Modular Varieties Over Function Fields and Arithmetic Applications

这个项目有两个主要目标。第一个目的是研究有限域上模变体的上同调群及其上有理点的个数随水平变化的渐近行为。它将建立模变体的一个相当重要的性质，即在适当选择的有限域上的模变体提供了与它们的Betti数相比具有许多有理点的变体的例子。这将对有限域上变异上有理点数目的Weil-Deligne界的最优性这一困难而又悬而未决的问题提供一些启示。这一结果也可能在编码理论中得到应用。第二个目标是开发研究函数场上模曲线的算法工具，并利用模曲线的参数化来研究函数场上的非等平凡椭圆曲线。Shimura引入的模变体和Drinfeld引入的函数场变体在当前的代数数论中占有绝对的中心地位。这些变体的主要应用之一是朗兰兹猜想，因为这些变体的上同调群提供了伽罗瓦和自同构表示之间的联系。该项目建立在PI先前获得的结果的基础上并对其进行扩展。所采用的方法来自于函数场上的朗兰兹猜想的证明、刚性解析几何和局部场上的表示理论。

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.