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Kernels of polarizations of abelian varieties over finite fields. (English) Zbl 0911.11031
Summary: Suppose \({\mathcal C}\) is an isogeny class of abelian varieties over a finite field \(k\). In this paper the author gives a partial answer to the question of which finite group schemes over \(k\) occur as kernels of polarizations of varieties of \({\mathcal C}\). He shows that there is an element \(I_{\mathcal C}\) of a finite two-torsion group that determines which Jordan-Hölder isomorphism classes of finite commutative group schemes over \(k\) contain kernels of polarizations. He indicates how the two-torsion group can be computed from the characteristic polynomial of the Frobenius endomorphism of the varieties in \({\mathcal C}\), and he gives some relatively weak sufficient conditions for the element \(I_{\mathcal C}\) to be zero. Using these conditions, it is shown that every isogeny class of simple odd-dimensional abelian varieties over a finite field contains a principally polarized variety. As a step in the proofs of these theorems, he proves that if \(K\) is a CM-field and \(A\) is a central simple \(K\)-algebra with an involution of the second kind, then every totally positive real element of \(K\) is the reduced norm of a positive symmetric element of \(A\).

MSC:
11G25 Varieties over finite and local fields
14K15 Arithmetic ground fields for abelian varieties
14G15 Finite ground fields in algebraic geometry
14K02 Isogeny
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