Geometric questions in the theory of Shimura varieties and applications

本研究方案涉及Shimura变种理论及其应用中的几何问题。预期的影响在代数几何、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，尽管其中一些问题可能会影响数论的计算方面，特别是在数学密码学的背景下。*Shimura变种是高度对称的代数变种。例如，对于这些变体上的函数，有一个傅里叶级数的概念，将它们表示为简单调和的和。此外，Shimura品种具有所谓的特殊性，其特征是固定在品种的许多对称性下。函数的傅里叶展开式及其在特定点的值都将几何、代数论和伽罗瓦表示联系在一起。他们的研究是数论的中心课题。*我们的建议涉及几个研究方向。对于特定类别的下村品种，即所谓的GSpin Shimura品种，博尔切德的菲尔兹奖章工作提供了一个具有杰出功能的品种。我们的目标是为由其在特定点的值产生的数找到一个因式分解公式。这将促进我们对下村变种的理解，并可能有助于阐明数论中的公开问题，如斯塔克猜想。*第二个方向涉及不同类别的下村品种，即所谓的单一下村品种。在我们最近工作的基础上，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。期望这些算子的作用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一种联系。*第三个方向是研究Shimura品种的动力过程。Shimura簇上的点在它的对称性(Hecke算子)下的映象与数论和刚性分析中的问题有关，刚性分析类似于复杂函数的分析，但是用广义数系来完成的。我们希望将我们对一维Shimura变种所做的工作扩展到任意维度。这将促进我们对数论和p-进动力系统的认识，并将对上面讨论的特值问题有应用。**

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. **