A p-Adic Riemann-Hilbert Correspondence and A p-Adic Theory of Mixed Hodge Modules

在这个项目中，研究者和他的合作者打算在复数上代数簇上的倒叶和D-模之间的Riemann-Hilbert对应的p-进环境下开发一个模拟。Hilbert对应推广了De Rham理论，并在给定的复代数簇(如在簇上的倒叶范畴中编码的那样)的拓扑和定义在簇上的微分算子系的行为(被编码为簇上的D-模；即，簇上的微分算子环层上的模簇)之间建立了深层次的联系。所提出的P-进制数类比将对P-进制数上的变种执行类似的功能。它将产生这样一个簇上的(目前猜想的)范畴“晶体倒叶”和(同样是猜想的)配备Frobenius算子的弱可容许滤子D-模范畴之间的等价性。晶体倒置叶轮的类别被认为携带着关于它所依附的种类的几何和算术信息，例如，它将是贝林森调节器理论的p进类比的自然成分。这给出了一些线索，表明了这样一类轮在丢番图方程组的局部分析中可以预期起到的重要作用。求解方程的问题是数学中最基本的问题之一，至少可以追溯到古希腊的数学家，如丢番图。他研究了求解整数方程的问题；这样的方程现在被称为丢番图方程。自从笛卡尔和费马的工作以来，人们已经认识到，几何为分析方程组提供了一个强大的工具，即使人们一开始从算术的角度对方程更感兴趣。因此，发展强大的几何工具对于丢番图方程理论的发展是重要的。在这个项目中，研究人员和他的合作者打算开发这样的工具，通过将通常所谓的阿基米德几何中的已知技术扩展到非阿基米德几何或p-ADYD几何的背景下。这种几何具有很强的算术色彩，为丢番图方程的分析提供了关键的几何设置，这些技术有望在分析中产生几个新的发展。这些发展很重要，不仅因为它们丰富了数学传统的核心部分之一，还因为丢番图方程理论与离散过程理论，特别是与码理论有着深刻的相互联系，因此丢番图方程理论的进步有望带来这些领域的进步。

Abstract In this project the investigator and his collaborator intend to develop an analogue in the p-adic setting of the Riemann-Hilbert correspondence between perverse sheaves and D-modules on algebraic varieties over the complex numbers. The Hilbert correspondence generalizes De Rham theory, and establishes a deep connection between the topology of a given complex algebraic variety (as encoded in the category of perverse sheaves on the variety) and the behavior of systems of differential operators defined on the variety (which are encoded as D-modules on the variety; that is, as sheaves of modules over the sheaf of rings of differential operators on the variety). The proposed p-adic analogue would perform a similar function for varieties over the p-adic numbers. It would yield an equivalence between the (currently conjectural) category of ``crystalline perverse sheaves'' on such a variety, and the (again conjectural) category of weakly admissible filtered D-modules equipped with a Frobenius operator. The category of crystalline perverse sheaves is believed to carry both geometric and also arithmetic information about the variety to which it is attached, and would for example be a natural ingredient in a p-adic analogue of Beilinson's theory of regulators. This gives some hint of the important role that such a category of sheaves can be expected to play in the local analysis at p of systems of Diophantine equations.The problem of solving equations is one of the most basic in mathematics, going back at least to the mathematicians of ancient Greece, such as Diophantus. He studied the problem of solving equations in whole numbers; such equations are now known as Diophantine equations. Since the work of Descartes and Fermat, it has been understood that geometry provides a powerful tool for analyzing systems of equations, even if one is at first more interested in the equations from an arithmetic point of view. For this reason, the development of powerful geometric tools is important for progress in the theory of Diophantine equations. In this project, the investigator and his collaborator intend to develop such tools, by extending known techniques in the usual so-called archimedean geometry to the context of non-archimedean, or p-adic, geometry. This geometry, which has a strong arithmetic flavor, provides a crucial geometric setting for the analysis of Diophantine equations, and these techniques are expected to yield several new developments in that analysis. Such developments are important not only because they enrich what continues to be one of the center pieces of the mathematical tradition, but because the theory of Diophantine equations has deep interconnections with the theory of discrete processes, and especially with the theory of codes, so that progress in theory of Diophantine equations can be expected to yield progress in these fields.