论文标题
等级类别中主要两极化的阿贝尔品种分布的变化
Variations in the distribution of principally polarized abelian varieties among isogeny classes
论文作者
论文摘要
我们表明，对于给定简单的普通同学类别中的有限型领域，主要两极化的阿贝尔品种的数量，并且具有内态r $等于0，或者与$ r $相关的类别比例，高达某些小型计算因素。这类戒指包括与等级基类相关的CM Field $ K $的最大顺序（已知结果已知），以及Frobenius和Verschiebung的$ \ Mathbf {Z} $生成的订单$ r $。 对于后一个顺序，我们可以使用louboutin的结果来估计基础场的大小和等级类别的Frobenius角度的适当比率。 The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vladut on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields.
We show that for a large class of rings $R$, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring $R$ is equal either to 0, or to a ratio of class numbers associated to $R$, up to some small computable factors. This class of rings includes the maximal order of the CM field $K$ associated to the isogeny class (for which the result was already known), as well as the order $R$ generated over $\mathbf{Z}$ by Frobenius and Verschiebung. For this latter order, we can use results of Louboutin to estimate the appropriate ratio of class numbers in terms of the size of the base field and the Frobenius angles of the isogeny class. The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vladut on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields.
