Geometric questions in the theory of Shimura varieties and applications

志村品种理论中的几何问题及应用

• 批准号: RGPIN-2019-03909
• 负责人: Goren, Eyal
• 金额: $ 2.77万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=687125
• 关键词: Geometric; questions; theory; Shimura; varieties

This research proposal is concerned with geometric questions in the theory of Shimura varieties and their applications. The expected impact is within the fields of algebraic geometry, number theory and dynamical systems. It is mostly theoretical research with little direct impact on technology, although some of the questions could have impact on computational aspects in number theory in particular in the context of mathematical cryptography.******Shimura varieties are algebraic varieties that are highly symmetric. For example, there is a notion of a Fourier series for functions on these varieties, expressing them as a sum of simple harmonics. Moreover, Shimura varieties are endowed with so-called special points, characterized by being fixed under many of the symmetries of the variety. Both the Fourier expansion of functions and their values at special points link between geometry, algebraic number theory and Galois representations. Their study is a central subject of number theory. ******Our proposal is concerned with several research directions. For a particular class of Shimura varieties, the so called GSpin Shimura varieties, one is provided by Borcherds' Fields medal work with a distinguished collection of functions. We are aiming to find a factorization formula for the numbers arising from their values at special points. This will advance our understanding of Shimura varieties and could shed light on open problems in number theory, such as Stark's conjecture. ******The second direction is concerned with a different class of Shimura varieties, the so called unitary Shimura varieties. Following on our recent work, we aim to study certain differential operators acting on functions on these spaces. It is expected that the action of these operators will have a very interesting counterpart in the theory of Galois representations. This is a connection we aim to prove. ******The third direction is the study of dynamical processes on Shimura varieties. The image of a point on a Shimura variety under its symmetries (Hecke operators) is related to questions in number theory and rigid analysis, the analogue of analysis of complex functions but done with generalized number systems. We wish to extend our work done for 1-dimensional Shimura varieties to arbitrary dimensions. This will advance our knowledge in number theory and p-adic dynamical systems and will have applications to the special values problem discussed above. **

本研究计划涉及志村品种理论及其应用中的几何问题。预期的影响是在代数几何、数论和动力系统领域。它主要是理论研究，对技术几乎没有直接影响，尽管有些问题可能会对数论中的计算方面产生影响，特别是在数学密码学的背景下。******志村簇是高度对称的代数簇。例如，这些变量的函数有一个傅里叶级数的概念，将它们表示为简单谐波的和。此外，志村品种被赋予了所谓的特殊点，其特征是在该品种的许多对称性下被固定。函数的傅立叶展开及其在特殊点上的值都将几何、代数数论和伽罗瓦表示联系在一起。他们的研究是数论的核心课题。 ******我们的提案涉及几个研究方向。对于一类特定的 Shimura 品种，即所谓的 GSpin Shimura 品种，Borcherds 的 Fields 奖章工作提供了一种具有卓越功能的集合。我们的目标是找到一个由特殊点的值产生的数字的因式分解公式。这将增进我们对志村簇的理解，并可能揭示数论中的开放问题，例如斯塔克猜想。 ******第二个方向涉及不同类别的志村品种，即所谓的单一志村品种。继我们最近的工作之后，我们的目标是研究作用于这些空间上的函数的某些微分算子。预计这些算子的作用将在伽罗瓦表示理论中有一个非常有趣的对应物。这是我们想要证明的联系。 ******第三个方向是志村品种动力学过程的研究。志村簇上的一点在其对称性（赫克算子）下的图像与数论和刚性分析中的问题有关，模拟复杂函数的分析，但使用广义数系统完成。我们希望将一维 Shimura 品种所做的工作扩展到任意维度。这将增进我们在数论和 p 进动力系统方面的知识，并将应用于上面讨论的特殊值问题。 **