论文标题
有限田的阿贝尔品种极化的核
Kernels of polarizations of abelian varieties over finite fields
论文作者
论文摘要
假设$ c $是有限场上$ k $的阿贝利亚品种的同一基础类别。在本文中，我们对以下问题的问题给出了部分答案，该问题是$ k $以上的有限集团方案，因为$ c $的品种的核心成果。我们表明，有一个有限的两种扭矩组的元素$ i_c $确定了哪个约旦 - 霍尔德同构类别类别的有限交换组方案类别$ k $包含极化的内核。我们指出了如何从$ c $中品种的frobenius内态的特征多项式中计算两种扭矩，我们给出了一些相对较弱的元素$ i_c $的条件，为零。使用这些条件，我们表明，有限场上的每一个简单的奇数阿贝尔品种都包含一个主要极化的品种。作为这些定理的证明的一步，我们证明，如果$ k $是CM-field，而$ a $是一个中心简单的$ k $ - 代价，则具有第二种的含量，那么$ k $的每一个完全正面的真实元素都是$ a $的正面对称元素的缩小规范。
Suppose $C$ is an isogeny class of abelian varieties over a finite field $k$. In this paper we give a partial answer to the question of which finite group schemes over $k$ occur as kernels of polarizations of varieties in $C$. We show that there is an element $I_C$ of a finite two-torsion group that determines which Jordan-Hölder isomorphism classes of finite commutative group schemes over $k$ contain kernels of polarizations. We indicate how the two-torsion group can be computed from the characteristic polynomial of the Frobenius endomorphism of the varieties in $C$, and we give some relatively weak sufficient conditions for the element $I_C$ to be zero. Using these conditions, we show 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, we prove 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 $A$.
