Curves Over Finite Fields and Deligne's Conjectures

我们建议研究与有限域上的代数曲线有关的某些群。所讨论的群可以看作是有限域上固定代数曲线的未分支覆盖的自同构群，它在整个讨论中也是固定的。当我们改变复盖时，我们得到了一个群系，称为曲线的代数基本群；这个概念是由Grothendieck引入的。代数基本群以一种迷人的方式混合了有限域的算术和所讨论的曲线的几何。具体地说，人们可以在曲线上指出这个群中的某个共轭类，它由所谓的Frobenius元素组成。我们想要回答的问题是：这些弗罗本尼乌斯元素在群中的(相对)位置是什么？该项目的总体研究领域是算术代数几何。粗略地说，代数指的是我们主要使用多项式来处理函数。我们通过几个这样的多项式来定义几何对象。这样的对象被称为代数集合或簇。事实证明，这些物体有令人惊讶的丰富的几何图形，特别是如果我们考虑更高维度和更高次数的方程。当我们只考虑那些以整数(或有理数)为系数的多项式时，算术方面就开始发挥作用了。这些物体在数论、几何和数据安全理论中有着广泛的应用。

deJong9970049We propose to study certain groups that are associated with algebraiccurves over finite fields. The groups in question can be viewed as theautomorphism groups of unramified coverings of a fixed algebraic curve over a finite field which is also fixed throughout the discussion. As we let vary the coverings we obtain a system of groups, which iscalled the algebraic fundamental group of the curve; this concept was introduced by Grothendieck. The algebraic fundamental group mixes ina fascinating way the arithmetic of the finite field and the geometryof the curve in question. In particular, one can associate to apoint on the curve a certain conjugacy class in this group, consistingof the so-called Frobenius elements. The question which we would liketo answer is: What is the (relative) position of these Frobenius elementsin the group? The general area of research of the project is Arithmetic AlgebraicGeometry. Roughly speaking, the Algebra refers to the fact that wework mainly with polynomials as far as functions are concerned.We define geometric objects byequating to zero a few of these polynomials. Such an object is called an algebraic set or a variety. It turns out that there is a surprisingly rich geometry of these objects, especially if we consider equations in higher dimensions and of higher degree. The arithmetical aspect comes into play when we consider only those polynomials which have integers(or rational numbers) as coefficients. These objects have wide-rangingapplications in number theory, geometry, and the theory of data security.