Geometric questions in the theory of Shimura varieties and applications

本研究计划涉及志村变量理论中的几何问题及其应用。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，尽管其中一些问题可能对数论的计算方面产生影响，特别是在数学密码学的背景下。志村变种是高度对称的代数变种。例如，对于这些变量上的函数，有一个傅里叶级数的概念，将它们表示为简谐波的和。此外，志村品种被赋予了所谓的特殊点，其特点是在品种的许多对称性下被固定。函数的傅里叶展开式及其在特殊点上的值与几何、代数数论和伽罗瓦表示法联系在一起。他们的研究是数论的中心课题。我们的提案涉及几个研究方向。对于一种特殊的志村品种，即所谓的GSpin志村品种，一个是由Borcherds' Fields奖章工作提供的具有杰出功能的集合。我们的目标是找到由它们在特殊点的值产生的数的因式分解公式。这将促进我们对志村变异的理解，并可能阐明数论中的开放问题，如斯塔克猜想。第二个方向涉及另一类志村变种，即所谓的酉志村变种。根据我们最近的工作，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。这是我们想要证明的联系。第三个方向是志村变异的动力学过程研究。Shimura变集上的点在其对称性（Hecke算子）下的像涉及到数论和刚性分析中的问题，这是复数函数分析的类似，但用广义数系统来完成。我们希望将一维志村变异的研究扩展到任意维度。这将提高我们在数论和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。

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.