 / February 2017
- Published online by Cambridge University Press:
- 06 February 2017, pp. 373-394
- Print publication:
- February 2017
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- Article
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Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field
$K$ and a 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.
