The distribution of torsion subschemes of abelian varieties

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 $\xi$ be a non-supersingular principally quasipolarized Barsotti-Tate group of level n. We classify the F_q-rational forms $\xi^\alpha$ of $\xi$. Among all principally polarized abelian varieties X/F_q of dimension g with p^n-torsion geometrically isomorphic to \xi, we compute the frequency with which X[p^n] is isomorphic to $\xi^\alpha$. The error in our estimate is bounded by $D/\sqrt{q}$ where D depends on g, n and p, but not on $\xi$.