Space filling curves over finite fields
Space filling curves over finite fields
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有限域上的空间填充曲线
DOI：
10.4310/mrl.1999.v6.n6.a2
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发表时间：
1999
影响因子：
1
通讯作者：
N. M. Katz
中科院分区：
文献类型：
--
作者：
N. M. Katz
在本文中，我们在有限域上构造曲线，在某种意义上，该曲线具有“很多”点，并对有限域上的曲线和阿贝尔簇的 zeta 函数给出一些应用。事实上，我们发现了引理 1 中给出的 A 中的曲线的基本构造，它穿过每个有理点，这是在 Drinfield-Vladut 界限意义上找到具有许多点的固定有限域上的生长亏格曲线的不成功尝试的一部分 [2]。沿着本笔记的思路应用这种构造的想法源于 1996 年 8 月与 Ofer Gabber 的一次对话，讨论有限域上的每个阿贝尔簇是否都是雅可比行列式的商，在此期间他临时构建了这一事实的证明。他的证明的一个变体出现在定理 11 中。我很高兴感谢他。
In this note, we construct curves over finite fields which have, in a certain sense, a “lot” of points, and give some applications to the zeta functions of curves and abelian varieties over finite fields. In fact, we found the basic construction, given in Lemma 1, of curves in A which go through every rational point, as part of an unsuccessful attempt to find curves of growing genus over a fixed finite field with lots of points in the sense of the Drinfield-Vladut bound [2]. The idea of applying that construction along the lines of this note grew out of an August 1996 conversation with Ofer Gabber about whether every abelian variety over a finite field was a quotient of a Jacobian, during which he constructed, on the fly, a proof of that fact. A variant of his proof appears here in Theorem 11. It is a pleasure to acknowledge my debt to him.