## Principally polarized ordinary abelian varieties over finite fields.(English)Zbl 0859.14016

Let $$k$$ be a finite field of characteristic $$p$$. An abelian variety $$A$$ over $$k$$ is ordinary if the $$p$$-rank of its group of geometric $$p$$-torsion points in equal to its dimension. P. Deligne [“Variétés abéliennes ordinaires sur un corps funi”, Invent. Math. 8, 238-243 (1969; Zbl 0179.26201)] showed that the category of ordinary abelian varieties over $$k$$ is equivalent to a category whose objects are pairs $$(T,F)$$, where $$T$$ is a free finite-rank $$\mathbb{Z}$$-module and $$F$$ is an endomorphism of $$T$$ that satisfies certain conditions; if one chooses an embedding $$\varphi:W\to \mathbb{C}$$ of the Witt vectors over $$k$$ into the complex numbers, then the $$T$$ associated to a given $$A$$ is the first integral homology group of the complex abelian variety obtained via $$\varphi$$ from the Serre-Tate canonical lift of $$A$$, and the $$F$$ associated to $$A$$ is the endomorphism of $$T$$ obtained from the lift of the Frobenius of $$A/k$$. One can view Deligne’s category equivalence as a dictionary that translates concepts from the category of ordinary abelian varieties over $$k$$ to the category of these “Deligne modules” $$(T,F)$$.
In the first part of the present paper, we add a few entries to Deligne’s dictionary by defining polarizations of Deligne modules and by showing how the group scheme structure of the kernel of an isogeny of abelian varieties can be determined from the corresponding isogeny of Deligne modules.
In the second part of the paper we address the following questions: Given an isogeny class $${\mathcal C}$$ of ordinary abelian varieties over $$k$$, how can one tell which group schemes occur as the kernels of polarizations of varieties in $${\mathcal C}$$? And more specifically, how can one tell whether $${\mathcal C}$$ contains a principally polarized variety? These questions become more tractable when rephrased in terms of Deligne modules. We find that there is an element of an obstruction group that determines the finite group schemes that occur as kernels of polarizations of varieties in $${\mathcal C}$$, up to semi-simplification. We show how the group and the element can be calculated from the characteristic polynomial of Frobenius of the varieties in $${\mathcal C}$$, and using this result we show that every simple odd-dimensional ordinary abelian variety over a finite field is isogenous to a principally polarized variety. [This last result remains true even without the “ordinary” hypothesis; see the author, “Kernels of polarizations of abelian varieties over finite fields”, J. Algebr. Geom. 5, No. 3, 583-608 (1996).]
To demonstrate the use of our techniques, in the third part of the paper we determine the characteristic polynomials of Frobenius of the isogeny classes of two-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties [compare to H.-G. Rück, Compos. Math. 76, No. 3, 351-366 (1990; Zbl 0742.14037)], and we produce the Weil numbers of several simple four-dimensional isogeny classes that do not contain principally polarized varieties.

### MSC:

 14K02 Isogeny 14G15 Finite ground fields in algebraic geometry 11G10 Abelian varieties of dimension $$> 1$$ 11G25 Varieties over finite and local fields 14K15 Arithmetic ground fields for abelian varieties

### Citations:

Zbl 0179.26201; Zbl 0742.14037
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### References:

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