Curves Over Finite Fields and Deligne's Conjectures

deJong 9970049我们提出研究有限域上与代数曲线相关的某些群。所讨论的群可以看作是有限域上的固定代数曲线的非分歧覆盖的自同构群，而有限域在整个讨论中也是固定的。当我们改变覆盖时，我们得到一个群系统，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曲线的几何。特别地，可以将曲线上的点与这个群中的某个共轭类相关联，该共轭类由所谓的Frobenius元素组成。我们想回答的问题是：这些Frobenius元素在群中的（相对）位置是什么？该项目的一般研究领域是算术代数几何。 粗略地说，代数指的是这样一个事实：就函数而言，我们主要是用多项式来工作的。我们通过将这些多项式中的几个等于零来定义几何对象。这样的对象称为代数集合或簇。事实证明，这些物体的几何结构非常丰富，特别是如果我们考虑高维和高阶的方程。当我们只考虑那些系数为整数（或有理数）的多项式时，算术方面就开始起作用了。 这些对象在数论、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.