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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Principally polarized ordinary abelian varieties over finite fields


Author: Everett W. Howe
Journal: Trans. Amer. Math. Soc. 347 (1995), 2361-2401
MSC: Primary 11G25; Secondary 11G10, 14K15
MathSciNet review: 1297531
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1297531-4
Article copyright: © Copyright 1995 American Mathematical Society