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THE DISTRIBUTION OF TORSION SUBSCHEMES OF ABELIAN VARIETIES
Published online by Cambridge University Press: 15 December 2014
Abstract
We consider the distribution of 
$p$-power group schemes among the torsion of abelian varieties over finite fields of characteristic 
$p$, as follows. Fix natural numbers 
$g$ and 
$n$, and let 
${\it\xi}$ be a non-supersingular principally quasipolarized Barsotti–Tate group of level 
$n$. We classify the 
$\mathbb{F}_{q}$-rational forms 
${\it\xi}^{{\it\alpha}}$ of 
${\it\xi}$. Among all principally polarized abelian varieties 
$X/\mathbb{F}_{q}$ of dimension 
$g$ with 
$X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$, we compute the frequency with which 
$X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$. The error in our estimate is bounded by 
$D/\sqrt{q}$, where 
$D$ depends on 
$g$, 
$n$, and 
$p$, but not on 
${\it\xi}$.
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 4 , October 2016 , pp. 693 - 710
- Copyright
- © Cambridge University Press 2014
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