Modular Varieties Over Function Fields and Arithmetic Applications

函数域和算术应用的模块化品种

• 批准号: 0801208
• 负责人: Mihran Papikian
• 金额: $ 11.42万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801208&HistoricalAwards=false
• 关键词: Modular; Varieties; Over; Function; Fields

This project has two main objectives. The first objective is to study the asymptotic behavior of the cohomology groups of modular varieties and of the number of rational points on such varieties over finite fields as the level varies. It will establish a rather important property of modular varieties, namely that the modular varieties over appropriately chosen finite fields provide examples of varieties with many rational points compared to their Betti numbers. This will shed some light on a difficult and largely open question of the optimality of the Weil-Deligne bound on the number of rational points on varieties over finite fields. This result might also find applications in coding theory. The second objective is to develop arithmetic tools for the study of modular curves over function fields and to use the parametrizations by modular curves to study non-isotrivial elliptic curves over function fields.Modular varieties introduced by Shimura and their function field counterparts introduced by Drinfeld play an absolutely central role in current algebraic number theory. One of the main applications of these varieties is to the Langlands conjectures, since the cohomology groups of these varieties provide a link between Galois and automorphic representations. The project builds on and extends the results obtained by the PI earlier. The methods which will be employed come from the ideas in the proofs of the Langlands conjectures over function fields, rigid-analytic geometry and representation theory over local fields.

该项目有两个主要目标。第一个目的是研究有限域上模簇的上同调群以及模簇上有理点个数随水平变化的渐近性态。它将建立模簇的一个相当重要的性质，即适当选择的有限域上的模簇提供了与它们的贝蒂数相比具有许多有理点的簇的例子。这将揭示一些困难的，很大程度上是开放的问题的最优性的韦尔-德利涅界的数量合理的点品种在有限领域。这一结果也可能在编码理论中找到应用。第二个目标是为研究函数域上的模曲线开发计算工具，并利用模曲线的参数化来研究函数域上的非等平凡椭圆曲线。由Shimura引入的模簇和由Drinfeld引入的函数域上的模簇在当前代数数论中起着绝对的核心作用。这些变种的主要应用之一是朗兰兹图，因为这些变种的上同调群提供了伽罗瓦和自守表示之间的联系。该项目建立在PI先前获得的结果的基础上并扩展了这些结果。将采用的方法来自的想法证明朗兰兹定理的功能领域，刚性解析几何和表示理论的地方领域。