Abelian Varieties over Finite Fields

2000 ◽  
pp. 63-65
Author(s):  
Serge Lang
Keyword(s):  
Finite Fields ◽  
Abelian Varieties

scholarly journals Counting Points on Curves and Abelian Varieties Over Finite Fields

2001 ◽  
Vol 32 (3) ◽  
pp. 171-189 ◽  
Author(s):  
Leonard M. Adleman ◽  
Ming-Deh Huang
Keyword(s):  
Finite Fields ◽  
Abelian Varieties ◽  
Counting Points

scholarly journals THE DISTRIBUTION OF TORSION SUBSCHEMES OF ABELIAN VARIETIES

2014 ◽  
Vol 15 (4) ◽  
pp. 693-710
Author(s):  
Jeffrey D. Achter
Keyword(s):  
Finite Fields ◽  
Abelian Varieties ◽  
Natural Numbers ◽  
Group Schemes ◽  
Tate Group ◽  
Power Group

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 ${\it\xi}$.

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