Kernels of polarizations of abelian varieties over finite fields.

*(English)*Zbl 0911.11031Summary: 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-HĂ¶lder 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 |