Combinatorics and Number Theory V

这项建议旨在研究数论和组合学中两个不同领域的问题。第一个项目是关于模形式在非同余子群上的算术性质。对于这些形式，肖尔附加了动机伽罗伊表示，根据朗兰兹哲学，这些表示被期望与自同形形式联系在一起。这些Galois表示与经典的Galois表示不同，一般不能分解为碎片，而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(势)自系法，探索自同构的结果，并研究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。定义在有限域上的簇的Zeta函数已经被很好地理解了。它们最著名的性质是由Weil猜想描述的，该猜想是由S在1970年代初建立的。有限单纯复形是这类簇的组合模拟。它们有望具有类似性质的Zeta函数，只是黎曼假设只适用于光谱最优的络合物。一维复形是一种图，它的Zeta函数自1966年Ihara的工作以来一直被研究。直到最近，当Pi和她的学生得到复形的Zeta函数的闭式表达式时，高维复形的Zeta函数才为人所知，这些复形是作为p-adad域上的某些二阶Chvalley群的建筑商而产生的。这些方法大多是表征论的。PI建议为来自其他基团的络合物寻找Zeta恒等式。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Zeta函数和自同构形之间建立联系。在数论方面进行基础研究，寻求数论在组合学和解决现实世界问题中的应用一直是PI的长期研究目标。对这些地区之间相互作用的研究已被证明是相当有成效的。这项建议是国际和平协会为追求同一总主题所作努力的延续。部分研究将由Pi的博士生进行。这项建议的结果将通过国际和平协会在研讨会、学术讨论会、会议、短期课程和讲习班上所作的演讲广泛传播。此外，他们亦会被纳入由该署开办的研究生课程内。每周将举行非正式研讨会，将研究与教育和教学相结合。国际和平协会还计划于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这项提议有关的成果。

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.