Combinatorics and Number Theory IV

这个建议包含三个项目，在数论及其应用组合。第一个项目是关于非同余子群的模形式的算法。Atkin和Swinnerton-Dyer对模曲线在有理数上有模的非同余子群提出了非常有趣的尖形的Fourier系数的同余关系，用来代替Hecke算子。另一方面，Scholl把伽罗瓦表示附加到非全等尖点形式空间上。ASD同余与Scholl表示的模性一起产生了非常有趣的同余和非同余形式的傅立叶系数之间的同余关系。一些例子是由PI和她的合著者构建的。继续她与合著者的联合工作，PI计划应用模块提升定理来研究Scholl表示何时从Hilbert模块形式中产生，并使用模块结果和ASD同余关系来建立一个猜想，该猜想认为非线性代数非同余形式的傅立叶系数具有无界的算子。第二个项目是从Bruhat-Tits建筑物的有限体积中构造复合体的zeta函数。这样的zeta函数应该是一个有理函数，它编码的拓扑和光谱信息的复杂性，并满足黎曼假设当且仅当该复杂性是Ramanujan。当维数为1时，这是Ihara zeta函数附加到图上。在最近的工作中，PI和一个博士生做了二维的案例。PI建议探索一般情况。第三个项目是关于低密度奇偶校验（LDPC）码。LDPC码具有非常高效的译码算法，这使得它在真实的世界中有着非常高的应用价值。解码错误的来源是伪码字。理解伪码字的一种方法是构造一个合适的无穷级数，称为zeta函数，每一项对应一个伪码字，这样的zeta函数应该是一个具有良好组合性质的有理函数。这是在PI的联合文件中间接完成的。PI主张追求一种更直接的方法，在数论中做基础研究，寻求数论在组合数学和编码理论中的应用，特别是解决真实的世界问题，一直是PI的长期研究目标。对这些区域之间相互作用的研究已经取得了丰硕的成果。本建议是PI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。他们也将被纳入由PI教授的研究生课程。每周将举行非正式研讨会，以整合研究与教育和教学。计划于2010年召开一次会议，以传播该项目的成果。

This proposal contains three projects, in number theory and itsapplications to combinatorics. The first project concerns thearithmetic of modular forms for noncongruence subgroups. Fornoncongruence subgroups whose associated modular curves have a modelover the rationals, Atkin and Swinnerton-Dyer suggested veryinteresting congruence relations on the Fourier coefficients of cuspforms which are meant to replace the Hecke operators. On the otherhand, Scholl has attached Galois representations to the space ofnoncongruence cusp forms. The ASD congruences together with themodularity of Scholl representations yield extremely interestingcongruence relations between the Fourier coefficients of congruenceand noncongruence forms. Some examples were constructed by the PIand her coauthors. Continuing her joint work with coauthors, the PIplans to apply the modularity lifting theorems to investigate whenScholl representations arise from Hilbert modular forms, and to usethe modularity result and ASD congruence relations to establish theconjecture which says that the Fourier coefficients of genuinelyalgebraic noncongruence forms have unbounded denominators. Thesecond project is to construct zeta functions of complexes arisingfrom finite quotients of the Bruhat-Tits buildings. Such zetafunction should be a rational function which encodes topological andspectral information of the complex, and which satisfies the RiemannHypothesis if and only if the complex is Ramanujan. When dimensionis one, this is the Ihara zeta function attached to a graph. In avery recent work, the PI and a PhD student did the 2-dimensionalcase. The PI proposes to explore the general case. The third projectconcerns low-density parity-check (LDPC) codes. The LDPC codes areequipped with very efficient decoding algorithms, which make themhighly desirable in real world applications. The source of decodingerrors is the pseudo-codewords. One way to understand thesepseudo-codewords is to construct a suitable infinite series, calleda zeta function, with each term corresponding to a pseudo-codeword.Such zeta function should be a rational function with goodcombinatorial property. This was done indirectly in a joint paper ofthe PI. The PI proposes to pursue a more direct approach.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 coding theory, especially to solve real worldproblems. The study of interplay between these areas has turned outto be quite fruitful. This proposal is a continuation of the PI's effort to pursue the same general theme. Part of the research will be carried out by the PI's Ph Dstudents. 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 taught by the PI. Weeklyinformal seminars will be conducted to integrate research witheducation and teaching. A conference is planned in 2010 todisseminate results from this project.