Combinatorics and Number Theory V

本计画主要研究数论与组合学两个不同领域的问题：第一个计画是关于非全等子群上模形式的算术性质。这些形式肖尔重视motivic伽罗瓦表示，预计将连接到自守形式根据朗兰兹哲学。这些伽罗瓦表示，不像他们的经典对应物，不能被分解成碎片一般，应该与自守形式的辛和正交群。在她过去的工作的基础上，PI将研究（潜在的）具有特殊对称性的Scholl表示的自同构，利用自同构的结果，研究Atkin和Swinnerton-Dyer提出的非全等形式的Fourier系数的全等性质。第二个项目是研究组合数学之间的相互作用，群论和数论通过相关的zeta函数。定义在有限域上的簇的Zeta函数是很好理解的，它们最著名的性质由20世纪70年代初建立的Weil定理描述。有限单纯复形是这些变种的组合类似物。他们预计有zeta函数享有类似的性质，除了黎曼假设将只适用于复杂的光谱最佳。一维复形是一种图形，自1966年Ihara的工作以来，人们一直在研究其zeta函数。高维复形的zeta函数直到最近PI和她的学生获得了p-adic域上某些秩为2的Chevalley群的建筑物的导数所产生的复形的zeta函数的封闭形式表达式后才为人所知。这些方法大多是代表理论。PI建议为来自其他群体的复合物找到Zeta身份。她还打算探索组合解释这些身份使用塞尔伯格迹公式，着眼于建立复杂的zeta函数和自守forms.It之间的连接一直是PI的长期研究目标做fundamentalresearch数论和寻求应用数论组合和解决真实的世界问题。对这些领域之间相互作用的研究已证明是相当富有成果的。这一建议是PI继续努力追求相同的总主题。部分研究将由PI的博士进行学生这一建议的结果将通过PI在研讨会、座谈会、会议、短期课程和讲习班上的演讲广泛传播。他们也将被纳入研究生课程将提供的PI。每周将举行非正式研讨会，以整合研究与教育和教学。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有关的结果。

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.