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

p-Adic黎曼-希尔伯特对应和混合Hodge模的p-Adic理论

• 批准号: 0070711
• 负责人: Matthew Emerton
• 金额: $ 7.93万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070711&HistoricalAwards=false
• 关键词: Adic; Riemann; Hilbert; Correspondence; Theory

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.

EMERTON-NSF 0070711的摘要该项目研究者及其合作者打算在反式滑轮和代数品种上的Riemann-Hilbert对应的P-ADIC环境中开发一个类似物。 希尔伯特对应关系概括了de rham理论，并建立了给定复杂的代数品种的拓扑结合（如多样性上的不正正支线束类别中编码）与在多样性上定义的差分运算符系统的行为（在多样性上编码为d-Modules;是在多样性上的d-Modules;是模块的sheeavers oferters oferters of shering of shee pose of she eaf of she eaf shorings of shering conse of she eaf shorings of sherings of shorings s rope shorings s rying shorings s rying shorings， 所提出的P-Adic类似物将对P-ADIC数字的品种执行相似的功能。 它将在这种品种上的``晶体变形带轮''的（当前猜想）类别与配备Frobenius操作员配备的弱化过滤的D-Modules的（再次猜测）类别的类别之间产生等效性。 据信，晶体变形带轮的类别既具有几何和算术信息，又具有算术信息，例如，在贝林森的调节剂理论的p addic类似物中，将是一种天然成分。 这表明了重要的作用，即可以期望这样一类系带系带的系带在二只方程系统的局部分析中发挥作用。解决方程式的问题是数学上最基本的问题之一，至少可以追溯到古希腊的数学家，例如养生。 他研究了整体求解方程的问题。现在，这些方程称为二磷剂方程。由于笛卡尔和费马特的工作，也已经理解，几何形状为分析方程式系统提供了有力的工具，即使起初从算术的角度对方程更感兴趣，也提供了一个强大的工具。 因此，强大的几何工具的开发对于二磷理论的进步很重要。 在这个项目中，调查员及其合作者打算通过将已知的Archimedean几何形状中的已知技术扩展到非Archimedean或P-ADIC几何形状的背景。这种几何形状具有强烈的算术风味，为分析二磷酸方程提供了至关重要的几何环境，并且这些技术有望在该分析中产生一些新的发展。 这些发展不仅是因为它们丰富了数学传统的中心部分之一，而且因为二聚体方程式的理论与离散过程理论，尤其是代码理论具有深厚的互连，因此可以预期，在这些领域中，可以预期，在这些领域的进展中，这种进展可以产生进展。