Arithmetic of L-Values

L 值的算术

• 批准号: 9701782
• 负责人: Glenn Stevens
• 金额: --
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9701782&HistoricalAwards=false
• 关键词: Arithmetic; Values

9701782 Stevens The principal investigator intends to continue his work on special values of L-functions and their relation to arithmetic geometry. The proposed research would improve our understanding of both automorphic forms and p-adic cohomology by providing new tools for studying the former and by providing concrete examples of the latter. The PI's recent proof of a conjecture of Mazur, Tate, Teitelbaum, and Coleman raises new questions about analytic families of automorphic forms, their p-adic L-functions, and families of galois representations and overconvergent F-crystals. The proposed research would clarify how such families degenerate at special non-crystalline points and would describe monodromy at such points in terms of a deformation in the weight direction, thus enlarging the standard picture of monodromy in potentially useful ways. This research would also complement Kato's recent work on values of L-functions and K_2 of modular curves by providing tools for deforming Kato's theory in p-adic analytic families. In related work, the PI hopes to develop a p-adic Eichler-Shimura correspondence that would relate his theory of overconvergent modular symbols to Katz's theory of overconvergent modular forms and to construct analytic families of non-ordinary half-integral weight modular forms by generalizing a p-adic theta lifting developed in earlier work of the PI. Finally, the PI intends to generalize these ideas to automorphic forms on other reductive algebraic groups. This research offers promising tools for the construction of p-adic analytic families of non-ordinary automorphic representations together with natural deformation spaces of Galois representations, and multivariable p-adic L-functions. This is connected with a number of new investigations, including p- adic monodromy, Jochnowitz's conjectures on the square root of theta operator on half-integral weight forms, and the p-adic deformation theory of Galois representations. This project falls into the general area of arithmetic geometry -a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

9701782史蒂文斯首席研究员打算继续他的工作的特殊价值的L-功能和他们的关系算术几何。拟议的研究将通过提供研究自守形式的新工具和提供后者的具体例子，提高我们对自守形式和p-adic上同调的理解。PI最近证明了Mazur、Tate、Teitelbaum和科尔曼的一个猜想，提出了关于自守形式的解析族、它们的p进L函数、伽罗瓦表示族和过收敛F晶体的新问题。拟议的研究将澄清这些家庭如何在特殊的非结晶点退化，并将在这些点上的重量方向的变形方面描述monodromy，从而扩大monodromy的标准图片在潜在的有用的方式。本文的研究也将补充加藤最近关于L-函数和模曲线K_2值的工作，为加藤理论在p-adic解析族中的变形提供工具。在相关的工作中，PI希望开发一个p-adic Eichler-Shimura对应，将他的过收敛模符号理论与Katz的过收敛模形式理论联系起来，并通过推广PI早期工作中开发的p-adic theta提升来构建非普通半整数权模形式的解析族。最后，PI打算将这些想法推广到其他约化代数群上的自守形式。这项研究提供了有前途的工具，为建设的p-adic分析家庭的非普通自守表示与自然变形空间的伽罗瓦表示，和多变量p-adic L-函数。这是与一些新的调查，包括p进monodromy，Jochnowitz的apturtures的平方根θ运营商的半积分的重量形式，和p进变形理论的伽罗瓦表示。这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。在它的许多后果是新的纠错码。这种代码对于现代计算机（硬盘）和光盘都是必不可少的。