Algebraic geometry over finite fields

有限域上的代数几何

• 批准号: 0600425
• 负责人: Aise de Jong
• 金额: $ 14.53万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600425&HistoricalAwards=false
• 关键词: Algebraic; geometry; over; finite; fields

The project will focus on three related areas of research in algebraicgeometry over finite fields. First of all we intend to attack Artin'sconjecture that the Brauer group of a projective surface over a finitefield is finite. Here do Jong will study especially elliptic surfaces andsurfaces with ample cotangent bundle. Secondly, de Jong will study thearithmetic fundamental groups of curves over finite fields. Here themain questions are concerning the dynamic of the Verschiebung on themoduli spaces of bundles, the growth of the fundamental group, and thedistribution of the Frobenius elements in the fundamental group. Andthirdly, in an ongoing collaboration with others (Starr, Hassett,Tschinkel, et al) de jong will study the geometry of moduli spaces ofrational curves on higher dimensional varieties. Here ofparticular interested are in finding applications to other areas ofresearch. Meanwhile, with the goal of making it easier for graduatestudents and newcomers to work on these problems, de jong intends to run aweb-site where collaborative development of introductory texts on thetopics is done.The area of mathematics that this proposal finds itself in has seen alot of progress in the last decade. Nonetheless there are manyimportant problems outstanding. Perhaps the most exiting of these isArtin's conjecture mentioned above. Among other things it implies theBirch-Swinnerton-Dyer conjecture for elliptic curves over functionfields of curves over finite fields. An example of such a field is thefield of rational functions in one variable over a finite field. Manyfamous classical number theoretical questions have their analogue forsuch function fields, and a number of these, such as the RiemannHypothesis, have been shown to be true in the function field case. Thereason for this is that people can study curves and more generally doalgebraic geometry over finite fields, to prove the conjectures. Inthis project we will study geometric approaches to Artin's conjecture,for example by thinking about moduli of vector bundles over curves andsurfaces over finite fields.

该项目将重点介绍有限领域的代数测量学研究的三个相关领域。首先，我们打算攻击Artin' -Sconjecture，即有限端口上的投影表面的Brauer组是有限的。 Jong将在这里特别研究椭圆形的表面和曲面，并带有足够的cotangent束。其次，de jong将研究有限场上的曲线基本曲线基本组。在这里，他们的问题是关于贝奇邦在束的束空间，基本群体的增长以及基本组中Frobenius元素的动态上的动态。最后，在与其他人的持续合作中（Starr，Hassett，Tschinkel等）De Jong将研究较高尺寸品种的模量空间的几何形状。在这里，有兴趣的特定是在研究其他领域的应用程序。同时，为了使毕业生和新移民更容易解决这些问题，de jong打算运行aweb site，其中完成了关于thetopics的介绍性文本的协作开发。该提案在过去十年中发现了很多进步的数学领域。尽管如此，还有许多重要的问题。也许上面提到的这些伊萨尔汀的猜想中最大的退出。除其他事项外，它暗示了Birch-Swinnerton-Dyer猜想，用于椭圆形曲线在有限场上的曲线功能场上。这样一个字段的一个示例是一个有限字段的一个变量中有理函数的字段。许多经典的经典理论问题具有它们的模拟函数函数字段，其中许多（例如riemannhypothesis）在函数字段案例中已被证明是正确的。因此，人们可以在有限田地上研究曲线，并且更通常是多层的几何形状，以证明猜想。在这个项目中，我们将研究Artin猜想的几何方法，例如，通过思考有限领域的曲线和曲面上的向量束模量。