Geometric questions in the theory of Shimura varieties and applications

本研究计划是关于Shimura簇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会影响数论中的计算方面，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，有一个概念的傅立叶级数的功能，这些品种，表示他们作为一个总和的简单谐波。此外，志村簇被赋予了所谓的特殊点，其特征在于在簇的许多对称性下被固定。函数的傅里叶展开式和它们在特殊点的值都将几何、代数数论和伽罗瓦表示联系起来。它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。对于一个特定类别的志村品种，所谓的GSpin志村品种，一个是由Borcherds的菲尔兹奖工作提供了一个杰出的功能集合。我们的目标是找到一个因式分解公式的数字所产生的值在特殊点。这将促进我们对志村变种的理解，并可能揭示数论中的开放问题，如斯塔克猜想。 第二个方向是关于一个不同的类志村品种，所谓的单一志村品种。在我们最近的工作之后，我们的目标是研究某些微分算子作用于这些空间上的函数。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。这是我们要证明的一种联系。 第三个方向是志村变种的动力学过程的研究。志村簇上的一个点在其对称性下的图像（Hecke算子）与数论和刚性分析中的问题有关，刚性分析类似于复变函数的分析，但与广义数系统有关。我们希望将我们对1维Shimura簇所做的工作推广到任意维。这将推进我们在数论和p-adic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.