Combinatorics and Number Theory V

本提案旨在研究数论和组合学两个不同领域的问题。第一个项目涉及模形式非同余子群的算术性质。对于这些形式，Scholl附加了动机伽罗表征，根据朗兰兹的哲学，这被认为是与自同构形式相联系的。这些伽罗瓦表示，不像它们的经典对应物，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有关。在她过去工作的基础上，PI将研究具有特殊对称性的Scholl表示的（潜在的）自同构，利用自同构的结果，并研究由Atkin和Swinnerton-Dyer提出的非同余形式的傅立叶系数的同余性质。第二个项目是通过相关的zeta函数来研究组合学、群论和数论之间的相互作用。在有限域上定义的变量的ζ函数是很容易理解的。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。有限简单复合体是这类变种的组合类似物。除了黎曼假设只适用于光谱最优的复合体外，它们预计具有具有类似性质的ζ函数。一维复合体是图形，其ζ函数自1966年Ihara的工作以来一直被研究。高维复合体的zeta函数直到最近才为人所知，当时PI和她的学生获得了复合体的zeta函数的封闭形式表达式，这些复合体是p进域上某些2阶Chevalley群的构筑物的商。这些方法大多是表征论的。PI建议寻找由其他群产生的复合体的ζ恒等式。她还打算使用Selberg迹公式探索这些恒等式的组合解释，着眼于建立复杂zeta函数和自同构形式之间的联系。PI的长期研究目标是对数论进行基础研究，并寻求数论在组合学和解决现实世界问题中的应用。对这些领域之间相互作用的研究已经证明是相当富有成效的。这一建议是PI努力追求相同主题的延续。部分研究将由PI的博士生进行。这项建议的成果将通过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。它们也将被纳入PI提供的研究生课程。每周举行非正式研讨会，将研究与教育和教学结合起来。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。

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.