Combinatorics and Number Theory IV

这份提案包含三个项目，分别是数论及其在组合学中的应用。第一个项目涉及非同余子群的模形式算法。对于模曲线模于有理数上的非同余子群，Atkin和Swinnerton-Dyer提出了非常有趣的关于尖形傅立叶系数的同余关系，用来代替Hecke算子。另一方面，Scholl将Galois表示附加到非同余尖点形式空间。ASD同余和肖尔表示的调制性一起产生了同余形式和非同余形式的傅立叶系数之间非常有趣的同余关系。一些例子是由PI和她的合著者构造的。继续她与合著者的合作，PI计划应用模提升定理来研究Scholl表示何时产生于Hilbert模形式，并利用模性结果和ASD同余关系来建立这样的猜想，即真代数非同余形式的傅立叶系数有无界分母。第二个项目是构造由Bruhat-Tits建筑物的有限商产生的复数的Zeta函数。这样的Zetaff函数应该是一个有理函数，它编码了复数的拓扑和光谱信息，并且满足黎曼假设当且仅当复数是Ramanujan。当维度为1时，这是附加到图形上的Ihara Zeta函数。在最近的一项工作中，PI和一名博士生完成了二维案例。私隐专员公署建议探讨一般情况。第三个项目涉及低密度奇偶校验(LDPC)码。LDPC码配备了非常高效的译码算法，这使得它们在现实世界的应用中具有非常高的要求。译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da Zeta函数，每个项对应一个伪码字，这样的Zeta函数应该是一个具有良好组合性的有理函数。这是在国际和平研究所的一份联合文件中间接完成的。国际数学家协会建议寻求一种更直接的方法。从事数论的基础研究，寻求数论在组合学和编码学中的应用，特别是解决现实世界的问题，一直是国际数学家协会的长期研究目标。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相同的总主题所作努力的延续。部分研究将由PI的Ph De学生进行。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。他们还将被纳入国际和平协会教授的研究生课程。每周将举行非正式研讨会，将研究与教育和教学相结合。计划在2010年召开一次会议，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.