L-functions and Eisenstein series: p-adic aspects and applications

L-函数和爱森斯坦级数：p-adic 方面和应用

• 批准号: 1249384
• 负责人: Ellen Eischen
• 金额: $ 9.8万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1249384&HistoricalAwards=false
• 关键词: functions; Eisenstein; series; adic; aspects

This grant concerns several projects motivated by p-adic aspects of L-functions, a central object of study in number theory. The first part of the project involves the development of certain p-adic differential operators and p-adic measures (Eisensteinmeasures), technical tools that the PI will use to construct p-adic L-functions. These p-adic differential operators and p-adic measures build on the PI's prior results on these topics. These Eisenstein measures are also conjectured to give a homotopy-theoretic invariant of certain manifolds that generalizes the Witten genus. The second part of the project concerns applications of the tools developed in the first part of the proposal. The main application is the construction of p-adic L-functions that p-adically interpolate the special values of L-functions attached to families of automorphic forms. One portion of this application is joint with Michael Harris, Jian-Shu Li, and Christopher Skinner. This part of the project also has applications to Iwasawa theory, a p-adic theory for studying certain arithmetic data, which in turn is expected to relate to the Birch and Swinnerton-Dyer Conjecture. L-functions are certain complex-valued functions that play an important role in number theory. Their values at certain points satisfy striking congruence conditions and relate to many open problems. One approach to studying L-functions uses p-adic numbers (depending on a prime number p), an extension of the rational numbers analogous to but different from the real numbers. This proposal involves further developing techniques for studying p-adic aspects of L-functions. The proposed research has expected consequences in algebraic number theory, analytic number theory, and algebraic topology.

这项赠款涉及的几个项目是由L功能的P-ADIC方面激励的，L功能是数量理论研究的核心对象。 该项目的第一部分涉及某些P-ADIC差异操作员和P-ADIC措施（EisensteinMeasures）的开发，即PI用来构建P-ADIC L功能的技术工具。 这些P-ADIC差异操作员和P-ADIC措施基于PI对这些主题的先前结果。这些爱森斯坦的措施还猜想是为某些歧管提供了概括属属的同质理论的不变性。 该项目的第二部分涉及提案第一部分中开发的工具的应用。主要的应用是构建P-ADIC L功能，该功能在P-Ady插值插入了与自动形式家族相关的L功能的特殊值。该申请的一部分是与迈克尔·哈里斯（Michael Harris），江·李（Jian-Shu Li）和克里斯托弗·斯金纳（Christopher Skinner）的联合。该项目的这一部分还针对Iwasawa理论，这是一种用于研究某些算术数据的P-ADIC理论，这反过来又与Birch和Swinnerton-Dyer猜想有关。 L功能是某些在数字理论中起重要作用的复杂功能。 他们在某些方面的价值满足了惊人的一致性条件，并且与许多开放问题有关。 研究L功能的一种方法使用P-ADIC数字（取决于素数P），这是类似于实际数字的合理数字的扩展。 该建议涉及进一步开发用于研究L功能的P-ADIC方面的技术。 拟议的研究预期了代数数理论，分析数理论和代数拓扑的后果。