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的代数变异上的逆束和d模之间的Riemann-Hilbert对应的p进设置中建立一个模拟。希尔伯特对应推广了De Rham理论，并在给定复代数变量的拓扑（编码在变量上的逆束范畴中）和定义在变量上的微分算子系统的行为（编码为变量上的d模，即在变量上的微分算子环的束上的模束）之间建立了深刻的联系。所提出的p进数类比将对p进数的变化执行类似的功能。它将产生这样一个变种上的（目前推测的）“结晶反常束”范畴和（同样推测的）配备Frobenius算子的弱可容许滤波d模范畴之间的等价。晶体反常束的范畴被认为携带着关于它所附着的品种的几何和算术信息，例如，它将是Beilinson的调节器理论的p进模拟中的天然成分。这就暗示了在丢番图方程系统的p点的局部分析中，这一类的轮系所能起到的重要作用。解方程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，比如丢番图。他研究解整数方程的问题；这样的方程现在被称为丢番图方程。自从笛卡儿和费马的工作以来，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，即使人们最初从算术的角度对方程更感兴趣。因此，强大的几何工具的发展对于丢番图方程理论的发展是非常重要的。在这个项目中，研究者和他的合作者打算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进几何的背景下，来开发这样的工具。这种几何，有很强的算术味道，为丢番图方程的分析提供了一个重要的几何背景，这些技术有望在分析中产生一些新的发展。这样的发展很重要，不仅因为它们丰富了数学传统的核心部分之一，而且因为丢番图方程理论与离散过程理论，特别是与密码理论有着深刻的联系，所以丢番图方程理论的进步可以期望在这些领域取得进展。

Abstract for Emerton-NSF 0070711In 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.