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Level structures on abelian varieties and Vojta’s conjecture

Part of: Arithmetic algebraic geometry Abelian varieties and schemes

Published online by Cambridge University Press:  06 February 2017

Dan Abramovich  and
Anthony Várilly-Alvarado
Dan Abramovich
Affiliation:
Department of Mathematics, Box 1917, Brown University, Providence, RI 02912, USA email abrmovic@math.brown.edu
Anthony Várilly-Alvarado
Affiliation:
Department of Mathematics, MS 136, Rice University, 6100 S. Main St., Houston, TX 77005, USA email av15@rice.edu
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Abstract

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.

Keywords

abelian varietiesrational pointsalgebraic stackstoroidal compactifications

MSC classification

Primary: 14K10: Algebraic moduli, classification 14K15: Arithmetic ground fields
Secondary: 11G35: Varieties over global fields 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 2 , February 2017 , pp. 373 - 394
Copyright
© The Authors 2017 

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