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. **

该研究建议涉及Shimura品种理论中的几何问题及其应用。预期的影响是在代数几何，数量理论和动力学系统的领域内。这主要是理论研究，对技术的直接影响很小，尽管其中一些问题可能会影响数字理论的计算方面，尤其是在数学加密的背景下。****** shimura品种是高度对称性的代数品种。例如，在这些品种上的功能有一个傅立叶序列的概念，将它们表示为简单谐波的总和。此外，Shimura品种具有所谓的特殊点，其特征是固定在该品种的许多对称性下。函数的傅立叶扩展及其在特殊点几何，代数数理论和GALOIS表示之间的特殊点链接。他们的研究是数字理论的中心主题。 *****我们的提议与多个研究方向有关。对于特定类别的Shimura品种，所谓的Gspin Shimura品种，由Borcherds的田野奖章提供，具有杰出的功能。我们的目标是找到一个因其在特殊点上产生的数字的分解公式。这将提高我们对Shimura品种的理解，并可以阐明数字理论中的开放问题，例如Stark的猜想。 ******第二个方向与不同类别的Shimura品种有关，即所谓的单一Shimura品种。在最近的工作之后，我们旨在研究某些在这些空间上功能的差异操作员。可以预期，这些运营商的行动将在加洛伊斯表示理论中具有非常有趣的对应物。这是我们旨在证明的联系。 ******第三个方向是研究Shimura品种的动态过程。在其对称性下（Hecke运算符）下，Shimura品种上的点的图像与数字理论和刚性分析的问题，复杂函数分析的类似物，但使用通用数字系统进行。我们希望将为一维Shimura品种的工作扩展到任意维度。这将提高我们在数字理论和P-ADIC动力学系统方面的知识，并将应用于上面讨论的特殊价值问题。 **