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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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