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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737241
• 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.

这项研究计划涉及志村簇理论中的几何问题及其应用，其预期影响是在代数几何、数论和动力系统领域，尽管有一些对技术的直接影响，但它主要是理论研究。这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，这些簇上的函数有一个傅里叶级数的概念，将它们表示为简单的总和此外，志村簇被赋予了所谓的特殊点，其特征是在簇的许多对称性下都是固定的，函数的傅立叶展开及其在特殊点上的值都连接着几何、代数数论和数学。伽罗瓦表示 他们的研究是数论的一个核心主题，我们的建议涉及到一类特定的 Shimura 变种，即所谓的 GSpin Shimura 变种，其中一个是由 Borcherds 的 Fields 奖章工作提供的。我们的目标是找到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 Shimura 簇，即所谓的酉 Shimura 簇，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。理论中有一个非常有趣的对应物伽罗瓦表示。这是我们要证明的联系。第三个方向是研究 Shimura 簇的动力学过程。Shimura 簇上的点在其对称性（赫克算子）下的图像与数论和刚性问题有关。分析，类似于复杂函数的分析，但使用广义数系统进行，我们希望将我们对一维 Shimura 簇所做的工作扩展到任意维度，这将提高我们在数论和 p-adic 动力系统方面的知识。应用到上面讨论的特殊值问题。