Principally Polarized Ordinary Abelian Varieties Over Finite Fields

Abstract

Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field $k$ to a category of ${\mathbf {Z}}$-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne’s category of ${\mathbf {Z}}$-modules. We use Deligne’s equivalence to characterize the finite group schemes over $k$ that occur as kernels of polarizations of ordinary abelian varieties in a given isogeny class over $k$. Our result shows that every isogeny class of simple odd-dimensional ordinary abelian varieties over a finite field contains a principally polarized variety. We use our result to completely characterize the Weil numbers of the isogeny classes of two-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties. We end by exhibiting the Weil numbers of several isogeny classes of absolutely simple four-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties.