Combinatorics and Number Theory V

组合学与数论 V

• 批准号: 1101368
• 负责人: Wen-Ching Li
• 金额: $ 12万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101368&HistoricalAwards=false
• 关键词: Combinatorics; Number; Theory

This proposal aims to investigate problems from two diverse areas in number theory andcombinatorics.The first project concerns arithmetic properties of modular forms onnoncongruence subgroups. To these forms Scholl attached motivic Galoisrepresentations, which are expected to be connected to automorphic forms according to Langlands philosophy. These Galois representations, unliketheir classical counterpart, cannot be broken into pieces in general,and should be related to automorphic forms on symplectic and orthogonal groups.Building upon her past work, PI will investigate the (potential) automorphyof Scholl representations with special symmetries, exploit consequencesof automorphy, and study congruence properties ofFourier coefficients of noncongruence forms as proposed by Atkin and Swinnerton-Dyer.The second project is to study the interplay between combinatorics,group theory and number theory through associated zeta functions. Zeta functions of varieties defined over finite fields are well-understood.Their most well-known properties are described by Weil conjectures, established in early1970's. Finite simplicial complexes are combinatorial analog of such varieties. They are expected to have zeta functions enjoying similar properties, except that Riemann Hypothesis will hold only for complexes which are spectrally optimal. One-dimensionalcomplexes are graphs, whose zeta functions have been studied since the work of Ihara in 1966.The zeta functions for higher dimensional complexes became known only recentlywhen PI and her students obtained closed form expressions for zeta functions ofcomplexes arising as quotients of the buildings of certain rank-2 Chevalley groups over p-adic fields. The approaches are mostly representation-theoretic. The PI proposes to find zeta identities for complexes arising from other groups. She also intends to explore combinatorial interpretations of these identities using the Selberg trace formula, with an eye towards establishing a connection between complex zeta functions and automorphic forms.It has been the PI's long term research goal to do fundamentalresearch in number theory and to seek applications of number theoryto combinatorics and to solve real world problems. The study of interplay between these areas has turned out to be quite fruitful. This proposal is a continuation ofthe PI's effort to pursue the same general theme. Part of the research will be carried out by PI's Ph.D. students. The results from this proposal will be disseminatedbroadly through the talks given by the PI in seminars, colloquia,conferences, short courses, and workshops. They will also beincorporated in the graduate courses to be offered by the PI. Weeklyinformal seminars will be conducted to integrate research witheducation and teaching. The PI also plans to co-organize a conference in 2013 at Banffto disseminate results related to this proposal obtained by her and her students.

该提案旨在研究数论和组合学两个不同领域的问题。第一个项目涉及非同余子群上模形式的算术性质。绍尔将动机伽罗瓦表示附加到这些形式上，根据朗兰兹哲学，这些表示预计将与自守形式联系起来。这些伽罗瓦表示与它们的经典对应物不同，一般不能被分解成碎片，并且应该与辛群和正交群上的自同构形式相关。在她过去的工作的基础上，PI 将研究具有特殊对称性的 Scholl 表示的（潜在）自同构，利用自同构的后果，并研究 Atkin 和 Swinnerton-Dyer 提出的非同余形式的傅里叶系数的同余性质。第二个项目是通过相关的 zeta 函数研究组合学、群论和数论之间的相互作用。在有限域上定义的簇的 Zeta 函数是众所周知的。它们最著名的性质是由 20 世纪 70 年代初建立的韦尔猜想描述的。有限单纯复形是此类变体的组合模拟。它们预计具有具有相似性质的 zeta 函数，但黎曼假设仅适用于光谱最优的配合物。一维复形是图，其 zeta 函数自 Ihara 于 1966 年的工作以来一直在研究。高维复形的 zeta 函数直到最近才为人所知，当时 PI 和她的学生获得了作为p-adic 域上的某些 2 阶 Chevalley 群。这些方法主要是表示理论的。 PI 建议寻找来自其他群体的复合物的 zeta 身份。她还打算利用 Selberg 迹公式探索这些恒等式的组合解释，着眼于在复杂的 zeta 函数和自同构形式之间建立联系。在数论方面进行基础研究并寻求应用一直是 PI 的长期研究目标从数论到组合数学并解决现实世界的问题。对这些领域之间相互作用的研究已证明是非常富有成果的。该提案是 PI 努力追求同一主题的延续。部分研究将由PI的博士进行。学生。该提案的结果将通过 PI 在研讨会、座谈会、会议、短期课程和讲习班中的演讲进行广泛传播。它们还将被纳入 PI 提供的研究生课程中。每周举行非正式研讨会，将研究与教育教学结合起来。 PI 还计划于 2013 年在班夫共同组织一次会议，以传播她和她的学生获得的与该提案相关的结果。