Combinatorics and Number Theory IV

组合数学与数论 IV

• 批准号: 0801096
• 负责人: Wen-Ching Li
• 金额: $ 15万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801096&HistoricalAwards=false
• 关键词: Combinatorics; Number; Theory; IV

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.

该提案包含三个项目，涉及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的算术。对于相关模曲线具有有理数模型的非同余子群，Atkin 和 Swinnerton-Dyer 提出了关于尖点形式的傅里叶系数的非常有趣的同余关系，旨在取代 Hecke 算子。另一方面，绍尔将伽罗瓦表示附加到非全等尖点形式的空间中。 ASD 同余与 Scholl 表示的模性一起在同余和非同余形式的傅立叶系数之间产生了极其有趣的同余关系。一些示例是由 PI 和她的合著者构建的。 PI 继续与合著者共同工作，计划应用模性提升定理来研究何时从希尔伯特模形式产生 Scholl 表示，并使用模性结果和 ASD 同余关系来建立猜想，即真正代数非同余形式的傅里叶系数具有无界分母。第二个项目是构造由 Bruhat-Tits 建筑物的有限商产生的复合体的 zeta 函数。这样的zeta函数应该是一个有理函数，它编码了配合物的拓扑和光谱信息，并且当且仅当配合物是拉马努金时，它满足黎曼假设。当维度为 1 时，这是附加到图形的 Ihara zeta 函数。在最近的一项工作中，PI 和一名博士生做了二维案例。 PI 建议探索一般情况。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解码算法，这使得它们在现实世界的应用中非常受欢迎。解码错误的根源是伪码字。理解这些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，其中每一项对应一个伪码字。这样的zeta函数应该是具有良好组合性质的有理函数。这是在 PI 的联合论文中间接完成的。 PI建议寻求更直接的方法。在数论方面进行基础研究并寻求数论在组合数学和编码理论中的应用，特别是解决现实世界的问题，一直是PI的长期研究目标。对这些领域之间相互作用的研究已证明是非常富有成果的。该提案是 PI 努力追求同一主题的延续。部分研究将由 PI 的博士生进行。该提案的结果将通过 PI 在研讨会、座谈会、会议、短期课程和讲习班中的演讲进行广泛传播。它们也将被纳入 PI 教授的研究生课程中。每周举行非正式研讨会，将研究与教育教学结合起来。计划于 2010 年召开一次会议来传播该项目的成果。